TPTP Problem File: SLH0646^1.p
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%------------------------------------------------------------------------------
% 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 : Universal_Hash_Families/0028_Field/prob_00153_005562__18292646_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1409 ( 523 unt; 137 typ; 0 def)
% Number of atoms : 3561 (1222 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 12816 ( 210 ~; 59 |; 168 &;10798 @)
% ( 0 <=>;1581 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 18 ( 17 usr)
% Number of type conns : 439 ( 439 >; 0 *; 0 +; 0 <<)
% Number of symbols : 123 ( 120 usr; 12 con; 0-4 aty)
% Number of variables : 3236 ( 61 ^;3044 !; 131 ?;3236 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:41:36.851
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
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% Explicit typings (120)
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_IntRing_OZFact,type,
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thf(sy_c_Nat_OSuc,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
collect_nat_set_int: ( ( nat > set_int ) > $o ) > set_nat_set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
collect_set_set_int: ( set_set_int > $o ) > set_set_set_int ).
thf(sy_c_Set_OPow_001t__Int__Oint,type,
pow_int: set_int > set_set_int ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Int__Oint_J,type,
pow_set_int: set_set_int > set_set_set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
set_or7260056672446632558et_int: ( nat > set_int ) > set_nat_set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Int__Oint_J,type,
set_or5935648273017318783et_int: set_int > set_set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_or4141871561713569845et_int: set_set_int > set_set_set_int ).
thf(sy_c_Subrings_Osubcring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
subcri1024317279029940167t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
subfie3888952257595785920t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_UnivPoly_Obound_001t__Nat__Onat,type,
bound_nat: nat > nat > ( nat > nat ) > $o ).
thf(sy_c_UnivPoly_Obound_001t__Set__Oset_It__Int__Oint_J,type,
bound_set_int: set_int > nat > ( nat > set_int ) > $o ).
thf(sy_c_UnivPoly_Oup_001t__Nat__Onat_001t__Product____Type__Ounit,type,
up_nat_Product_unit: partia4692342223508353374t_unit > set_nat_nat ).
thf(sy_c_UnivPoly_Oup_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
up_set1168727741560211120t_unit: partia4934656038542163276t_unit > set_nat_set_int ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
member_nat_set_int: ( nat > set_int ) > set_nat_set_int > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J,type,
member_set_int_nat: ( set_int > nat ) > set_set_int_nat > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
member5205197933313416826et_int: ( set_int > set_int ) > set_set_int_set_int > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
member_set_set_int: set_set_int > set_set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
member7356822600254261989et_int: set_set_set_int > set_set_set_set_int > $o ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_x_H____,type,
x: nat ).
thf(sy_v_x____,type,
x2: set_int ).
thf(sy_v_y_H____,type,
y: nat ).
thf(sy_v_y____,type,
y2: set_int ).
% Relevant facts (1268)
thf(fact_0_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_1_x_H__def,axiom,
( x
= ( zfact_iso_inv @ n @ x2 ) ) ).
% x'_def
thf(fact_2_y_H__def,axiom,
( y
= ( zfact_iso_inv @ n @ y2 ) ) ).
% y'_def
thf(fact_3_n__ge__1,axiom,
ord_less_nat @ one_one_nat @ n ).
% n_ge_1
thf(fact_4_a,axiom,
( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ x2 @ y2 )
= ( ring_zfact_iso @ n @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ x @ y ) ) ) ).
% a
thf(fact_5_n__ge__0,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% n_ge_0
thf(fact_6_ring_Ofold__congs_I5_J,axiom,
! [R: partia4692342223508353374t_unit,R2: partia4692342223508353374t_unit,V: nat > nat > nat,F: ( nat > nat > nat ) > nat > nat > nat,F2: ( nat > nat > nat ) > nat > nat > nat] :
( ( R = R2 )
=> ( ( ( add_nat_Product_unit @ R2 )
= V )
=> ( ! [V2: nat > nat > nat] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( add_up707286280682557645t_unit @ F @ R )
= ( add_up707286280682557645t_unit @ F2 @ R2 ) ) ) ) ) ).
% ring.fold_congs(5)
thf(fact_7_ring_Ofold__congs_I5_J,axiom,
! [R: partia4934656038542163276t_unit,R2: partia4934656038542163276t_unit,V: set_int > set_int > set_int,F: ( set_int > set_int > set_int ) > set_int > set_int > set_int,F2: ( set_int > set_int > set_int ) > set_int > set_int > set_int] :
( ( R = R2 )
=> ( ( ( add_se5859248395121729892t_unit @ R2 )
= V )
=> ( ! [V2: set_int > set_int > set_int] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( add_up3191207752591510331t_unit @ F @ R )
= ( add_up3191207752591510331t_unit @ F2 @ R2 ) ) ) ) ) ).
% ring.fold_congs(5)
thf(fact_8_ring_Ounfold__congs_I5_J,axiom,
! [R: partia4692342223508353374t_unit,R2: partia4692342223508353374t_unit,V: nat > nat > nat,F: ( nat > nat > nat ) > nat > nat > nat,F2: ( nat > nat > nat ) > nat > nat > nat] :
( ( R = R2 )
=> ( ( ( add_nat_Product_unit @ R2 )
= V )
=> ( ! [V2: nat > nat > nat] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( add_up707286280682557645t_unit @ F @ R )
= ( add_up707286280682557645t_unit @ F2 @ R2 ) ) ) ) ) ).
% ring.unfold_congs(5)
thf(fact_9_ring_Ounfold__congs_I5_J,axiom,
! [R: partia4934656038542163276t_unit,R2: partia4934656038542163276t_unit,V: set_int > set_int > set_int,F: ( set_int > set_int > set_int ) > set_int > set_int > set_int,F2: ( set_int > set_int > set_int ) > set_int > set_int > set_int] :
( ( R = R2 )
=> ( ( ( add_se5859248395121729892t_unit @ R2 )
= V )
=> ( ! [V2: set_int > set_int > set_int] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( add_up3191207752591510331t_unit @ F @ R )
= ( add_up3191207752591510331t_unit @ F2 @ R2 ) ) ) ) ) ).
% ring.unfold_congs(5)
thf(fact_10_r_Opoly__add_Ocases,axiom,
! [X: produc6642921209907155909et_int] :
~ ! [P1: list_set_int,P2: list_set_int] :
( X
!= ( produc1446815805028224253et_int @ P1 @ P2 ) ) ).
% r.poly_add.cases
thf(fact_11_r_OSpan_Ocases,axiom,
! [X: produc6891024271396224491et_int] :
~ ! [K: set_set_int,Us: list_set_int] :
( X
!= ( produc4282412693294859939et_int @ K @ Us ) ) ).
% r.Span.cases
thf(fact_12_finite__lessThan,axiom,
! [K2: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K2 ) ) ).
% finite_lessThan
thf(fact_13_lessThan__iff,axiom,
! [I: set_int,K2: set_int] :
( ( member_set_int @ I @ ( set_or5935648273017318783et_int @ K2 ) )
= ( ord_less_set_int @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_14_lessThan__iff,axiom,
! [I: nat > set_int,K2: nat > set_int] :
( ( member_nat_set_int @ I @ ( set_or7260056672446632558et_int @ K2 ) )
= ( ord_less_nat_set_int @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_15_lessThan__iff,axiom,
! [I: set_set_int,K2: set_set_int] :
( ( member_set_set_int @ I @ ( set_or4141871561713569845et_int @ K2 ) )
= ( ord_less_set_set_int @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_16_lessThan__iff,axiom,
! [I: int,K2: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K2 ) )
= ( ord_less_int @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_17_lessThan__iff,axiom,
! [I: nat,K2: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K2 ) )
= ( ord_less_nat @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_18_y__carr,axiom,
member_set_int @ y2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% y_carr
thf(fact_19_x__carr,axiom,
member_set_int @ x2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% x_carr
thf(fact_20_bounded__nat__set__is__finite,axiom,
! [N: set_nat,N2: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ord_less_nat @ X2 @ N2 ) )
=> ( finite_finite_nat @ N ) ) ).
% bounded_nat_set_is_finite
thf(fact_21_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ord_less_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_22_lessThan__strict__subset__iff,axiom,
! [M2: int,N2: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M2 ) @ ( set_ord_lessThan_int @ N2 ) )
= ( ord_less_int @ M2 @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_23_lessThan__strict__subset__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_24_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_25_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_26_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_27_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_28_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_29_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_30_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_31_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1316708129612266289at_nat @ N2 )
= one_one_nat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_32_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1314217659103216013at_int @ N2 )
= one_one_int )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_33_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_34_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_35_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_36_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_37_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_38_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_39_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_40_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_41_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_42_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_43_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_44_of__nat__eq__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M2 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_45_r_Oadd_Or__cancel,axiom,
! [A: set_int,C: set_int,B: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ C ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( A = B ) ) ) ) ) ).
% r.add.r_cancel
thf(fact_46_r_Oadd_Om__lcomm,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Z ) ) ) ) ) ) ).
% r.add.m_lcomm
thf(fact_47_r_Oadd_Om__comm,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X ) ) ) ) ).
% r.add.m_comm
thf(fact_48_r_Oadd_Om__assoc,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) ) ) ) ) ) ).
% r.add.m_assoc
thf(fact_49_r_Oadd_Ol__cancel,axiom,
! [C: set_int,A: set_int,B: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C @ A )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C @ B ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( A = B ) ) ) ) ) ).
% r.add.l_cancel
thf(fact_50_r_Oadd_Oright__cancel,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% r.add.right_cancel
thf(fact_51_r_Oadd_Om__closed,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.m_closed
thf(fact_52_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_53_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_54_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ( P @ M3 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_55_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A: set_int,P: set_int > $o] :
( ( member_set_int @ A @ ( collect_set_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_57_mem__Collect__eq,axiom,
! [A: nat > set_int,P: ( nat > set_int ) > $o] :
( ( member_nat_set_int @ A @ ( collect_nat_set_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_58_mem__Collect__eq,axiom,
! [A: set_set_int,P: set_set_int > $o] :
( ( member_set_set_int @ A @ ( collect_set_set_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_set_int] :
( ( collect_set_int
@ ^ [X3: set_int] : ( member_set_int @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A2: set_nat_set_int] :
( ( collect_nat_set_int
@ ^ [X3: nat > set_int] : ( member_nat_set_int @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A2: set_set_set_int] :
( ( collect_set_set_int
@ ^ [X3: set_set_int] : ( member_set_set_int @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_63_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_64_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_65_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_66_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_67_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_68_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_69_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_70_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_71_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_72_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_73_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_74_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_75_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_76_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_77_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_78_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_79_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_80_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_81_r_Oorder__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_4716970363388151434t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.order_gt_0_iff_finite
thf(fact_82_r_Oonepideal,axiom,
princi8860937869964495385t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.onepideal
thf(fact_83_r_Ofinite__carr__imp__char__ge__0,axiom,
( ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.finite_carr_imp_char_ge_0
thf(fact_84__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Acarrier_A_IZFact_A_Iint_An_J_J_A_092_060Longrightarrow_062_Azfact__iso__inv_An_Ax_A_092_060in_062_Acarrier_A_Imod__ring_An_J_092_060close_062,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_nat @ ( zfact_iso_inv @ n @ X ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ).
% \<open>\<And>x. x \<in> carrier (ZFact (int n)) \<Longrightarrow> zfact_iso_inv n x \<in> carrier (mod_ring n)\<close>
thf(fact_85_fin__zfact,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% fin_zfact
thf(fact_86_r_Osemiring__axioms,axiom,
semiri8708897239777792527t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.semiring_axioms
thf(fact_87_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_88_r_Ocarrier__is__subcring,axiom,
subcri1024317279029940167t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.carrier_is_subcring
thf(fact_89_r_Oadd_Oint__pow__mult__distrib,axiom,
! [X: set_int,Y: set_int,I: int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ Y ) ) ) ) ) ) ).
% r.add.int_pow_mult_distrib
thf(fact_90_r_Oadd_Oint__pow__distrib,axiom,
! [X: set_int,Y: set_int,I: int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ Y ) ) ) ) ) ).
% r.add.int_pow_distrib
thf(fact_91_r_Oadd_Oint__pow__closed,axiom,
! [X: set_int,I: int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.add.int_pow_closed
thf(fact_92_r_Oadd_Oint__pow__1,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ one_one_int @ X )
= X ) ) ).
% r.add.int_pow_1
thf(fact_93_semiring_Osemiring__simprules_I12_J,axiom,
! [R3: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X @ ( add_se5859248395121729892t_unit @ R3 @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ R3 @ Y @ ( add_se5859248395121729892t_unit @ R3 @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_94_semiring_Osemiring__simprules_I12_J,axiom,
! [R3: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X @ ( add_nat_Product_unit @ R3 @ Y @ Z ) )
= ( add_nat_Product_unit @ R3 @ Y @ ( add_nat_Product_unit @ R3 @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_95_semiring_Osemiring__simprules_I7_J,axiom,
! [R3: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X @ Y )
= ( add_se5859248395121729892t_unit @ R3 @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_96_semiring_Osemiring__simprules_I7_J,axiom,
! [R3: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X @ Y )
= ( add_nat_Product_unit @ R3 @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_97_semiring_Osemiring__simprules_I5_J,axiom,
! [R3: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( add_se5859248395121729892t_unit @ R3 @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R3 @ X @ ( add_se5859248395121729892t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_98_semiring_Osemiring__simprules_I5_J,axiom,
! [R3: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( add_nat_Product_unit @ R3 @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ R3 @ X @ ( add_nat_Product_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_99_semiring_Osemiring__simprules_I1_J,axiom,
! [R3: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R3 @ X @ Y ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_100_semiring_Osemiring__simprules_I1_J,axiom,
! [R3: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( add_nat_Product_unit @ R3 @ X @ Y ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_101_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_102_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_103_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_104_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_105_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_106_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_107_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_108_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_109_r_Ocgenideal__is__principalideal,axiom,
! [I: set_int] :
( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( princi8860937869964495385t_unit @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.cgenideal_is_principalideal
thf(fact_110_r_Oadd_Oint__pow__mult,axiom,
! [X: set_int,I: int,J: int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( plus_plus_int @ I @ J ) @ X )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ J @ X ) ) ) ) ).
% r.add.int_pow_mult
thf(fact_111_r_Oadd_Oint__pow__pow,axiom,
! [X: set_int,M2: int,N2: int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M2 @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ X ) )
= ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( times_times_int @ N2 @ M2 ) @ X ) ) ) ).
% r.add.int_pow_pow
thf(fact_112_r_Ocgenideal__self,axiom,
! [I: set_int] :
( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ I @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I ) ) ) ).
% r.cgenideal_self
thf(fact_113_r_Oadd_Oinv__comm,axiom,
! [X: set_int,Y: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.add.inv_comm
thf(fact_114_r_Oadd_Ol__inv__ex,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.l_inv_ex
thf(fact_115_r_Oadd_Oone__unique,axiom,
! [U: set_int] :
( ( member_set_int @ U @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.one_unique
thf(fact_116_r_Oadd_Or__inv__ex,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ X2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.r_inv_ex
thf(fact_117_r_Ominus__unique,axiom,
! [Y: set_int,X: set_int,Y2: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% r.minus_unique
thf(fact_118_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_119_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_120_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_121_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_122_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_123_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_124_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_125_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_126_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_127_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_128_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_129_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_130_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_131_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_132_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_133_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_134_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_135_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_136_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_137_of__nat__add,axiom,
! [M2: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_add
thf(fact_138_of__nat__add,axiom,
! [M2: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_add
thf(fact_139_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_140_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_141_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_142_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_143_of__nat__mult,axiom,
! [M2: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_mult
thf(fact_144_of__nat__mult,axiom,
! [M2: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_mult
thf(fact_145_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_146_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_147_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_148_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_149_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_150_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_151_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_152_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_153_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_154_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_155_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_156_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_157_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_158_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_159_r_Ozero__closed,axiom,
member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% r.zero_closed
thf(fact_160_r_Oadd_Oint__pow__one,axiom,
! [Z: int] :
( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.add.int_pow_one
thf(fact_161_r_Or__zero,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= X ) ) ).
% r.r_zero
thf(fact_162_r_Ol__zero,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X )
= X ) ) ).
% r.l_zero
thf(fact_163_r_Oadd_Or__cancel__one_H,axiom,
! [X: set_int,A: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( X
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ X ) )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.add.r_cancel_one'
thf(fact_164_r_Oadd_Or__cancel__one,axiom,
! [X: set_int,A: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ X )
= X )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.add.r_cancel_one
thf(fact_165_r_Oadd_Ol__cancel__one_H,axiom,
! [X: set_int,A: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( X
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ A ) )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.add.l_cancel_one'
thf(fact_166_r_Oadd_Ol__cancel__one,axiom,
! [X: set_int,A: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ A )
= X )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.add.l_cancel_one
thf(fact_167_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_168_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_169_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_170_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_171_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_int @ I @ K2 )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_172_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I @ K2 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_173_group__cancel_Oadd1,axiom,
! [A2: int,K2: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_174_group__cancel_Oadd1,axiom,
! [A2: nat,K2: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_175_group__cancel_Oadd2,axiom,
! [B2: int,K2: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K2 @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_176_group__cancel_Oadd2,axiom,
! [B2: nat,K2: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K2 @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_177_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_178_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_179_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_180_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_181_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_182_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_183_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_184_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_185_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_186_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_187_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_188_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_189_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_190_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_191_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_192_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_193_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_194_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_195_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_196_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_197_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_198_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_199_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_200_add__less__imp__less__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_imp_less_right
thf(fact_201_add__less__imp__less__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_imp_less_right
thf(fact_202_add__less__imp__less__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_imp_less_left
thf(fact_203_add__less__imp__less__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_imp_less_left
thf(fact_204_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_205_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_206_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_207_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_208_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_209_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_210_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_211_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K2 = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_212_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_213_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_214_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_215_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_216_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_217_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_218_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_219_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_220_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_221_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_222_ring__hom__closed,axiom,
! [H: set_int > set_int,R3: partia4934656038542163276t_unit,S2: partia4934656038542163276t_unit,X: set_int] :
( ( member5205197933313416826et_int @ H @ ( ring_h3404898052528352314t_unit @ R3 @ S2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( H @ X ) @ ( partia966996272515721803t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_223_ring__hom__closed,axiom,
! [H: set_int > nat,R3: partia4934656038542163276t_unit,S2: partia4692342223508353374t_unit,X: set_int] :
( ( member_set_int_nat @ H @ ( ring_h9051218956466089420t_unit @ R3 @ S2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_nat @ ( H @ X ) @ ( partia3499330772048238685t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_224_ring__hom__closed,axiom,
! [H: nat > set_int,R3: partia4692342223508353374t_unit,S2: partia4934656038542163276t_unit,X: nat] :
( ( member_nat_set_int @ H @ ( ring_h4752909569380436264t_unit @ R3 @ S2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_set_int @ ( H @ X ) @ ( partia966996272515721803t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_225_ring__hom__closed,axiom,
! [H: nat > nat,R3: partia4692342223508353374t_unit,S2: partia4692342223508353374t_unit,X: nat] :
( ( member_nat_nat @ H @ ( ring_h4412161176302437050t_unit @ R3 @ S2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( H @ X ) @ ( partia3499330772048238685t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_226_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_227_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_228_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_229_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_230_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_231_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_232_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_233_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_234_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_235_ring__hom__add,axiom,
! [H: set_int > nat,R3: partia4934656038542163276t_unit,S2: partia4692342223508353374t_unit,X: set_int,Y: set_int] :
( ( member_set_int_nat @ H @ ( ring_h9051218956466089420t_unit @ R3 @ S2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( H @ ( add_se5859248395121729892t_unit @ R3 @ X @ Y ) )
= ( add_nat_Product_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_236_ring__hom__add,axiom,
! [H: set_int > set_int,R3: partia4934656038542163276t_unit,S2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( member5205197933313416826et_int @ H @ ( ring_h3404898052528352314t_unit @ R3 @ S2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( H @ ( add_se5859248395121729892t_unit @ R3 @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_237_ring__hom__add,axiom,
! [H: nat > nat,R3: partia4692342223508353374t_unit,S2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( member_nat_nat @ H @ ( ring_h4412161176302437050t_unit @ R3 @ S2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( H @ ( add_nat_Product_unit @ R3 @ X @ Y ) )
= ( add_nat_Product_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_238_ring__hom__add,axiom,
! [H: nat > set_int,R3: partia4692342223508353374t_unit,S2: partia4934656038542163276t_unit,X: nat,Y: nat] :
( ( member_nat_set_int @ H @ ( ring_h4752909569380436264t_unit @ R3 @ S2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( H @ ( add_nat_Product_unit @ R3 @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_239_semiring_Osemiring__simprules_I11_J,axiom,
! [R3: partia4934656038542163276t_unit,X: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X @ ( zero_s6269048424454532197t_unit @ R3 ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_240_semiring_Osemiring__simprules_I11_J,axiom,
! [R3: partia4692342223508353374t_unit,X: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X @ ( zero_n5149899317435570679t_unit @ R3 ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_241_semiring_Osemiring__simprules_I6_J,axiom,
! [R3: partia4934656038542163276t_unit,X: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( zero_s6269048424454532197t_unit @ R3 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_242_semiring_Osemiring__simprules_I6_J,axiom,
! [R3: partia4692342223508353374t_unit,X: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( zero_n5149899317435570679t_unit @ R3 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_243_zfact__iso__inv__0,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( zfact_iso_inv @ N2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N2 ) ) ) )
= zero_zero_nat ) ) ).
% zfact_iso_inv_0
thf(fact_244_zfact__iso__0,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ring_zfact_iso @ N2 @ zero_zero_nat )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zfact_iso_0
thf(fact_245_r_OboundD__carrier,axiom,
! [N2: nat,F: nat > set_int,M2: nat] :
( ( bound_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ N2 @ F )
=> ( ( ord_less_nat @ N2 @ M2 )
=> ( member_set_int @ ( F @ M2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.boundD_carrier
thf(fact_246_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_247_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_248_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_249_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_250_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_251_r_Ochar__bound_I2_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.char_bound(2)
thf(fact_252_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_253_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_254_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_255_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_256_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_257_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_258_r_Oint__embed__closed,axiom,
! [K2: int] : ( member_set_int @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.int_embed_closed
thf(fact_259_r_Oint__embed__zero,axiom,
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ zero_zero_int )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.int_embed_zero
thf(fact_260_r_Oint__embed__add,axiom,
! [X: int,Y: int] :
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( plus_plus_int @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ).
% r.int_embed_add
thf(fact_261_r_Oembed__char__eq__0,axiom,
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( semiri1314217659103216013at_int @ ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.embed_char_eq_0
thf(fact_262_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_263_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_264_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_265_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_266_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_267_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_268_nat__add__left__cancel__less,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_269_mult__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_270_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_271_mult__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N2 ) )
= ( ( M2 = N2 )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_272_mult__cancel2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ K2 )
= ( times_times_nat @ N2 @ K2 ) )
= ( ( M2 = N2 )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_273_nat__1__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_274_nat__mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_275_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_276_mult__less__cancel2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N2 @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_277_nat__0__less__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_278_add__mult__distrib,axiom,
! [M2: nat,N2: nat,K2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N2 ) @ K2 )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N2 @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_279_add__mult__distrib2,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K2 @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_280_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_281_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_282_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_283_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_284_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_285_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_286_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_287_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_288_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_289_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_290_less__add__eq__less,axiom,
! [K2: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K2 @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_291_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_292_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_293_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_294_mult__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_295_mult__less__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_296_mult__eq__self__implies__10,axiom,
! [M2: nat,N2: nat] :
( ( M2
= ( times_times_nat @ M2 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_297_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_298_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_299_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_300_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_301_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_302_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_303_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_304_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_305_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_306_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_307_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_308_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_309_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_310_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_311_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_312_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_313_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_314_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_315_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_316_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_317_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_318_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_319_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_320_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_321_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_322_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_323_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_324_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_325_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_326_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_327_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_328_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_329_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_330_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_331_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_332_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_333_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_334_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_335_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_336_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_337_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_338_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_339_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_340_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_341_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_342_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_343_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_344_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_345_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_346_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_347_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_348_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_349_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_350_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_351_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_352_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_353_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_354_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_355_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_356_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_357_less__1__mult,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ M2 )
=> ( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_358_less__1__mult,axiom,
! [M2: int,N2: int] :
( ( ord_less_int @ one_one_int @ M2 )
=> ( ( ord_less_int @ one_one_int @ N2 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M2 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_359_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_360_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_361_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_362_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_363_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_364_r_Ochar__bound_I1_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_less_eq_nat @ ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X ) ) ) ).
% r.char_bound(1)
thf(fact_365_r_Obound__upD,axiom,
! [F: nat > set_int] :
( ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [N4: nat] : ( bound_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ N4 @ F ) ) ).
% r.bound_upD
thf(fact_366_bound_Ointro,axiom,
! [N2: nat,F: nat > set_int,Z: set_int] :
( ! [M4: nat] :
( ( ord_less_nat @ N2 @ M4 )
=> ( ( F @ M4 )
= Z ) )
=> ( bound_set_int @ Z @ N2 @ F ) ) ).
% bound.intro
thf(fact_367_r_Oembed__char__eq__0__iff,axiom,
! [N2: int] :
( ( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) @ N2 ) ) ).
% r.embed_char_eq_0_iff
thf(fact_368_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_369_r_Oadd_Onat__pow__mult,axiom,
! [X: set_int,N2: nat,M2: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ X ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M2 @ X ) )
= ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( plus_plus_nat @ N2 @ M2 ) @ X ) ) ) ).
% r.add.nat_pow_mult
thf(fact_370_r_Oint__embed__mult,axiom,
! [X: int,Y: int] :
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( times_times_int @ X @ Y ) )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ).
% r.int_embed_mult
thf(fact_371_r_Om__lcomm,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Z ) ) ) ) ) ) ).
% r.m_lcomm
thf(fact_372_r_Om__comm,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X ) ) ) ) ).
% r.m_comm
thf(fact_373_r_Om__assoc,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) ) ) ) ) ) ).
% r.m_assoc
thf(fact_374_r_Or__distr,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ X ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ Y ) ) ) ) ) ) ).
% r.r_distr
thf(fact_375_r_Ol__distr,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Z ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) ) ) ) ) ) ).
% r.l_distr
thf(fact_376_r_Oadd_Opow__mult__distrib,axiom,
! [X: set_int,Y: set_int,N2: nat] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ X ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ Y ) ) ) ) ) ) ).
% r.add.pow_mult_distrib
thf(fact_377_r_Oadd_Onat__pow__distrib,axiom,
! [X: set_int,Y: set_int,N2: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ X ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ Y ) ) ) ) ) ).
% r.add.nat_pow_distrib
thf(fact_378_r_Oadd_Onat__pow__comm,axiom,
! [X: set_int,N2: nat,M2: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ X ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M2 @ X ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M2 @ X ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ X ) ) ) ) ).
% r.add.nat_pow_comm
thf(fact_379_r_Oadd_Ogroup__commutes__pow,axiom,
! [X: set_int,Y: set_int,N2: nat] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ X ) @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ X ) ) ) ) ) ) ).
% r.add.group_commutes_pow
thf(fact_380_r_Oadd_Onat__pow__pow,axiom,
! [X: set_int,M2: nat,N2: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M2 @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ X ) )
= ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( times_times_nat @ N2 @ M2 ) @ X ) ) ) ).
% r.add.nat_pow_pow
thf(fact_381_r_Oadd__pow__rdistr,axiom,
! [A: set_int,B: set_int,K2: nat] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ B ) )
= ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ) ) ) ).
% r.add_pow_rdistr
thf(fact_382_r_Oadd__pow__ldistr,axiom,
! [A: set_int,B: set_int,K2: nat] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ A ) @ B )
= ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ) ) ) ).
% r.add_pow_ldistr
thf(fact_383_r_Oadd__pow__rdistr__int,axiom,
! [A: set_int,B: set_int,K2: int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ B ) )
= ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ) ) ) ).
% r.add_pow_rdistr_int
thf(fact_384_r_Oadd__pow__ldistr__int,axiom,
! [A: set_int,B: set_int,K2: int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ A ) @ B )
= ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ) ) ) ).
% r.add_pow_ldistr_int
thf(fact_385_r_Oint__embed__mult__aux,axiom,
! [X: int,Y: nat] :
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( times_times_int @ X @ ( semiri1314217659103216013at_int @ Y ) ) )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).
% r.int_embed_mult_aux
thf(fact_386_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_387_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_388_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_389_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_390_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_391_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_392_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_393_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_394_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_395_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_396_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_397_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_398_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_399_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_400_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_401_lessThan__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_402_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_403_nat__add__left__cancel__le,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_404_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_405_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_406_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_407_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_408_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_409_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_410_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_411_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_412_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_413_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_414_dvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_415_dvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_416_dvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_417_dvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_418_dvd__mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_419_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_420_dvd__add__times__triv__right__iff,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_421_dvd__add__times__triv__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_422_dvd__add__times__triv__left__iff,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_423_dvd__add__times__triv__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_424_unit__prod,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_prod
thf(fact_425_unit__prod,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_prod
thf(fact_426_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_427_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_428_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_429_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_430_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_431_mult__le__cancel2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N2 @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_432_mem__upI,axiom,
! [F: nat > set_int,R3: partia4934656038542163276t_unit] :
( ! [N4: nat] : ( member_set_int @ ( F @ N4 ) @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ? [N5: nat] : ( bound_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ N5 @ F )
=> ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ R3 ) ) ) ) ).
% mem_upI
thf(fact_433_mem__upI,axiom,
! [F: nat > nat,R3: partia4692342223508353374t_unit] :
( ! [N4: nat] : ( member_nat @ ( F @ N4 ) @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ? [N5: nat] : ( bound_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ N5 @ F )
=> ( member_nat_nat @ F @ ( up_nat_Product_unit @ R3 ) ) ) ) ).
% mem_upI
thf(fact_434_r_Om__closed,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.m_closed
thf(fact_435_r_Oadd_Onat__pow__closed,axiom,
! [X: set_int,N2: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ X ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.add.nat_pow_closed
thf(fact_436_r_Oadd_Onat__pow__one,axiom,
! [N2: nat] :
( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.add.nat_pow_one
thf(fact_437_r_Or__null,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.r_null
thf(fact_438_r_Ol__null,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.l_null
thf(fact_439_r_Oadd_Onat__pow__0,axiom,
! [X: set_int] :
( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ zero_zero_nat @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.add.nat_pow_0
thf(fact_440_r_Oadd_Onat__pow__eone,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ one_one_nat @ X )
= X ) ) ).
% r.add.nat_pow_eone
thf(fact_441_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_442_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_443_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_444_dvd__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_trans
thf(fact_445_zdvd__mult__cancel,axiom,
! [K2: int,M2: int,N2: int] :
( ( dvd_dvd_int @ ( times_times_int @ K2 @ M2 ) @ ( times_times_int @ K2 @ N2 ) )
=> ( ( K2 != zero_zero_int )
=> ( dvd_dvd_int @ M2 @ N2 ) ) ) ).
% zdvd_mult_cancel
thf(fact_446_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_447_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_448_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_449_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_450_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_451_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_452_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_453_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_454_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_455_mem__upD,axiom,
! [F: nat > set_int,R3: partia4934656038542163276t_unit,N2: nat] :
( ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ R3 ) )
=> ( member_set_int @ ( F @ N2 ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ).
% mem_upD
thf(fact_456_mem__upD,axiom,
! [F: nat > nat,R3: partia4692342223508353374t_unit,N2: nat] :
( ( member_nat_nat @ F @ ( up_nat_Product_unit @ R3 ) )
=> ( member_nat @ ( F @ N2 ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ).
% mem_upD
thf(fact_457_bound__below,axiom,
! [Z: set_int,M2: nat,F: nat > set_int,N2: nat] :
( ( bound_set_int @ Z @ M2 @ F )
=> ( ( ( F @ N2 )
!= Z )
=> ( ord_less_eq_nat @ N2 @ M2 ) ) ) ).
% bound_below
thf(fact_458_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_459_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_460_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_461_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_462_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_463_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_464_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_465_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_466_dvd__triv__right,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% dvd_triv_right
thf(fact_467_dvd__triv__right,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_468_dvd__mult__right,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ B @ C ) ) ).
% dvd_mult_right
thf(fact_469_dvd__mult__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_470_mult__dvd__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_471_mult__dvd__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_472_dvd__triv__left,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% dvd_triv_left
thf(fact_473_dvd__triv__left,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_474_dvd__mult__left,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ A @ C ) ) ).
% dvd_mult_left
thf(fact_475_dvd__mult__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ).
% dvd_mult_left
thf(fact_476_dvd__mult2,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_477_dvd__mult2,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_478_dvd__mult,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult
thf(fact_479_dvd__mult,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_480_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B3: int,A3: int] :
? [K4: int] :
( A3
= ( times_times_int @ B3 @ K4 ) ) ) ) ).
% dvd_def
thf(fact_481_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B3: nat,A3: nat] :
? [K4: nat] :
( A3
= ( times_times_nat @ B3 @ K4 ) ) ) ) ).
% dvd_def
thf(fact_482_dvdI,axiom,
! [A: int,B: int,K2: int] :
( ( A
= ( times_times_int @ B @ K2 ) )
=> ( dvd_dvd_int @ B @ A ) ) ).
% dvdI
thf(fact_483_dvdI,axiom,
! [A: nat,B: nat,K2: nat] :
( ( A
= ( times_times_nat @ B @ K2 ) )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% dvdI
thf(fact_484_dvdE,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ~ ! [K3: int] :
( A
!= ( times_times_int @ B @ K3 ) ) ) ).
% dvdE
thf(fact_485_dvdE,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ~ ! [K3: nat] :
( A
!= ( times_times_nat @ B @ K3 ) ) ) ).
% dvdE
thf(fact_486_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_487_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_488_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_489_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_490_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_491_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_492_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_493_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_494_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_495_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_496_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_497_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_498_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_499_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_500_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_501_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_502_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_503_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_504_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_505_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_506_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_507_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_508_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_509_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_510_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_511_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_512_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_513_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_514_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_515_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_516_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_517_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_518_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N6: nat] :
( ( ord_less_nat @ M @ N6 )
| ( M = N6 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_519_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_520_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N6: nat] :
( ( ord_less_eq_nat @ M @ N6 )
& ( M != N6 ) ) ) ) ).
% nat_less_le
thf(fact_521_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N6: nat] :
? [K4: nat] :
( N6
= ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_522_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_523_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_524_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_525_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_526_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K2 @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_527_add__leD2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ K2 @ N2 ) ) ).
% add_leD2
thf(fact_528_add__leD1,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_529_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_530_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_531_add__leE,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K2 @ N2 ) ) ) ).
% add_leE
thf(fact_532_mult__le__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_533_mult__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_534_mult__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_535_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_536_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_537_zdvd__reduce,axiom,
! [K2: int,N2: int,M2: int] :
( ( dvd_dvd_int @ K2 @ ( plus_plus_int @ N2 @ ( times_times_int @ K2 @ M2 ) ) )
= ( dvd_dvd_int @ K2 @ N2 ) ) ).
% zdvd_reduce
thf(fact_538_zdvd__period,axiom,
! [A: int,D: int,X: int,T: int,C: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
= ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_539_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_540_semiring_Oadd__pow__rdistr,axiom,
! [R3: partia4934656038542163276t_unit,A: set_int,B: set_int,K2: nat] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ A @ ( add_po7583499734880473159it_nat @ R3 @ K2 @ B ) )
= ( add_po7583499734880473159it_nat @ R3 @ K2 @ ( mult_s3864001451298473021t_unit @ R3 @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_rdistr
thf(fact_541_semiring_Oadd__pow__rdistr,axiom,
! [R3: partia4692342223508353374t_unit,A: nat,B: nat,K2: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ A @ ( add_po2422570615063001561it_nat @ R3 @ K2 @ B ) )
= ( add_po2422570615063001561it_nat @ R3 @ K2 @ ( mult_n6028127365542633569t_unit @ R3 @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_rdistr
thf(fact_542_semiring_Oadd__pow__ldistr,axiom,
! [R3: partia4934656038542163276t_unit,A: set_int,B: set_int,K2: nat] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( add_po7583499734880473159it_nat @ R3 @ K2 @ A ) @ B )
= ( add_po7583499734880473159it_nat @ R3 @ K2 @ ( mult_s3864001451298473021t_unit @ R3 @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_ldistr
thf(fact_543_semiring_Oadd__pow__ldistr,axiom,
! [R3: partia4692342223508353374t_unit,A: nat,B: nat,K2: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( add_po2422570615063001561it_nat @ R3 @ K2 @ A ) @ B )
= ( add_po2422570615063001561it_nat @ R3 @ K2 @ ( mult_n6028127365542633569t_unit @ R3 @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_ldistr
thf(fact_544_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_545_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_546_unit__mult__right__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_547_unit__mult__right__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ B @ A )
= ( times_times_nat @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_548_unit__mult__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_549_unit__mult__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_550_mult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_551_mult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_552_dvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_553_dvd__mult__unit__iff_H,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_554_mult__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_555_mult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_556_dvd__mult__unit__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_557_dvd__mult__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_558_is__unit__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% is_unit_mult_iff
thf(fact_559_is__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% is_unit_mult_iff
thf(fact_560_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_561_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_562_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_563_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_564_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_565_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_566_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_567_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_568_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_569_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_570_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_571_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_572_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_573_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_574_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_575_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_576_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_577_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_578_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_579_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_580_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_581_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_582_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_583_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_584_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_585_mult__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 @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_586_mult__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 @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_587_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_588_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_589_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_590_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_591_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_592_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_593_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_594_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_595_mult__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 @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_596_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_597_split__mult__pos__le,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 @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_598_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_599_mult__mono_H,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 @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_600_mult__mono_H,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 @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_601_mult__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 @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_602_mult__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 @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_603_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_604_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_605_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_606_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_607_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_608_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_609_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_610_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_611_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_612_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_613_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_614_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_615_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_616_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_617_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_618_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_619_nat__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_620_ring__hom__mult,axiom,
! [H: set_int > set_int,R3: partia4934656038542163276t_unit,S2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( member5205197933313416826et_int @ H @ ( ring_h3404898052528352314t_unit @ R3 @ S2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( H @ ( mult_s3864001451298473021t_unit @ R3 @ X @ Y ) )
= ( mult_s3864001451298473021t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_621_ring__hom__mult,axiom,
! [H: set_int > nat,R3: partia4934656038542163276t_unit,S2: partia4692342223508353374t_unit,X: set_int,Y: set_int] :
( ( member_set_int_nat @ H @ ( ring_h9051218956466089420t_unit @ R3 @ S2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( H @ ( mult_s3864001451298473021t_unit @ R3 @ X @ Y ) )
= ( mult_n6028127365542633569t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_622_ring__hom__mult,axiom,
! [H: nat > set_int,R3: partia4692342223508353374t_unit,S2: partia4934656038542163276t_unit,X: nat,Y: nat] :
( ( member_nat_set_int @ H @ ( ring_h4752909569380436264t_unit @ R3 @ S2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( H @ ( mult_n6028127365542633569t_unit @ R3 @ X @ Y ) )
= ( mult_s3864001451298473021t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_623_ring__hom__mult,axiom,
! [H: nat > nat,R3: partia4692342223508353374t_unit,S2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( member_nat_nat @ H @ ( ring_h4412161176302437050t_unit @ R3 @ S2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( H @ ( mult_n6028127365542633569t_unit @ R3 @ X @ Y ) )
= ( mult_n6028127365542633569t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_624_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_625_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K2: nat] :
( ! [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_626_semiring_Osemiring__simprules_I8_J,axiom,
! [R3: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ X @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ R3 @ X @ ( mult_s3864001451298473021t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_627_semiring_Osemiring__simprules_I8_J,axiom,
! [R3: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ X @ Y ) @ Z )
= ( mult_n6028127365542633569t_unit @ R3 @ X @ ( mult_n6028127365542633569t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_628_semiring_Osemiring__simprules_I3_J,axiom,
! [R3: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R3 @ X @ Y ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_629_semiring_Osemiring__simprules_I3_J,axiom,
! [R3: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R3 @ X @ Y ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_630_unit__dvdE,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [C2: int] :
( B
!= ( times_times_int @ A @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_631_unit__dvdE,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [C2: nat] :
( B
!= ( times_times_nat @ A @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_632_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_633_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_634_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_635_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_636_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_637_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_638_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_639_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_640_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_641_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_642_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_643_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_644_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_645_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_646_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_647_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_648_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_649_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_650_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_651_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_652_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_653_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_654_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_655_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_656_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_657_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_658_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_659_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_660_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_661_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_662_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_663_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_664_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_665_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_666_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_667_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_668_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_669_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_670_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_671_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_672_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_673_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_674_int__int__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M2 = N2 ) ) ).
% int_int_eq
thf(fact_675_plus__int__code_I1_J,axiom,
! [K2: int] :
( ( plus_plus_int @ K2 @ zero_zero_int )
= K2 ) ).
% plus_int_code(1)
thf(fact_676_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_677_zmult__zless__mono2,axiom,
! [I: int,J: int,K2: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ord_less_int @ ( times_times_int @ K2 @ I ) @ ( times_times_int @ K2 @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_678_times__int__code_I1_J,axiom,
! [K2: int] :
( ( times_times_int @ K2 @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_679_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_680_semiring_Or__null,axiom,
! [R3: partia4934656038542163276t_unit,X: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ X @ ( zero_s6269048424454532197t_unit @ R3 ) )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% semiring.r_null
thf(fact_681_semiring_Or__null,axiom,
! [R3: partia4692342223508353374t_unit,X: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ X @ ( zero_n5149899317435570679t_unit @ R3 ) )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ).
% semiring.r_null
thf(fact_682_semiring_Ol__null,axiom,
! [R3: partia4934656038542163276t_unit,X: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( zero_s6269048424454532197t_unit @ R3 ) @ X )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% semiring.l_null
thf(fact_683_semiring_Ol__null,axiom,
! [R3: partia4692342223508353374t_unit,X: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( zero_n5149899317435570679t_unit @ R3 ) @ X )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ).
% semiring.l_null
thf(fact_684_semiring_Ol__distr,axiom,
! [R3: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( add_se5859248395121729892t_unit @ R3 @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ X @ Z ) @ ( mult_s3864001451298473021t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_685_semiring_Ol__distr,axiom,
! [R3: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( add_nat_Product_unit @ R3 @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ X @ Z ) @ ( mult_n6028127365542633569t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_686_semiring_Or__distr,axiom,
! [R3: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ Z @ ( add_se5859248395121729892t_unit @ R3 @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ Z @ X ) @ ( mult_s3864001451298473021t_unit @ R3 @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_687_semiring_Or__distr,axiom,
! [R3: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ Z @ ( add_nat_Product_unit @ R3 @ X @ Y ) )
= ( add_nat_Product_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ Z @ X ) @ ( mult_n6028127365542633569t_unit @ R3 @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_688_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V3: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V3 )
=> ( ( ( plus_plus_int @ U @ V3 )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V3 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_689_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_690_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_691_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_692_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_693_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_694_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_695_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_696_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_697_convex__bound__lt,axiom,
! [X: int,A: int,Y: int,U: int,V3: int] :
( ( ord_less_int @ X @ A )
=> ( ( ord_less_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V3 )
=> ( ( ( plus_plus_int @ U @ V3 )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V3 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_698_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_699_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_700_int__gr__induct,axiom,
! [K2: int,I: int,P: int > $o] :
( ( ord_less_int @ K2 @ I )
=> ( ( P @ ( plus_plus_int @ K2 @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K2 @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_701_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_702_int__distrib_I1_J,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% int_distrib(1)
thf(fact_703_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% int_distrib(2)
thf(fact_704_pos__zmult__eq__1__iff,axiom,
! [M2: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N2 )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N2 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_705_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N2 ) )
= ( ( K2 = zero_zero_nat )
| ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_706_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K2 ) ) ).
% left_add_mult_distrib
thf(fact_707_bound__def,axiom,
( bound_set_int
= ( ^ [Z3: set_int,N6: nat,F3: nat > set_int] :
! [M: nat] :
( ( ord_less_nat @ N6 @ M )
=> ( ( F3 @ M )
= Z3 ) ) ) ) ).
% bound_def
thf(fact_708_bound_Obound,axiom,
! [Z: set_int,N2: nat,F: nat > set_int,M2: nat] :
( ( bound_set_int @ Z @ N2 @ F )
=> ( ( ord_less_nat @ N2 @ M2 )
=> ( ( F @ M2 )
= Z ) ) ) ).
% bound.bound
thf(fact_709_zadd__int__left,axiom,
! [M2: nat,N2: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N2 ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_710_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
& ( K2
= ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_711_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N4: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% pos_int_cases
thf(fact_712_nat__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_713_nat__mult__eq__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N2 ) )
= ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_714_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K2: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_715_r_Omonoid__cancelI,axiom,
( ! [A4: set_int,B4: set_int,C2: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C2 @ A4 )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C2 @ B4 ) )
=> ( ( member_set_int @ A4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( A4 = B4 ) ) ) ) )
=> ( ! [A4: set_int,B4: set_int,C2: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A4 @ C2 )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B4 @ C2 ) )
=> ( ( member_set_int @ A4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( A4 = B4 ) ) ) ) )
=> ( monoid497721730651901107t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.monoid_cancelI
thf(fact_716__092_060open_062_092_060And_062y_Ax_O_A_092_060lbrakk_062x_A_092_060in_062_Acarrier_A_IZFact_A_Iint_An_J_J_059_Ay_A_092_060in_062_Acarrier_A_IZFact_A_Iint_An_J_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Azfact__iso__inv_An_A_Ix_A_092_060otimes_062_092_060_094bsub_062ZFact_A_Iint_An_J_092_060_094esub_062_Ay_J_A_061_Azfact__iso__inv_An_Ax_A_092_060otimes_062_092_060_094bsub_062mod__ring_An_092_060_094esub_062_Azfact__iso__inv_An_Ay_092_060close_062,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( zfact_iso_inv @ n @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( zfact_iso_inv @ n @ X ) @ ( zfact_iso_inv @ n @ Y ) ) ) ) ) ).
% \<open>\<And>y x. \<lbrakk>x \<in> carrier (ZFact (int n)); y \<in> carrier (ZFact (int n))\<rbrakk> \<Longrightarrow> zfact_iso_inv n (x \<otimes>\<^bsub>ZFact (int n)\<^esub> y) = zfact_iso_inv n x \<otimes>\<^bsub>mod_ring n\<^esub> zfact_iso_inv n y\<close>
thf(fact_717_r_Oline__extension__mem__iff,axiom,
! [U: set_int,K5: set_set_int,A: set_int,E2: set_set_int] :
( ( member_set_int @ U @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K5 @ A @ E2 ) )
= ( ? [X3: set_int] :
( ( member_set_int @ X3 @ K5 )
& ? [Y5: set_int] :
( ( member_set_int @ Y5 @ E2 )
& ( U
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X3 @ A ) @ Y5 ) ) ) ) ) ) ).
% r.line_extension_mem_iff
thf(fact_718_r_Oadd_Opower__order__eq__one,axiom,
! [A: set_int] :
( ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( finite_card_set_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) @ A )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.power_order_eq_one
thf(fact_719_r_Oadd_Opow__eq__div2,axiom,
! [X: set_int,M2: nat,N2: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M2 @ X )
= ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ X ) )
=> ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( minus_minus_nat @ M2 @ N2 ) @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.pow_eq_div2
thf(fact_720_r_Oadd_Onat__pow__Suc2,axiom,
! [X: set_int,N2: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( suc @ N2 ) @ X )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ X ) ) ) ) ).
% r.add.nat_pow_Suc2
thf(fact_721_r_Ocgenideal__prod,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( set_mu2785919024023201382t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B ) )
= ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ) ) ) ).
% r.cgenideal_prod
thf(fact_722_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_723_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_724_nat__dvd__1__iff__1,axiom,
! [M2: nat] :
( ( dvd_dvd_nat @ M2 @ one_one_nat )
= ( M2 = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_725_r_Oline__extension__in__carrier,axiom,
! [K5: set_set_int,A: set_int,E2: set_set_int] :
( ( ord_le4403425263959731960et_int @ K5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ E2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K5 @ A @ E2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.line_extension_in_carrier
thf(fact_726_r_Oset__mult__closed,axiom,
! [H2: set_set_int,K5: set_set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( set_mu2785919024023201382t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 @ K5 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.set_mult_closed
thf(fact_727_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_728_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_729_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_730_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_731_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_732_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_733_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_734_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_735_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_736_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_737_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_738_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_739_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_740_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_741_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_742_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_743_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_744_dvd__1__iff__1,axiom,
! [M2: nat] :
( ( dvd_dvd_nat @ M2 @ ( suc @ zero_zero_nat ) )
= ( M2
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_745_dvd__1__left,axiom,
! [K2: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K2 ) ).
% dvd_1_left
thf(fact_746_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_747_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_748_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_749_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_750_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_751_Suc__le__mono,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% Suc_le_mono
thf(fact_752_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_753_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_754_add__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_755_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_756_nat__mult__dvd__cancel__disj,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) )
= ( ( K2 = zero_zero_nat )
| ( dvd_dvd_nat @ M2 @ N2 ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_757_diff__diff__left,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_758_int__dvd__int__iff,axiom,
! [M2: nat,N2: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( dvd_dvd_nat @ M2 @ N2 ) ) ).
% int_dvd_int_iff
thf(fact_759_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_760_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_761_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_762_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_763_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_764_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_765_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_766_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_767_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_768_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_769_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_770_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_771_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_772_one__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_773_mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_774_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_775_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_776_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_777_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_778_mult__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( times_times_nat @ M2 @ ( suc @ N2 ) )
= ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_779_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_780_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_781_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_782_one__le__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_783_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_784_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_785_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_786_r_Oadd_Onat__pow__Suc,axiom,
! [N2: nat,X: set_int] :
( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( suc @ N2 ) @ X )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ X ) @ X ) ) ).
% r.add.nat_pow_Suc
thf(fact_787_dvd__diffD,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N2 ) )
=> ( ( dvd_dvd_nat @ K2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( dvd_dvd_nat @ K2 @ M2 ) ) ) ) ).
% dvd_diffD
thf(fact_788_dvd__diffD1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N2 ) )
=> ( ( dvd_dvd_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( dvd_dvd_nat @ K2 @ N2 ) ) ) ) ).
% dvd_diffD1
thf(fact_789_Suc__diff__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_790_dvd__diff__nat,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( dvd_dvd_nat @ K2 @ M2 )
=> ( ( dvd_dvd_nat @ K2 @ N2 )
=> ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% dvd_diff_nat
thf(fact_791_less__eq__dvd__minus,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( dvd_dvd_nat @ M2 @ N2 )
= ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_792_dvd__minus__self,axiom,
! [M2: nat,N2: nat] :
( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ( ord_less_nat @ N2 @ M2 )
| ( dvd_dvd_nat @ M2 @ N2 ) ) ) ).
% dvd_minus_self
thf(fact_793_Suc__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_794_diff__less__Suc,axiom,
! [M2: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_795_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_796_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I: nat] :
( ( P @ K2 )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_797_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_798_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_799_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_800_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_801_cancel__ab__semigroup__add__class_Odiff__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 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_802_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_803_of__nat__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M2 @ N2 ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).
% of_nat_diff
thf(fact_804_of__nat__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M2 @ N2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% of_nat_diff
thf(fact_805_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_806_eq__iff__diff__eq__0,axiom,
( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
= ( ^ [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_807_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_808_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_809_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_810_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_811_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_812_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_813_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_814_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_815_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_816_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_817_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_818_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_819_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_820_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_821_group__cancel_Osub1,axiom,
! [A2: int,K2: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K2 @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_822_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_823_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_824_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_825_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_826_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_827_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_828_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_829_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_830_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_831_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_832_dvd__diff,axiom,
! [X: int,Y: int,Z: int] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( dvd_dvd_int @ X @ Z )
=> ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_833_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_834_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_835_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_836_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_837_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_838_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_839_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] :
( ( P @ X2 @ Y3 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
=> ( P @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_840_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_841_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_842_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_843_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_844_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_845_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_846_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_847_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_848_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_849_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_850_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_851_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_852_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_853_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ N2 )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_854_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_855_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M2 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_856_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_857_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N2 @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_858_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_859_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_860_less__trans__Suc,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_861_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_862_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_863_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_864_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_865_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N2: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_866_transitive__stepwise__le,axiom,
! [M2: nat,N2: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ! [X2: nat] : ( R3 @ X2 @ X2 )
=> ( ! [X2: nat,Y3: nat,Z5: nat] :
( ( R3 @ X2 @ Y3 )
=> ( ( R3 @ Y3 @ Z5 )
=> ( R3 @ X2 @ Z5 ) ) )
=> ( ! [N4: nat] : ( R3 @ N4 @ ( suc @ N4 ) )
=> ( R3 @ M2 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_867_nat__induct__at__least,axiom,
! [M2: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ M2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_868_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N4 )
=> ( P @ M3 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_869_not__less__eq__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_870_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_871_le__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M2 @ N2 )
| ( M2
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_872_Suc__le__D,axiom,
! [N2: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M7 )
=> ? [M4: nat] :
( M7
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_873_le__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_874_le__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( M2
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_875_Suc__leD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_leD
thf(fact_876_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_877_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_878_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_879_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_880_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_881_le__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_882_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K2 )
= ( minus_minus_nat @ N2 @ K2 ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_883_nat__arith_Osuc1,axiom,
! [A2: nat,K2: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_884_add__Suc,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc
thf(fact_885_add__Suc__shift,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus_nat @ M2 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_886_Nat_Odiff__cancel,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_887_diff__cancel2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_888_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_889_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_890_Suc__mult__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ ( suc @ K2 ) @ M2 )
= ( times_times_nat @ ( suc @ K2 ) @ N2 ) )
= ( M2 = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_891_diff__mult__distrib,axiom,
! [M2: nat,N2: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N2 @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_892_diff__mult__distrib2,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M2 @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_893_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_894_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_895_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M: nat,N6: nat] : ( if_nat @ ( M = zero_zero_nat ) @ N6 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N6 ) ) ) ) ) ).
% add_eq_if
thf(fact_896_dvd__minus__add,axiom,
! [Q: nat,N2: nat,R: nat,M2: nat] :
( ( ord_less_eq_nat @ Q @ N2 )
=> ( ( ord_less_eq_nat @ Q @ ( times_times_nat @ R @ M2 ) )
=> ( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N2 @ Q ) )
= ( dvd_dvd_nat @ M2 @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R @ M2 ) @ Q ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_897_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_898_add__le__imp__le__diff,axiom,
! [I: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_899_add__le__imp__le__diff,axiom,
! [I: int,K2: int,N2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N2 )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N2 @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_900_add__le__add__imp__diff__le,axiom,
! [I: nat,K2: nat,N2: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_901_add__le__add__imp__diff__le,axiom,
! [I: int,K2: int,N2: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N2 )
=> ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K2 ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N2 )
=> ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K2 ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N2 @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_902_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_903_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_904_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_905_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_906_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_907_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_908_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_909_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_910_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_911_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_912_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_913_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_914_eq__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_915_eq__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_916_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_917_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_918_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_919_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_920_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_921_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_922_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B4: nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_923_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_set_int,R: set_int > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: set_int] :
( ( member_set_int @ A4 @ A2 )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: set_int,A22: set_int,B4: nat] :
( ( member_set_int @ A1 @ A2 )
=> ( ( member_set_int @ A22 @ A2 )
=> ( ( member_nat @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_924_card__le__if__inj__on__rel,axiom,
! [B2: set_set_int,A2: set_nat,R: nat > set_int > $o] :
( ( finite6197958912794628473et_int @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B5: set_int] :
( ( member_set_int @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B4: set_int] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_set_int @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_set_int @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_925_card__le__if__inj__on__rel,axiom,
! [B2: set_set_set_int,A2: set_nat,R: nat > set_set_int > $o] :
( ( finite4249678464180374575et_int @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B5: set_set_int] :
( ( member_set_set_int @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B4: set_set_int] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_set_set_int @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite7882580182802147440et_int @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_926_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_set_set_int,R: set_set_int > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: set_set_int] :
( ( member_set_set_int @ A4 @ A2 )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: set_set_int,A22: set_set_int,B4: nat] :
( ( member_set_set_int @ A1 @ A2 )
=> ( ( member_set_set_int @ A22 @ A2 )
=> ( ( member_nat @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite7882580182802147440et_int @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_927_card__le__if__inj__on__rel,axiom,
! [B2: set_set_int,A2: set_set_int,R: set_int > set_int > $o] :
( ( finite6197958912794628473et_int @ B2 )
=> ( ! [A4: set_int] :
( ( member_set_int @ A4 @ A2 )
=> ? [B5: set_int] :
( ( member_set_int @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: set_int,A22: set_int,B4: set_int] :
( ( member_set_int @ A1 @ A2 )
=> ( ( member_set_int @ A22 @ A2 )
=> ( ( member_set_int @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ A2 ) @ ( finite_card_set_int @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_928_card__le__if__inj__on__rel,axiom,
! [B2: set_set_set_int,A2: set_set_int,R: set_int > set_set_int > $o] :
( ( finite4249678464180374575et_int @ B2 )
=> ( ! [A4: set_int] :
( ( member_set_int @ A4 @ A2 )
=> ? [B5: set_set_int] :
( ( member_set_set_int @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: set_int,A22: set_int,B4: set_set_int] :
( ( member_set_int @ A1 @ A2 )
=> ( ( member_set_int @ A22 @ A2 )
=> ( ( member_set_set_int @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ A2 ) @ ( finite7882580182802147440et_int @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_929_card__le__if__inj__on__rel,axiom,
! [B2: set_nat_set_int,A2: set_nat,R: nat > ( nat > set_int ) > $o] :
( ( finite7455725759970522984et_int @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B5: nat > set_int] :
( ( member_nat_set_int @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B4: nat > set_int] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat_set_int @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite538658252611494441et_int @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_930_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat_set_int,R: ( nat > set_int ) > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: nat > set_int] :
( ( member_nat_set_int @ A4 @ A2 )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: nat > set_int,A22: nat > set_int,B4: nat] :
( ( member_nat_set_int @ A1 @ A2 )
=> ( ( member_nat_set_int @ A22 @ A2 )
=> ( ( member_nat @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite538658252611494441et_int @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_931_card__le__if__inj__on__rel,axiom,
! [B2: set_set_int,A2: set_set_set_int,R: set_set_int > set_int > $o] :
( ( finite6197958912794628473et_int @ B2 )
=> ( ! [A4: set_set_int] :
( ( member_set_set_int @ A4 @ A2 )
=> ? [B5: set_int] :
( ( member_set_int @ B5 @ B2 )
& ( R @ A4 @ B5 ) ) )
=> ( ! [A1: set_set_int,A22: set_set_int,B4: set_int] :
( ( member_set_set_int @ A1 @ A2 )
=> ( ( member_set_set_int @ A22 @ A2 )
=> ( ( member_set_int @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite7882580182802147440et_int @ A2 ) @ ( finite_card_set_int @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_932_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_933_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_934_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z3: int] :
? [N6: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N6 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_935_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N7: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_less_eq_nat @ ( F @ N7 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_936_lift__Suc__antimono__le,axiom,
! [F: nat > set_set_int,N2: nat,N7: nat] :
( ! [N4: nat] : ( ord_le4403425263959731960et_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_le4403425263959731960et_int @ ( F @ N7 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_937_lift__Suc__antimono__le,axiom,
! [F: nat > int,N2: nat,N7: nat] :
( ! [N4: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_less_eq_int @ ( F @ N7 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_938_lift__Suc__antimono__le,axiom,
! [F: nat > set_set_set_int,N2: nat,N7: nat] :
( ! [N4: nat] : ( ord_le4317611570275147438et_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_le4317611570275147438et_int @ ( F @ N7 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_939_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N7: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_940_lift__Suc__mono__le,axiom,
! [F: nat > set_set_int,N2: nat,N7: nat] :
( ! [N4: nat] : ( ord_le4403425263959731960et_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_le4403425263959731960et_int @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_941_lift__Suc__mono__le,axiom,
! [F: nat > int,N2: nat,N7: nat] :
( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_942_lift__Suc__mono__le,axiom,
! [F: nat > set_set_set_int,N2: nat,N7: nat] :
( ! [N4: nat] : ( ord_le4317611570275147438et_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N7 )
=> ( ord_le4317611570275147438et_int @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_943_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N7: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N7 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_944_lift__Suc__mono__less,axiom,
! [F: nat > int,N2: nat,N7: nat] :
( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N7 )
=> ( ord_less_int @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_945_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M2: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_946_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N2: nat,M2: nat] :
( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_947_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ~ ! [N4: nat] :
( K2
!= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% nonneg_int_cases
thf(fact_948_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ? [N4: nat] :
( K2
= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_949_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z3: int] :
? [N6: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N6 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_950_nat__dvd__not__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_nat @ M2 @ N2 )
=> ~ ( dvd_dvd_nat @ N2 @ M2 ) ) ) ).
% nat_dvd_not_less
thf(fact_951_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_952_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_953_int__ge__induct,axiom,
! [K2: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K2 @ I )
=> ( ( P @ K2 )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K2 @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_954_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_955_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_956_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M: nat] :
( N2
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_957_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_958_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_959_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_960_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_961_Suc__leI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_leI
thf(fact_962_Suc__le__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_eq
thf(fact_963_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_964_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_965_Suc__le__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_966_le__less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_967_less__Suc__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_968_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N6: nat] : ( ord_less_eq_nat @ ( suc @ N6 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_969_le__imp__less__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_970_add__is__1,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_971_one__is__add,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_972_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_973_less__natE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ! [Q2: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_974_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_975_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_976_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N6: nat] :
? [K4: nat] :
( N6
= ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_977_less__imp__Suc__add,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_978_less__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_979_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_980_zle__int,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% zle_int
thf(fact_981_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_982_Suc__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_983_less__diff__conv,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_984_add__diff__inverse__nat,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less_nat @ M2 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_985_Suc__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_986_Suc__eq__plus1,axiom,
( suc
= ( ^ [N6: nat] : ( plus_plus_nat @ N6 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_987_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_988_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_989_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_990_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_991_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_992_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_993_le__diff__conv,axiom,
! [J: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_994_mult__Suc,axiom,
! [M2: nat,N2: nat] :
( ( times_times_nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus_nat @ N2 @ ( times_times_nat @ M2 @ N2 ) ) ) ).
% mult_Suc
thf(fact_995_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_996_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_997_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_998_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_999_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_1000_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1001_dvd__imp__le,axiom,
! [K2: nat,N2: nat] :
( ( dvd_dvd_nat @ K2 @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ K2 @ N2 ) ) ) ).
% dvd_imp_le
thf(fact_1002_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N2 )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1003_dvd__mult__cancel,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( dvd_dvd_nat @ M2 @ N2 ) ) ) ).
% dvd_mult_cancel
thf(fact_1004_nat__mult__dvd__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) )
= ( dvd_dvd_nat @ M2 @ N2 ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_1005_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1006_one__less__mult,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_1007_n__less__m__mult__n,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1008_n__less__n__mult__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1009_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_1010_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1011_less__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1012_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1013_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1014_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1015_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_1016_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( M2
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1017_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1018_dvd__mult__cancel1,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M2 @ N2 ) @ M2 )
= ( N2 = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_1019_dvd__mult__cancel2,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M2 ) @ M2 )
= ( N2 = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_1020_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1021_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_1022_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M: nat,N6: nat] : ( if_nat @ ( M = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N6 @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N6 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1023_card__zfact__carr,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( finite_card_set_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N2 ) ) ) )
= N2 ) ) ).
% card_zfact_carr
thf(fact_1024_r_Oa__card__cosets__equal,axiom,
! [C: set_set_int,H2: set_set_int] :
( ( member_set_set_int @ C @ ( a_RCOS5559887075240879033t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 ) )
=> ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( finite_card_set_int @ C )
= ( finite_card_set_int @ H2 ) ) ) ) ) ).
% r.a_card_cosets_equal
thf(fact_1025_card_Oinfinite,axiom,
! [A2: set_set_set_int] :
( ~ ( finite4249678464180374575et_int @ A2 )
=> ( ( finite7882580182802147440et_int @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1026_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1027_card_Oinfinite,axiom,
! [A2: set_set_int] :
( ~ ( finite6197958912794628473et_int @ A2 )
=> ( ( finite_card_set_int @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1028_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1029_r_Oa__lcos__m__assoc,axiom,
! [M5: set_set_int,G: set_int,H: set_int] :
( ( ord_le4403425263959731960et_int @ M5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ G @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ G @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H @ M5 ) )
= ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ G @ H ) @ M5 ) ) ) ) ) ).
% r.a_lcos_m_assoc
thf(fact_1030_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_1031_finite__Diff2,axiom,
! [B2: set_set_int,A2: set_set_int] :
( ( finite6197958912794628473et_int @ B2 )
=> ( ( finite6197958912794628473et_int @ ( minus_8897228262479074673et_int @ A2 @ B2 ) )
= ( finite6197958912794628473et_int @ A2 ) ) ) ).
% finite_Diff2
thf(fact_1032_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_1033_finite__Diff,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ( finite6197958912794628473et_int @ A2 )
=> ( finite6197958912794628473et_int @ ( minus_8897228262479074673et_int @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_1034_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_1035_r_Oa__l__coset__subset__G,axiom,
! [H2: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ H2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.a_l_coset_subset_G
thf(fact_1036_r_Oa__lcos__mult__one,axiom,
! [M5: set_set_int] :
( ( ord_le4403425263959731960et_int @ M5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ M5 )
= M5 ) ) ).
% r.a_lcos_mult_one
thf(fact_1037_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_1038_dvd__antisym,axiom,
! [M2: nat,N2: nat] :
( ( dvd_dvd_nat @ M2 @ N2 )
=> ( ( dvd_dvd_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% dvd_antisym
thf(fact_1039_int__le__induct,axiom,
! [I: int,K2: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K2 )
=> ( ( P @ K2 )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K2 )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1040_Diff__infinite__finite,axiom,
! [T2: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1041_Diff__infinite__finite,axiom,
! [T2: set_set_int,S2: set_set_int] :
( ( finite6197958912794628473et_int @ T2 )
=> ( ~ ( finite6197958912794628473et_int @ S2 )
=> ~ ( finite6197958912794628473et_int @ ( minus_8897228262479074673et_int @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1042_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N4: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% int_diff_cases
thf(fact_1043_int__distrib_I3_J,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z2 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1044_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z2 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% int_distrib(4)
thf(fact_1045_int__induct,axiom,
! [P: int > $o,K2: int,I: int] :
( ( P @ K2 )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K2 @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K2 )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1046_card__less__sym__Diff,axiom,
! [A2: set_set_set_int,B2: set_set_set_int] :
( ( finite4249678464180374575et_int @ A2 )
=> ( ( finite4249678464180374575et_int @ B2 )
=> ( ( ord_less_nat @ ( finite7882580182802147440et_int @ A2 ) @ ( finite7882580182802147440et_int @ B2 ) )
=> ( ord_less_nat @ ( finite7882580182802147440et_int @ ( minus_6857623457997529383et_int @ A2 @ B2 ) ) @ ( finite7882580182802147440et_int @ ( minus_6857623457997529383et_int @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1047_card__less__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1048_card__less__sym__Diff,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ( finite6197958912794628473et_int @ A2 )
=> ( ( finite6197958912794628473et_int @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_int @ A2 ) @ ( finite_card_set_int @ B2 ) )
=> ( ord_less_nat @ ( finite_card_set_int @ ( minus_8897228262479074673et_int @ A2 @ B2 ) ) @ ( finite_card_set_int @ ( minus_8897228262479074673et_int @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1049_card__le__sym__Diff,axiom,
! [A2: set_set_set_int,B2: set_set_set_int] :
( ( finite4249678464180374575et_int @ A2 )
=> ( ( finite4249678464180374575et_int @ B2 )
=> ( ( ord_less_eq_nat @ ( finite7882580182802147440et_int @ A2 ) @ ( finite7882580182802147440et_int @ B2 ) )
=> ( ord_less_eq_nat @ ( finite7882580182802147440et_int @ ( minus_6857623457997529383et_int @ A2 @ B2 ) ) @ ( finite7882580182802147440et_int @ ( minus_6857623457997529383et_int @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1050_card__le__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1051_card__le__sym__Diff,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ( finite6197958912794628473et_int @ A2 )
=> ( ( finite6197958912794628473et_int @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_int @ A2 ) @ ( finite_card_set_int @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ ( minus_8897228262479074673et_int @ A2 @ B2 ) ) @ ( finite_card_set_int @ ( minus_8897228262479074673et_int @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1052_int__less__induct,axiom,
! [I: int,K2: int,P: int > $o] :
( ( ord_less_int @ I @ K2 )
=> ( ( P @ ( minus_minus_int @ K2 @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K2 )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1053_card__Diff__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1054_card__Diff__subset,axiom,
! [B2: set_set_int,A2: set_set_int] :
( ( finite6197958912794628473et_int @ B2 )
=> ( ( ord_le4403425263959731960et_int @ B2 @ A2 )
=> ( ( finite_card_set_int @ ( minus_8897228262479074673et_int @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_set_int @ A2 ) @ ( finite_card_set_int @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1055_card__Diff__subset,axiom,
! [B2: set_set_set_int,A2: set_set_set_int] :
( ( finite4249678464180374575et_int @ B2 )
=> ( ( ord_le4317611570275147438et_int @ B2 @ A2 )
=> ( ( finite7882580182802147440et_int @ ( minus_6857623457997529383et_int @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite7882580182802147440et_int @ A2 ) @ ( finite7882580182802147440et_int @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1056_diff__card__le__card__Diff,axiom,
! [B2: set_set_set_int,A2: set_set_set_int] :
( ( finite4249678464180374575et_int @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite7882580182802147440et_int @ A2 ) @ ( finite7882580182802147440et_int @ B2 ) ) @ ( finite7882580182802147440et_int @ ( minus_6857623457997529383et_int @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1057_diff__card__le__card__Diff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1058_diff__card__le__card__Diff,axiom,
! [B2: set_set_int,A2: set_set_int] :
( ( finite6197958912794628473et_int @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_int @ A2 ) @ ( finite_card_set_int @ B2 ) ) @ ( finite_card_set_int @ ( minus_8897228262479074673et_int @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1059_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_1060_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1061_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_1062_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1063_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1064_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1065_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_1066_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_1067_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_1068_finite__has__maximal2,axiom,
! [A2: set_nat_set_int,A: nat > set_int] :
( ( finite7455725759970522984et_int @ A2 )
=> ( ( member_nat_set_int @ A @ A2 )
=> ? [X2: nat > set_int] :
( ( member_nat_set_int @ X2 @ A2 )
& ( ord_le3704955753469811889et_int @ A @ X2 )
& ! [Xa: nat > set_int] :
( ( member_nat_set_int @ Xa @ A2 )
=> ( ( ord_le3704955753469811889et_int @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1069_finite__has__maximal2,axiom,
! [A2: set_set_int,A: set_int] :
( ( finite6197958912794628473et_int @ A2 )
=> ( ( member_set_int @ A @ A2 )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ A2 )
& ( ord_less_eq_set_int @ A @ X2 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A2 )
=> ( ( ord_less_eq_set_int @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1070_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1071_finite__has__maximal2,axiom,
! [A2: set_set_set_int,A: set_set_int] :
( ( finite4249678464180374575et_int @ A2 )
=> ( ( member_set_set_int @ A @ A2 )
=> ? [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A2 )
& ( ord_le4403425263959731960et_int @ A @ X2 )
& ! [Xa: set_set_int] :
( ( member_set_set_int @ Xa @ A2 )
=> ( ( ord_le4403425263959731960et_int @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1072_finite__has__maximal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( ord_less_eq_int @ A @ X2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1073_finite__has__maximal2,axiom,
! [A2: set_set_set_set_int,A: set_set_set_int] :
( ( finite1113392736978984933et_int @ A2 )
=> ( ( member7356822600254261989et_int @ A @ A2 )
=> ? [X2: set_set_set_int] :
( ( member7356822600254261989et_int @ X2 @ A2 )
& ( ord_le4317611570275147438et_int @ A @ X2 )
& ! [Xa: set_set_set_int] :
( ( member7356822600254261989et_int @ Xa @ A2 )
=> ( ( ord_le4317611570275147438et_int @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1074_finite__has__minimal2,axiom,
! [A2: set_nat_set_int,A: nat > set_int] :
( ( finite7455725759970522984et_int @ A2 )
=> ( ( member_nat_set_int @ A @ A2 )
=> ? [X2: nat > set_int] :
( ( member_nat_set_int @ X2 @ A2 )
& ( ord_le3704955753469811889et_int @ X2 @ A )
& ! [Xa: nat > set_int] :
( ( member_nat_set_int @ Xa @ A2 )
=> ( ( ord_le3704955753469811889et_int @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1075_finite__has__minimal2,axiom,
! [A2: set_set_int,A: set_int] :
( ( finite6197958912794628473et_int @ A2 )
=> ( ( member_set_int @ A @ A2 )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ A2 )
& ( ord_less_eq_set_int @ X2 @ A )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A2 )
=> ( ( ord_less_eq_set_int @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1076_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1077_finite__has__minimal2,axiom,
! [A2: set_set_set_int,A: set_set_int] :
( ( finite4249678464180374575et_int @ A2 )
=> ( ( member_set_set_int @ A @ A2 )
=> ? [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A2 )
& ( ord_le4403425263959731960et_int @ X2 @ A )
& ! [Xa: set_set_int] :
( ( member_set_set_int @ Xa @ A2 )
=> ( ( ord_le4403425263959731960et_int @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1078_finite__has__minimal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( ord_less_eq_int @ X2 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1079_finite__has__minimal2,axiom,
! [A2: set_set_set_set_int,A: set_set_set_int] :
( ( finite1113392736978984933et_int @ A2 )
=> ( ( member7356822600254261989et_int @ A @ A2 )
=> ? [X2: set_set_set_int] :
( ( member7356822600254261989et_int @ X2 @ A2 )
& ( ord_le4317611570275147438et_int @ X2 @ A )
& ! [Xa: set_set_set_int] :
( ( member7356822600254261989et_int @ Xa @ A2 )
=> ( ( ord_le4317611570275147438et_int @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1080_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1081_rev__finite__subset,axiom,
! [B2: set_set_int,A2: set_set_int] :
( ( finite6197958912794628473et_int @ B2 )
=> ( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ( finite6197958912794628473et_int @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1082_rev__finite__subset,axiom,
! [B2: set_set_set_int,A2: set_set_set_int] :
( ( finite4249678464180374575et_int @ B2 )
=> ( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ( finite4249678464180374575et_int @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1083_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_1084_infinite__super,axiom,
! [S2: set_set_int,T2: set_set_int] :
( ( ord_le4403425263959731960et_int @ S2 @ T2 )
=> ( ~ ( finite6197958912794628473et_int @ S2 )
=> ~ ( finite6197958912794628473et_int @ T2 ) ) ) ).
% infinite_super
thf(fact_1085_infinite__super,axiom,
! [S2: set_set_set_int,T2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ S2 @ T2 )
=> ( ~ ( finite4249678464180374575et_int @ S2 )
=> ~ ( finite4249678464180374575et_int @ T2 ) ) ) ).
% infinite_super
thf(fact_1086_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_1087_finite__subset,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ( ( finite6197958912794628473et_int @ B2 )
=> ( finite6197958912794628473et_int @ A2 ) ) ) ).
% finite_subset
thf(fact_1088_finite__subset,axiom,
! [A2: set_set_set_int,B2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ( ( finite4249678464180374575et_int @ B2 )
=> ( finite4249678464180374575et_int @ A2 ) ) ) ).
% finite_subset
thf(fact_1089_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [B6: set_nat] :
( ( ord_less_set_nat @ B6 @ A5 )
=> ( P @ B6 ) )
=> ( P @ A5 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_1090_finite__psubset__induct,axiom,
! [A2: set_set_int,P: set_set_int > $o] :
( ( finite6197958912794628473et_int @ A2 )
=> ( ! [A5: set_set_int] :
( ( finite6197958912794628473et_int @ A5 )
=> ( ! [B6: set_set_int] :
( ( ord_less_set_set_int @ B6 @ A5 )
=> ( P @ B6 ) )
=> ( P @ A5 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_1091_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1092_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_1093_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N2 )
& ( ord_less_eq_set_nat @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1094_infinite__arbitrarily__large,axiom,
! [A2: set_set_int,N2: nat] :
( ~ ( finite6197958912794628473et_int @ A2 )
=> ? [B7: set_set_int] :
( ( finite6197958912794628473et_int @ B7 )
& ( ( finite_card_set_int @ B7 )
= N2 )
& ( ord_le4403425263959731960et_int @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1095_infinite__arbitrarily__large,axiom,
! [A2: set_set_set_int,N2: nat] :
( ~ ( finite4249678464180374575et_int @ A2 )
=> ? [B7: set_set_set_int] :
( ( finite4249678464180374575et_int @ B7 )
& ( ( finite7882580182802147440et_int @ B7 )
= N2 )
& ( ord_le4317611570275147438et_int @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1096_card__subset__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_1097_card__subset__eq,axiom,
! [B2: set_set_int,A2: set_set_int] :
( ( finite6197958912794628473et_int @ B2 )
=> ( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ( ( ( finite_card_set_int @ A2 )
= ( finite_card_set_int @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_1098_card__subset__eq,axiom,
! [B2: set_set_set_int,A2: set_set_set_int] :
( ( finite4249678464180374575et_int @ B2 )
=> ( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ( ( ( finite7882580182802147440et_int @ A2 )
= ( finite7882580182802147440et_int @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_1099_finite__if__finite__subsets__card__bdd,axiom,
! [F4: set_nat,C4: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F4 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C4 ) ) )
=> ( ( finite_finite_nat @ F4 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F4 ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1100_finite__if__finite__subsets__card__bdd,axiom,
! [F4: set_set_int,C4: nat] :
( ! [G2: set_set_int] :
( ( ord_le4403425263959731960et_int @ G2 @ F4 )
=> ( ( finite6197958912794628473et_int @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ G2 ) @ C4 ) ) )
=> ( ( finite6197958912794628473et_int @ F4 )
& ( ord_less_eq_nat @ ( finite_card_set_int @ F4 ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1101_finite__if__finite__subsets__card__bdd,axiom,
! [F4: set_set_set_int,C4: nat] :
( ! [G2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ G2 @ F4 )
=> ( ( finite4249678464180374575et_int @ G2 )
=> ( ord_less_eq_nat @ ( finite7882580182802147440et_int @ G2 ) @ C4 ) ) )
=> ( ( finite4249678464180374575et_int @ F4 )
& ( ord_less_eq_nat @ ( finite7882580182802147440et_int @ F4 ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1102_obtain__subset__with__card__n,axiom,
! [N2: nat,S2: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S2 ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S2 )
=> ( ( ( finite_card_nat @ T3 )
= N2 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1103_obtain__subset__with__card__n,axiom,
! [N2: nat,S2: set_set_int] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_set_int @ S2 ) )
=> ~ ! [T3: set_set_int] :
( ( ord_le4403425263959731960et_int @ T3 @ S2 )
=> ( ( ( finite_card_set_int @ T3 )
= N2 )
=> ~ ( finite6197958912794628473et_int @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1104_obtain__subset__with__card__n,axiom,
! [N2: nat,S2: set_set_set_int] :
( ( ord_less_eq_nat @ N2 @ ( finite7882580182802147440et_int @ S2 ) )
=> ~ ! [T3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ T3 @ S2 )
=> ( ( ( finite7882580182802147440et_int @ T3 )
= N2 )
=> ~ ( finite4249678464180374575et_int @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1105_exists__subset__between,axiom,
! [A2: set_nat,N2: nat,C4: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C4 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C4 )
=> ( ( finite_finite_nat @ C4 )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C4 )
& ( ( finite_card_nat @ B7 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1106_exists__subset__between,axiom,
! [A2: set_set_int,N2: nat,C4: set_set_int] :
( ( ord_less_eq_nat @ ( finite_card_set_int @ A2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_set_int @ C4 ) )
=> ( ( ord_le4403425263959731960et_int @ A2 @ C4 )
=> ( ( finite6197958912794628473et_int @ C4 )
=> ? [B7: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B7 )
& ( ord_le4403425263959731960et_int @ B7 @ C4 )
& ( ( finite_card_set_int @ B7 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1107_exists__subset__between,axiom,
! [A2: set_set_set_int,N2: nat,C4: set_set_set_int] :
( ( ord_less_eq_nat @ ( finite7882580182802147440et_int @ A2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite7882580182802147440et_int @ C4 ) )
=> ( ( ord_le4317611570275147438et_int @ A2 @ C4 )
=> ( ( finite4249678464180374575et_int @ C4 )
=> ? [B7: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B7 )
& ( ord_le4317611570275147438et_int @ B7 @ C4 )
& ( ( finite7882580182802147440et_int @ B7 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1108_card__seteq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_1109_card__seteq,axiom,
! [B2: set_set_int,A2: set_set_int] :
( ( finite6197958912794628473et_int @ B2 )
=> ( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_int @ B2 ) @ ( finite_card_set_int @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_1110_card__seteq,axiom,
! [B2: set_set_set_int,A2: set_set_set_int] :
( ( finite4249678464180374575et_int @ B2 )
=> ( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite7882580182802147440et_int @ B2 ) @ ( finite7882580182802147440et_int @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_1111_card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_1112_card__mono,axiom,
! [B2: set_set_int,A2: set_set_int] :
( ( finite6197958912794628473et_int @ B2 )
=> ( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ A2 ) @ ( finite_card_set_int @ B2 ) ) ) ) ).
% card_mono
thf(fact_1113_card__mono,axiom,
! [B2: set_set_set_int,A2: set_set_set_int] :
( ( finite4249678464180374575et_int @ B2 )
=> ( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite7882580182802147440et_int @ A2 ) @ ( finite7882580182802147440et_int @ B2 ) ) ) ) ).
% card_mono
thf(fact_1114_card__ge__0__finite,axiom,
! [A2: set_set_set_int] :
( ( ord_less_nat @ zero_zero_nat @ ( finite7882580182802147440et_int @ A2 ) )
=> ( finite4249678464180374575et_int @ A2 ) ) ).
% card_ge_0_finite
thf(fact_1115_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_1116_card__ge__0__finite,axiom,
! [A2: set_set_int] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_int @ A2 ) )
=> ( finite6197958912794628473et_int @ A2 ) ) ).
% card_ge_0_finite
thf(fact_1117_psubset__card__mono,axiom,
! [B2: set_set_set_int,A2: set_set_set_int] :
( ( finite4249678464180374575et_int @ B2 )
=> ( ( ord_le4562804192517611682et_int @ A2 @ B2 )
=> ( ord_less_nat @ ( finite7882580182802147440et_int @ A2 ) @ ( finite7882580182802147440et_int @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_1118_psubset__card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_1119_psubset__card__mono,axiom,
! [B2: set_set_int,A2: set_set_int] :
( ( finite6197958912794628473et_int @ B2 )
=> ( ( ord_less_set_set_int @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_set_int @ A2 ) @ ( finite_card_set_int @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_1120_card__le__Suc0__iff__eq,axiom,
! [A2: set_set_set_int] :
( ( finite4249678464180374575et_int @ A2 )
=> ( ( ord_less_eq_nat @ ( finite7882580182802147440et_int @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: set_set_int] :
( ( member_set_set_int @ X3 @ A2 )
=> ! [Y5: set_set_int] :
( ( member_set_set_int @ Y5 @ A2 )
=> ( X3 = Y5 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1121_card__le__Suc0__iff__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ A2 )
=> ( X3 = Y5 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1122_card__le__Suc0__iff__eq,axiom,
! [A2: set_set_int] :
( ( finite6197958912794628473et_int @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_int @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: set_int] :
( ( member_set_int @ X3 @ A2 )
=> ! [Y5: set_int] :
( ( member_set_int @ Y5 @ A2 )
=> ( X3 = Y5 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1123_card__psubset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_1124_card__psubset,axiom,
! [B2: set_set_int,A2: set_set_int] :
( ( finite6197958912794628473et_int @ B2 )
=> ( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_int @ A2 ) @ ( finite_card_set_int @ B2 ) )
=> ( ord_less_set_set_int @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_1125_card__psubset,axiom,
! [B2: set_set_set_int,A2: set_set_set_int] :
( ( finite4249678464180374575et_int @ B2 )
=> ( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite7882580182802147440et_int @ A2 ) @ ( finite7882580182802147440et_int @ B2 ) )
=> ( ord_le4562804192517611682et_int @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_1126_r_Oa__lagrange,axiom,
! [H2: set_set_int] :
( ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( additi7073586575563672860t_unit @ H2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( times_times_nat @ ( finite7882580182802147440et_int @ ( a_RCOS5559887075240879033t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 ) ) @ ( finite_card_set_int @ H2 ) )
= ( order_4716970363388151434t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.a_lagrange
thf(fact_1127_r_Oint__embed__diff,axiom,
! [X: int,Y: int] :
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( minus_minus_int @ X @ Y ) )
= ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ).
% r.int_embed_diff
thf(fact_1128_r_Ominus__closed,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.minus_closed
thf(fact_1129_r_Or__right__minus__eq,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( A = B ) ) ) ) ).
% r.r_right_minus_eq
thf(fact_1130_r_Ocarrier__is__subalgebra,axiom,
! [K5: set_set_int] :
( ( ord_le4403425263959731960et_int @ K5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( embedd2743979684206749024t_unit @ K5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.carrier_is_subalgebra
thf(fact_1131_r_Osubalgebra__in__carrier,axiom,
! [K5: set_set_int,V4: set_set_int] :
( ( embedd2743979684206749024t_unit @ K5 @ V4 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ord_le4403425263959731960et_int @ V4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.subalgebra_in_carrier
thf(fact_1132_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1133_r_Osubset__Idl__subset,axiom,
! [I5: set_set_int,H2: set_set_int] :
( ( ord_le4403425263959731960et_int @ I5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ H2 @ I5 )
=> ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 ) @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I5 ) ) ) ) ).
% r.subset_Idl_subset
thf(fact_1134_r_Ogenideal__self,axiom,
! [S2: set_set_int] :
( ( ord_le4403425263959731960et_int @ S2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ S2 @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S2 ) ) ) ).
% r.genideal_self
thf(fact_1135_minf_I7_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_1136_minf_I7_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ~ ( ord_less_int @ T @ X4 ) ) ).
% minf(7)
thf(fact_1137_minf_I5_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_1138_minf_I5_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ord_less_int @ X4 @ T ) ) ).
% minf(5)
thf(fact_1139_minf_I4_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_1140_minf_I4_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_1141_minf_I3_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_1142_minf_I3_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_1143_minf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z6 )
=> ( ( Q3 @ X2 )
= ( Q4 @ X2 ) ) )
=> ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( ( P @ X4 )
| ( Q3 @ X4 ) )
= ( ( P3 @ X4 )
| ( Q4 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1144_minf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q3: int > $o,Q4: int > $o] :
( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z6 )
=> ( ( Q3 @ X2 )
= ( Q4 @ X2 ) ) )
=> ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( ( P @ X4 )
| ( Q3 @ X4 ) )
= ( ( P3 @ X4 )
| ( Q4 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1145_minf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z6 )
=> ( ( Q3 @ X2 )
= ( Q4 @ X2 ) ) )
=> ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( ( P @ X4 )
& ( Q3 @ X4 ) )
= ( ( P3 @ X4 )
& ( Q4 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1146_minf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q3: int > $o,Q4: int > $o] :
( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z6 )
=> ( ( Q3 @ X2 )
= ( Q4 @ X2 ) ) )
=> ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( ( P @ X4 )
& ( Q3 @ X4 ) )
= ( ( P3 @ X4 )
& ( Q4 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1147_pinf_I7_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_1148_pinf_I7_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ord_less_int @ T @ X4 ) ) ).
% pinf(7)
thf(fact_1149_pinf_I5_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_1150_pinf_I5_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ~ ( ord_less_int @ X4 @ T ) ) ).
% pinf(5)
thf(fact_1151_pinf_I4_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_1152_pinf_I4_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_1153_pinf_I3_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_1154_pinf_I3_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_1155_pinf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z6 @ X2 )
=> ( ( Q3 @ X2 )
= ( Q4 @ X2 ) ) )
=> ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( ( P @ X4 )
| ( Q3 @ X4 ) )
= ( ( P3 @ X4 )
| ( Q4 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1156_pinf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q3: int > $o,Q4: int > $o] :
( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ Z6 @ X2 )
=> ( ( Q3 @ X2 )
= ( Q4 @ X2 ) ) )
=> ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( ( P @ X4 )
| ( Q3 @ X4 ) )
= ( ( P3 @ X4 )
| ( Q4 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1157_pinf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z6 @ X2 )
=> ( ( Q3 @ X2 )
= ( Q4 @ X2 ) ) )
=> ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( ( P @ X4 )
& ( Q3 @ X4 ) )
= ( ( P3 @ X4 )
& ( Q4 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1158_pinf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q3: int > $o,Q4: int > $o] :
( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ Z6 @ X2 )
=> ( ( Q3 @ X2 )
= ( Q4 @ X2 ) ) )
=> ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( ( P @ X4 )
& ( Q3 @ X4 ) )
= ( ( P3 @ X4 )
& ( Q4 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1159_pinf_I6_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_1160_pinf_I6_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ T ) ) ).
% pinf(6)
thf(fact_1161_pinf_I8_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_1162_pinf_I8_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ord_less_eq_int @ T @ X4 ) ) ).
% pinf(8)
thf(fact_1163_minf_I6_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_1164_minf_I6_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ord_less_eq_int @ X4 @ T ) ) ).
% minf(6)
thf(fact_1165_minf_I8_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_1166_minf_I8_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ~ ( ord_less_eq_int @ T @ X4 ) ) ).
% minf(8)
thf(fact_1167_inf__period_I1_J,axiom,
! [P: int > $o,D3: int,Q3: int > $o] :
( ! [X2: int,K3: int] :
( ( P @ X2 )
= ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ( ! [X2: int,K3: int] :
( ( Q3 @ X2 )
= ( Q3 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ! [X4: int,K6: int] :
( ( ( P @ X4 )
& ( Q3 @ X4 ) )
= ( ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K6 @ D3 ) ) )
& ( Q3 @ ( minus_minus_int @ X4 @ ( times_times_int @ K6 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1168_inf__period_I2_J,axiom,
! [P: int > $o,D3: int,Q3: int > $o] :
( ! [X2: int,K3: int] :
( ( P @ X2 )
= ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ( ! [X2: int,K3: int] :
( ( Q3 @ X2 )
= ( Q3 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ! [X4: int,K6: int] :
( ( ( P @ X4 )
| ( Q3 @ X4 ) )
= ( ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K6 @ D3 ) ) )
| ( Q3 @ ( minus_minus_int @ X4 @ ( times_times_int @ K6 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1169_minf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_1170_minf_I10_J,axiom,
! [D: int,S: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_1171_minf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ).
% minf(9)
thf(fact_1172_minf_I9_J,axiom,
! [D: int,S: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ).
% minf(9)
thf(fact_1173_pinf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_1174_pinf_I10_J,axiom,
! [D: int,S: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_1175_pinf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ).
% pinf(9)
thf(fact_1176_pinf_I9_J,axiom,
! [D: int,S: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ).
% pinf(9)
thf(fact_1177_zdvd__mono,axiom,
! [K2: int,M2: int,T: int] :
( ( K2 != zero_zero_int )
=> ( ( dvd_dvd_int @ M2 @ T )
= ( dvd_dvd_int @ ( times_times_int @ K2 @ M2 ) @ ( times_times_int @ K2 @ T ) ) ) ) ).
% zdvd_mono
thf(fact_1178_unity__coeff__ex,axiom,
! [P: int > $o,L: int] :
( ( ? [X3: int] : ( P @ ( times_times_int @ L @ X3 ) ) )
= ( ? [X3: int] :
( ( dvd_dvd_int @ L @ ( plus_plus_int @ X3 @ zero_zero_int ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_1179_unity__coeff__ex,axiom,
! [P: nat > $o,L: nat] :
( ( ? [X3: nat] : ( P @ ( times_times_nat @ L @ X3 ) ) )
= ( ? [X3: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X3 @ zero_zero_nat ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_1180_inf__period_I3_J,axiom,
! [D: int,D3: int,T: int] :
( ( dvd_dvd_int @ D @ D3 )
=> ! [X4: int,K6: int] :
( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X4 @ ( times_times_int @ K6 @ D3 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_1181_inf__period_I4_J,axiom,
! [D: int,D3: int,T: int] :
( ( dvd_dvd_int @ D @ D3 )
=> ! [X4: int,K6: int] :
( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X4 @ ( times_times_int @ K6 @ D3 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_1182_minusinfinity,axiom,
! [D: int,P12: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int,K3: int] :
( ( P12 @ X2 )
= ( P12 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P12 @ X2 ) ) )
=> ( ? [X_1: int] : ( P12 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_1183_plusinfinity,axiom,
! [D: int,P3: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int,K3: int] :
( ( P3 @ X2 )
= ( P3 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [X_1: int] : ( P3 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_1184_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K2: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1185_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K2: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1186_r_Orcosets__subset__PowG,axiom,
! [H2: set_set_int] :
( ( additi7073586575563672860t_unit @ H2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ord_le4317611570275147438et_int @ ( a_RCOS5559887075240879033t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 ) @ ( pow_set_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.rcosets_subset_PowG
thf(fact_1187_r_Oline__extension__smult__closed,axiom,
! [K5: set_set_int,E2: set_set_int,A: set_int,K2: set_int,U: set_int] :
( ( subfie3888952257595785920t_unit @ K5 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ! [K3: set_int,V2: set_int] :
( ( member_set_int @ K3 @ K5 )
=> ( ( member_set_int @ V2 @ E2 )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K3 @ V2 ) @ E2 ) ) )
=> ( ( ord_le4403425263959731960et_int @ E2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ K2 @ K5 )
=> ( ( member_set_int @ U @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K5 @ A @ E2 ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ U ) @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K5 @ A @ E2 ) ) ) ) ) ) ) ) ).
% r.line_extension_smult_closed
thf(fact_1188_r_Osubring__props_I2_J,axiom,
! [K5: set_set_int] :
( ( subfie3888952257595785920t_unit @ K5 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ K5 ) ) ).
% r.subring_props(2)
thf(fact_1189_r_Osubring__props_I7_J,axiom,
! [K5: set_set_int,H1: set_int,H22: set_int] :
( ( subfie3888952257595785920t_unit @ K5 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ H1 @ K5 )
=> ( ( member_set_int @ H22 @ K5 )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H1 @ H22 ) @ K5 ) ) ) ) ).
% r.subring_props(7)
thf(fact_1190_r_Osubring__props_I6_J,axiom,
! [K5: set_set_int,H1: set_int,H22: set_int] :
( ( subfie3888952257595785920t_unit @ K5 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ H1 @ K5 )
=> ( ( member_set_int @ H22 @ K5 )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H1 @ H22 ) @ K5 ) ) ) ) ).
% r.subring_props(6)
thf(fact_1191_r_Osubring__props_I1_J,axiom,
! [K5: set_set_int] :
( ( subfie3888952257595785920t_unit @ K5 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ord_le4403425263959731960et_int @ K5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.subring_props(1)
thf(fact_1192_finite__Pow__iff,axiom,
! [A2: set_nat] :
( ( finite1152437895449049373et_nat @ ( pow_nat @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_Pow_iff
thf(fact_1193_finite__Pow__iff,axiom,
! [A2: set_set_int] :
( ( finite4249678464180374575et_int @ ( pow_set_int @ A2 ) )
= ( finite6197958912794628473et_int @ A2 ) ) ).
% finite_Pow_iff
thf(fact_1194_finite__Pow__iff,axiom,
! [A2: set_int] :
( ( finite6197958912794628473et_int @ ( pow_int @ A2 ) )
= ( finite_finite_int @ A2 ) ) ).
% finite_Pow_iff
thf(fact_1195_r_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K5: set_set_int,E2: set_set_int,V4: set_set_int] :
( ( subfie3888952257595785920t_unit @ K5 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K5 @ E2 )
=> ( ( embedd2743979684206749024t_unit @ K5 @ V4 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ord_le4403425263959731960et_int @ V4 @ E2 )
=> ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K5 @ V4 ) ) ) ) ) ).
% r.subalbegra_incl_imp_finite_dimension
thf(fact_1196_r_Ofinite__dimension__imp__subalgebra,axiom,
! [K5: set_set_int,E2: set_set_int] :
( ( subfie3888952257595785920t_unit @ K5 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K5 @ E2 )
=> ( embedd2743979684206749024t_unit @ K5 @ E2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.finite_dimension_imp_subalgebra
thf(fact_1197_r_Otelescopic__base__dim_I1_J,axiom,
! [K5: set_set_int,F4: set_set_int,E2: set_set_int] :
( ( subfie3888952257595785920t_unit @ K5 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( subfie3888952257595785920t_unit @ F4 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K5 @ F4 )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F4 @ E2 )
=> ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K5 @ E2 ) ) ) ) ) ).
% r.telescopic_base_dim(1)
thf(fact_1198_psubsetI,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_set_int @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_1199_psubsetI,axiom,
! [A2: set_set_set_int,B2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le4562804192517611682et_int @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_1200_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1201_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_1202_nat__int__comparison_I1_J,axiom,
( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [A3: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1203_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1204_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1205_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1206_psubsetD,axiom,
! [A2: set_set_int,B2: set_set_int,C: set_int] :
( ( ord_less_set_set_int @ A2 @ B2 )
=> ( ( member_set_int @ C @ A2 )
=> ( member_set_int @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1207_psubsetD,axiom,
! [A2: set_nat_set_int,B2: set_nat_set_int,C: nat > set_int] :
( ( ord_le2931775347370382171et_int @ A2 @ B2 )
=> ( ( member_nat_set_int @ C @ A2 )
=> ( member_nat_set_int @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1208_psubsetD,axiom,
! [A2: set_set_set_int,B2: set_set_set_int,C: set_set_int] :
( ( ord_le4562804192517611682et_int @ A2 @ B2 )
=> ( ( member_set_set_int @ C @ A2 )
=> ( member_set_set_int @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1209_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1210_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_1211_verit__comp__simplify1_I3_J,axiom,
! [B8: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B8 @ A6 ) )
= ( ord_less_nat @ A6 @ B8 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1212_verit__comp__simplify1_I3_J,axiom,
! [B8: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B8 @ A6 ) )
= ( ord_less_int @ A6 @ B8 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1213_psubsetE,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B2 )
=> ~ ( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ( ord_le4403425263959731960et_int @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_1214_psubsetE,axiom,
! [A2: set_set_set_int,B2: set_set_set_int] :
( ( ord_le4562804192517611682et_int @ A2 @ B2 )
=> ~ ( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ( ord_le4317611570275147438et_int @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_1215_psubset__eq,axiom,
( ord_less_set_set_int
= ( ^ [A7: set_set_int,B9: set_set_int] :
( ( ord_le4403425263959731960et_int @ A7 @ B9 )
& ( A7 != B9 ) ) ) ) ).
% psubset_eq
thf(fact_1216_psubset__eq,axiom,
( ord_le4562804192517611682et_int
= ( ^ [A7: set_set_set_int,B9: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A7 @ B9 )
& ( A7 != B9 ) ) ) ) ).
% psubset_eq
thf(fact_1217_psubset__imp__subset,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B2 )
=> ( ord_le4403425263959731960et_int @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_1218_psubset__imp__subset,axiom,
! [A2: set_set_set_int,B2: set_set_set_int] :
( ( ord_le4562804192517611682et_int @ A2 @ B2 )
=> ( ord_le4317611570275147438et_int @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_1219_psubset__subset__trans,axiom,
! [A2: set_set_set_int,B2: set_set_set_int,C4: set_set_set_int] :
( ( ord_le4562804192517611682et_int @ A2 @ B2 )
=> ( ( ord_le4317611570275147438et_int @ B2 @ C4 )
=> ( ord_le4562804192517611682et_int @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_1220_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1221_int__plus,axiom,
! [N2: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_1222_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1223_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1224_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1225_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1226_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1227_int__Suc,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1228_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1229_r_Oadd_Oint__pow__diff,axiom,
! [X: set_int,N2: int,M2: int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( minus_minus_int @ N2 @ M2 ) @ X )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ X ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M2 @ X ) ) ) ) ) ).
% r.add.int_pow_diff
thf(fact_1230_r_Osubring__props_I5_J,axiom,
! [K5: set_set_int,H: set_int] :
( ( subfie3888952257595785920t_unit @ K5 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ H @ K5 )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H ) @ K5 ) ) ) ).
% r.subring_props(5)
thf(fact_1231_r_Oa__transpose__inv,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= Z )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ Z )
= Y ) ) ) ) ) ).
% r.a_transpose_inv
thf(fact_1232_r_Oadd_Oinv__mult__group,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) ) ) ) ) ).
% r.add.inv_mult_group
thf(fact_1233_r_Oadd_Oinv__solve__left,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( A
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B ) @ C ) )
= ( C
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ A ) ) ) ) ) ) ).
% r.add.inv_solve_left
thf(fact_1234_r_Oadd_Oinv__solve__left_H,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B ) @ C )
= A )
= ( C
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ A ) ) ) ) ) ) ).
% r.add.inv_solve_left'
thf(fact_1235_r_Oadd_Oinv__solve__right,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( A
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C ) ) )
= ( B
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C ) ) ) ) ) ) ).
% r.add.inv_solve_right
thf(fact_1236_r_Oadd_Oinv__solve__right_H,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C ) )
= A )
= ( B
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C ) ) ) ) ) ) ).
% r.add.inv_solve_right'
thf(fact_1237_r_Ominus__add,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ) ) ).
% r.minus_add
thf(fact_1238_r_Or__neg1,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= Y ) ) ) ).
% r.r_neg1
thf(fact_1239_r_Or__neg2,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ Y ) )
= Y ) ) ) ).
% r.r_neg2
thf(fact_1240_r_Ol__minus,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ Y )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) ) ) ) ) ).
% r.l_minus
thf(fact_1241_r_Or__minus,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) ) ) ) ) ).
% r.r_minus
thf(fact_1242_r_Oadd_Onat__pow__inv,axiom,
! [X: set_int,I: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X ) ) ) ) ).
% r.add.nat_pow_inv
thf(fact_1243_r_Oadd_Oint__pow__inv,axiom,
! [X: set_int,I: int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X ) ) ) ) ).
% r.add.int_pow_inv
thf(fact_1244_r_Ominus__eq,axiom,
! [X: set_int,Y: set_int] :
( ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ).
% r.minus_eq
thf(fact_1245_r_Ol__neg,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.l_neg
thf(fact_1246_r_Ominus__equality,axiom,
! [Y: set_int,X: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X )
= Y ) ) ) ) ).
% r.minus_equality
thf(fact_1247_r_Or__neg,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.r_neg
thf(fact_1248_r_Osum__zero__eq__neg,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( X
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ) ) ).
% r.sum_zero_eq_neg
thf(fact_1249_r_Oadd_Oinv__closed,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.add.inv_closed
thf(fact_1250_r_Ominus__minus,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) )
= X ) ) ).
% r.minus_minus
thf(fact_1251_r_Ominus__zero,axiom,
( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.minus_zero
thf(fact_1252_r_Oadd_Oinv__eq__1__iff,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( X
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.inv_eq_1_iff
thf(fact_1253_r_Oadd_Oone__in__subset,axiom,
! [H2: set_set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( H2 != bot_bot_set_set_int )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ H2 )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ H2 ) )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ H2 )
=> ! [Xa2: set_int] :
( ( member_set_int @ Xa2 @ H2 )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Xa2 ) @ H2 ) ) )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ H2 ) ) ) ) ) ).
% r.add.one_in_subset
thf(fact_1254_r_Oadd_Oint__pow__neg__int,axiom,
! [X: set_int,N2: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ X )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ X ) ) ) ) ).
% r.add.int_pow_neg_int
thf(fact_1255_r_Ocarrier__not__empty,axiom,
( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= bot_bot_set_set_int ) ).
% r.carrier_not_empty
thf(fact_1256_r_Osubring__props_I4_J,axiom,
! [K5: set_set_int] :
( ( subfie3888952257595785920t_unit @ K5 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( K5 != bot_bot_set_set_int ) ) ).
% r.subring_props(4)
thf(fact_1257_r_Oint__embed__inv,axiom,
! [X: int] :
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( uminus_uminus_int @ X ) )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) ) ) ).
% r.int_embed_inv
thf(fact_1258_r_Oadd_Oint__pow__neg,axiom,
! [X: set_int,I: int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( uminus_uminus_int @ I ) @ X )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X ) ) ) ) ).
% r.add.int_pow_neg
thf(fact_1259_negative__eq__positive,axiom,
! [N2: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N2 = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1260_negative__zle,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zle
thf(fact_1261_negative__zless,axiom,
! [N2: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_1262_int__cases2,axiom,
! [Z: int] :
( ! [N4: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% int_cases2
thf(fact_1263_not__int__zless__negative,axiom,
! [N2: nat,M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_1264_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N4: nat] : ( P @ ( semiri1314217659103216013at_int @ N4 ) )
=> ( ! [N4: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_1265_int__cases,axiom,
! [Z: int] :
( ! [N4: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% int_cases
thf(fact_1266_zmult__eq__1__iff,axiom,
! [M2: int,N2: int] :
( ( ( times_times_int @ M2 @ N2 )
= one_one_int )
= ( ( ( M2 = one_one_int )
& ( N2 = one_one_int ) )
| ( ( M2
= ( uminus_uminus_int @ one_one_int ) )
& ( N2
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1267_pos__zmult__eq__1__iff__lemma,axiom,
! [M2: int,N2: int] :
( ( ( times_times_int @ M2 @ N2 )
= one_one_int )
=> ( ( M2 = one_one_int )
| ( M2
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_nat @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ x @ y ) @ ( set_ord_lessThan_nat @ n ) ).
%------------------------------------------------------------------------------