TPTP Problem File: SLH0650^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Universal_Hash_Families/0028_Field/prob_00029_000980__18215256_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1380 ( 459 unt; 108 typ; 0 def)
% Number of atoms : 3650 (1181 equ; 0 cnn)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 13785 ( 299 ~; 36 |; 175 &;11435 @)
% ( 0 <=>;1840 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 435 ( 435 >; 0 *; 0 +; 0 <<)
% Number of symbols : 103 ( 100 usr; 12 con; 0-4 aty)
% Number of variables : 3223 ( 74 ^;3004 !; 145 ?;3223 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:37:42.838
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_t__Int__Oint,type,
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% Explicit typings (100)
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thf(sy_c_Nat_OSuc,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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up_set1168727741560211120t_unit: partia4934656038542163276t_unit > set_nat_set_int ).
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_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_v_n,type,
n: nat ).
% Relevant facts (1268)
thf(fact_0_assms,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% assms
thf(fact_1_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_2_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_3_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_4_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_5_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_7_s_Osemiring__axioms,axiom,
semiri8708897239777792527t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% s.semiring_axioms
thf(fact_8_s_Oabelian__monoid__axioms,axiom,
abelia3815030880812984441t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% s.abelian_monoid_axioms
thf(fact_9_s_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ N )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.nat_pow_zero
thf(fact_10_s_Osubring__props_I2_J,axiom,
! [K: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ K ) ) ).
% s.subring_props(2)
thf(fact_11_s_Ozero__divides,axiom,
! [A: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ A )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.zero_divides
thf(fact_12_s_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_int,F: int > set_int] :
( ! [X: int] :
( ( member_int @ X @ A2 )
=> ( ( F @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
=> ( ( finsum5140864651718467066it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.add.finprod_one_eqI
thf(fact_13_s_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_set_set_int,F: set_set_int > set_int] :
( ! [X: set_set_int] :
( ( member_set_set_int @ X @ A2 )
=> ( ( F @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
=> ( ( finsum1798061883449901670et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.add.finprod_one_eqI
thf(fact_14_s_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_nat,F: nat > set_int] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( F @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
=> ( ( finsum5143355122227517342it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.add.finprod_one_eqI
thf(fact_15_s_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_set_int,F: set_int > set_int] :
( ! [X: set_int] :
( ( member_set_int @ X @ A2 )
=> ( ( F @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
=> ( ( finsum2830217253030306736et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.add.finprod_one_eqI
thf(fact_16_s_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_nat_set_int,F: ( nat > set_int ) > set_int] :
( ! [X: nat > set_int] :
( ( member_nat_set_int @ X @ A2 )
=> ( ( F @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
=> ( ( finsum1765041520989711135et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.add.finprod_one_eqI
thf(fact_17_s_Oring__axioms,axiom,
ring_s5316885176909347197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% s.ring_axioms
thf(fact_18_s_Ois__abelian__group,axiom,
abelia23968383328945916t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% s.is_abelian_group
thf(fact_19_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_20_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_21_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_22_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_23_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_24_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_25_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_26_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_27_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_28_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_29_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_30_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_31_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_32_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_33_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_34_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_35_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_36_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_37_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_38_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_39_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_40_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_41_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_42_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_43_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_44_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_45_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_46_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_47_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_48_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_49_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_50_s_Odivides__zero,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.divides_zero
thf(fact_51_s_Otelescopic__base__dim_I1_J,axiom,
! [K: set_set_int,F2: set_set_int,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( subfie3888952257595785920t_unit @ F2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ F2 )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F2 @ E )
=> ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ E ) ) ) ) ) ).
% s.telescopic_base_dim(1)
thf(fact_52_semiring_Onat__pow__zero,axiom,
! [R: partia4934656038542163276t_unit,N: nat] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( N != zero_zero_nat )
=> ( ( pow_se2518650051167492506it_nat @ R @ ( zero_s6269048424454532197t_unit @ R ) @ N )
= ( zero_s6269048424454532197t_unit @ R ) ) ) ) ).
% semiring.nat_pow_zero
thf(fact_53_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_54_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_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_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_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_60_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_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_62_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_63_Collect__cong,axiom,
! [P: ( nat > set_int ) > $o,Q: ( nat > set_int ) > $o] :
( ! [X: nat > set_int] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat_set_int @ P )
= ( collect_nat_set_int @ Q ) ) ) ).
% Collect_cong
thf(fact_64_Collect__cong,axiom,
! [P: set_int > $o,Q: set_int > $o] :
( ! [X: set_int] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_set_int @ P )
= ( collect_set_int @ Q ) ) ) ).
% Collect_cong
thf(fact_65_s_Osubring__props_I4_J,axiom,
! [K: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( K != bot_bot_set_set_int ) ) ).
% s.subring_props(4)
thf(fact_66_s_Odivides__trans,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ C )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C ) ) ) ) ).
% s.divides_trans
thf(fact_67_s_Oproperfactor__divides,axiom,
! [A: set_int,B: set_int] :
( ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ).
% s.properfactor_divides
thf(fact_68_s_Osubring__props_I7_J,axiom,
! [K: set_set_int,H1: set_int,H2: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ H1 @ K )
=> ( ( member_set_int @ H2 @ K )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H1 @ H2 ) @ K ) ) ) ) ).
% s.subring_props(7)
thf(fact_69_s_Oint__embed__zero,axiom,
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ zero_zero_int )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% s.int_embed_zero
thf(fact_70_s_Osubring__props_I6_J,axiom,
! [K: set_set_int,H1: set_int,H2: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ H1 @ K )
=> ( ( member_set_int @ H2 @ K )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H1 @ H2 ) @ K ) ) ) ) ).
% s.subring_props(6)
thf(fact_71_s_Ospace__subgroup__props_I2_J,axiom,
! [K: set_set_int,N: nat,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ E ) ) ) ).
% s.space_subgroup_props(2)
thf(fact_72_s_Osubring__props_I3_J,axiom,
! [K: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ K ) ) ).
% s.subring_props(3)
thf(fact_73_s_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 ) ) ) ) ) ).
% s.add.l_cancel
thf(fact_74_s_Oadd_Om__assoc,axiom,
! [X2: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X2 @ ( 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 ) ) @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) ) ) ) ) ) ).
% s.add.m_assoc
thf(fact_75_s_Oadd_Om__comm,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 ) ) ) ) ).
% s.add.m_comm
thf(fact_76_s_Oadd_Om__lcomm,axiom,
! [X2: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X2 @ ( 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 ) ) @ X2 @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Z ) ) ) ) ) ) ).
% s.add.m_lcomm
thf(fact_77_s_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 ) ) ) ) ) ).
% s.add.r_cancel
thf(fact_78_s_Ocarrier__not__empty,axiom,
( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= bot_bot_set_set_int ) ).
% s.carrier_not_empty
thf(fact_79_s_Om__assoc,axiom,
! [X2: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X2 @ ( 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 ) ) @ X2 @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) ) ) ) ) ) ).
% s.m_assoc
thf(fact_80_s_Oint__embed__closed,axiom,
! [K2: int] : ( member_set_int @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% s.int_embed_closed
thf(fact_81_s_Odimension__is__inj,axiom,
! [K: set_set_int,N: nat,E: set_set_int,M: nat] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ K @ E )
=> ( N = M ) ) ) ) ).
% s.dimension_is_inj
thf(fact_82_s_Ofinite__dimensionE_H,axiom,
! [K: set_set_int,E: set_set_int] :
( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ E )
=> ~ ! [N2: nat] :
~ ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ K @ E ) ) ).
% s.finite_dimensionE'
thf(fact_83_s_Ofinite__dimensionI,axiom,
! [N: nat,K: set_set_int,E: set_set_int] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ E ) ) ).
% s.finite_dimensionI
thf(fact_84_s_Ofinite__dimension__def,axiom,
! [K: set_set_int,E: set_set_int] :
( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ E )
= ( ? [N3: nat] : ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N3 @ K @ E ) ) ) ).
% s.finite_dimension_def
thf(fact_85_s_Oadd_Oinv__comm,axiom,
! [X2: set_int,Y: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% s.add.inv_comm
thf(fact_86_s_Oadd_Ol__inv__ex,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ X2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% s.add.l_inv_ex
thf(fact_87_s_Oadd_Oone__unique,axiom,
! [U: set_int] :
( ( member_set_int @ U @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ U @ X )
= X ) )
=> ( U
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% s.add.one_unique
thf(fact_88_s_Oadd_Or__inv__ex,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% s.add.r_inv_ex
thf(fact_89_s_Ominus__unique,axiom,
! [Y: set_int,X2: set_int,Y2: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X2 @ ( 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 ) ) ) ) ) ) ).
% s.minus_unique
thf(fact_90_s_Ol__distr,axiom,
! [X2: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X2 @ ( 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 ) ) @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Z ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) ) ) ) ) ) ).
% s.l_distr
thf(fact_91_s_Or__distr,axiom,
! [X2: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X2 @ ( 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 ) ) @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ X2 ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ Y ) ) ) ) ) ) ).
% s.r_distr
thf(fact_92_s_Oinv__unique,axiom,
! [Y: set_int,X2: set_int,Y2: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y2 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X2 @ ( 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 ) ) ) ) ) ) ).
% s.inv_unique
thf(fact_93_s_Oone__unique,axiom,
! [U: set_int] :
( ( member_set_int @ U @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ U @ X )
= X ) )
=> ( U
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% s.one_unique
thf(fact_94_s_Ospace__subgroup__props_I3_J,axiom,
! [K: set_set_int,N: nat,E: set_set_int,V1: set_int,V2: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( ( member_set_int @ V1 @ E )
=> ( ( member_set_int @ V2 @ E )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ V1 @ V2 ) @ E ) ) ) ) ) ).
% s.space_subgroup_props(3)
thf(fact_95_s_Ogroup__commutes__pow,axiom,
! [X2: set_int,Y: set_int,N: nat] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ N ) @ Y )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ N ) ) ) ) ) ) ).
% s.group_commutes_pow
thf(fact_96_s_Onat__pow__comm,axiom,
! [X2: set_int,N: nat,M: nat] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ N ) @ ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ M ) )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ M ) @ ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ N ) ) ) ) ).
% s.nat_pow_comm
thf(fact_97_s_Opow__mult__distrib,axiom,
! [X2: set_int,Y: set_int,N: nat] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) @ N )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ N ) @ ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ N ) ) ) ) ) ) ).
% s.pow_mult_distrib
thf(fact_98_s_Odivides__mult,axiom,
! [A: set_int,C: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C @ A ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C @ B ) ) ) ) ) ).
% s.divides_mult
thf(fact_99_s_Odivides__prod__r,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ C ) ) ) ) ) ).
% s.divides_prod_r
thf(fact_100_s_Oone__divides,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ A ) ) ).
% s.one_divides
thf(fact_101_s_Ospace__subgroup__props_I5_J,axiom,
! [K: set_set_int,N: nat,E: set_set_int,K2: set_int,V: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( ( member_set_int @ K2 @ K )
=> ( ( member_set_int @ V @ E )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ V ) @ E ) ) ) ) ) ).
% s.space_subgroup_props(5)
thf(fact_102_s_Oproperfactor__prod__r,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ 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 ) ) ) )
=> ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ C ) ) ) ) ) ) ).
% s.properfactor_prod_r
thf(fact_103_s_Ounique__dimension,axiom,
! [K: set_set_int,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ E )
=> ? [X: nat] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ K @ E )
& ! [Y3: nat] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y3 @ K @ E )
=> ( Y3 = X ) ) ) ) ) ).
% s.unique_dimension
thf(fact_104_s_Oproperfactor__trans1,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ C )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C ) ) ) ) ) ).
% s.properfactor_trans1
thf(fact_105_s_Oproperfactor__trans2,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( factor5186451337065598620t_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 ) ) ) )
=> ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C ) ) ) ) ) ).
% s.properfactor_trans2
thf(fact_106_s_OS_Ozero__closed,axiom,
member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% s.S.zero_closed
thf(fact_107_s_Oadd_Om__closed,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( 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 ) ) @ X2 @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% s.add.m_closed
thf(fact_108_s_Oadd_Oright__cancel,axiom,
! [X2: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X2 @ ( 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 @ X2 )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ X2 ) )
= ( Y = Z ) ) ) ) ) ).
% s.add.right_cancel
thf(fact_109_s_Om__closed,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( 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 ) ) @ X2 @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% s.m_closed
thf(fact_110_s_Oone__closed,axiom,
member_set_int @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% s.one_closed
thf(fact_111_s_Onat__pow__closed,axiom,
! [X2: set_int,N: nat] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ N ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.nat_pow_closed
thf(fact_112_s_Odivides__refl,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ A ) ) ).
% s.divides_refl
thf(fact_113_s_Onat__pow__one,axiom,
! [N: nat] :
( ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ N )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% s.nat_pow_one
thf(fact_114_s_Oadd_Ol__cancel__one,axiom,
! [X2: set_int,A: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ A )
= X2 )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% s.add.l_cancel_one
thf(fact_115_s_Oadd_Ol__cancel__one_H,axiom,
! [X2: set_int,A: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( X2
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ A ) )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% s.add.l_cancel_one'
thf(fact_116_s_Oadd_Or__cancel__one,axiom,
! [X2: set_int,A: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ X2 )
= X2 )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% s.add.r_cancel_one
thf(fact_117_s_Oadd_Or__cancel__one_H,axiom,
! [X2: set_int,A: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( X2
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ X2 ) )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% s.add.r_cancel_one'
thf(fact_118_s_Ol__zero,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X2 )
= X2 ) ) ).
% s.l_zero
thf(fact_119_s_Or__zero,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= X2 ) ) ).
% s.r_zero
thf(fact_120_s_Ol__null,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.l_null
thf(fact_121_s_Or__null,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.r_null
thf(fact_122_s_Ofinsum__empty,axiom,
! [F: int > set_int] :
( ( finsum5140864651718467066it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ bot_bot_set_int )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% s.finsum_empty
thf(fact_123_s_Ofinsum__empty,axiom,
! [F: set_set_int > set_int] :
( ( finsum1798061883449901670et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ bot_bo2384636101374064866et_int )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% s.finsum_empty
thf(fact_124_s_Ofinsum__empty,axiom,
! [F: nat > set_int] :
( ( finsum5143355122227517342it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ bot_bot_set_nat )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% s.finsum_empty
thf(fact_125_s_Ofinsum__empty,axiom,
! [F: ( nat > set_int ) > set_int] :
( ( finsum1765041520989711135et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ bot_bo8417611410066262939et_int )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% s.finsum_empty
thf(fact_126_s_Ofinsum__empty,axiom,
! [F: set_int > set_int] :
( ( finsum2830217253030306736et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ bot_bot_set_set_int )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% s.finsum_empty
thf(fact_127_s_Ol__one,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X2 )
= X2 ) ) ).
% s.l_one
thf(fact_128_s_Or__one,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= X2 ) ) ).
% s.r_one
thf(fact_129_s_Odivides__mult__lI,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C @ A ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C @ B ) ) ) ) ) ).
% s.divides_mult_lI
thf(fact_130_s_Onat__pow__0,axiom,
! [X2: set_int] :
( ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ zero_zero_nat )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% s.nat_pow_0
thf(fact_131_abelian__groupE_I4_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ R @ Y @ X2 ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_132_abelian__groupE_I3_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( abelia23968383328945916t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ ( add_se5859248395121729892t_unit @ R @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R @ X2 @ ( add_se5859248395121729892t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_133_abelian__groupE_I1_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_134_abelian__monoidE_I5_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia3815030880812984441t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ R @ Y @ X2 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_135_abelian__monoidE_I3_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( abelia3815030880812984441t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ ( add_se5859248395121729892t_unit @ R @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R @ X2 @ ( add_se5859248395121729892t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_136_abelian__monoidE_I1_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia3815030880812984441t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_137_ring_Oring__simprules_I23_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ Z @ ( add_se5859248395121729892t_unit @ R @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ R @ ( mult_s3864001451298473021t_unit @ R @ Z @ X2 ) @ ( mult_s3864001451298473021t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_138_ring_Oring__simprules_I22_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ X2 @ ( add_se5859248395121729892t_unit @ R @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ R @ Y @ ( add_se5859248395121729892t_unit @ R @ X2 @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_139_ring_Oring__simprules_I13_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ ( add_se5859248395121729892t_unit @ R @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R @ ( mult_s3864001451298473021t_unit @ R @ X2 @ Z ) @ ( mult_s3864001451298473021t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_140_ring_Oring__simprules_I12_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ ( one_se8065767436706823081t_unit @ R ) @ X2 )
= X2 ) ) ) ).
% ring.ring_simprules(12)
thf(fact_141_ring_Oring__simprules_I11_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ ( mult_s3864001451298473021t_unit @ R @ X2 @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ R @ X2 @ ( mult_s3864001451298473021t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_142_ring_Oring__simprules_I10_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ R @ Y @ X2 ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_143_ring_Oring__simprules_I7_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ ( add_se5859248395121729892t_unit @ R @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R @ X2 @ ( add_se5859248395121729892t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_144_ring_Oring__simprules_I6_J,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ R ) @ ( partia966996272515721803t_unit @ R ) ) ) ).
% ring.ring_simprules(6)
thf(fact_145_ring_Oring__simprules_I5_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_146_ring_Oring__simprules_I1_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_147_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ X2 @ ( add_se5859248395121729892t_unit @ R @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ R @ Y @ ( add_se5859248395121729892t_unit @ R @ X2 @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_148_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ ( one_se8065767436706823081t_unit @ R ) @ X2 )
= X2 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_149_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ ( mult_s3864001451298473021t_unit @ R @ X2 @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ R @ X2 @ ( mult_s3864001451298473021t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_150_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ R @ Y @ X2 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_151_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ ( add_se5859248395121729892t_unit @ R @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R @ X2 @ ( add_se5859248395121729892t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_152_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ R ) @ ( partia966996272515721803t_unit @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_153_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_154_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_155_semiring_Ol__distr,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ ( add_se5859248395121729892t_unit @ R @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R @ ( mult_s3864001451298473021t_unit @ R @ X2 @ Z ) @ ( mult_s3864001451298473021t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_156_semiring_Or__distr,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ Z @ ( add_se5859248395121729892t_unit @ R @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ R @ ( mult_s3864001451298473021t_unit @ R @ Z @ X2 ) @ ( mult_s3864001451298473021t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_157_abelian__monoid_Oa__comm,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ G @ Y @ X2 ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_158_abelian__monoid_Oa__assoc,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ ( add_se5859248395121729892t_unit @ G @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ G @ X2 @ ( add_se5859248395121729892t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_159_abelian__monoid_Oa__lcomm,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ X2 @ ( add_se5859248395121729892t_unit @ G @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ G @ Y @ ( add_se5859248395121729892t_unit @ G @ X2 @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_160_abelian__monoid_Oa__closed,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ G @ X2 @ Y ) @ ( partia966996272515721803t_unit @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_161_ring_Oring__simprules_I8_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ ( zero_s6269048424454532197t_unit @ R ) @ X2 )
= X2 ) ) ) ).
% ring.ring_simprules(8)
thf(fact_162_ring_Oring__simprules_I15_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ X2 @ ( zero_s6269048424454532197t_unit @ R ) )
= X2 ) ) ) ).
% ring.ring_simprules(15)
thf(fact_163_ring_Oring__simprules_I24_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ ( zero_s6269048424454532197t_unit @ R ) @ X2 )
= ( zero_s6269048424454532197t_unit @ R ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_164_ring_Oring__simprules_I25_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ X2 @ ( zero_s6269048424454532197t_unit @ R ) )
= ( zero_s6269048424454532197t_unit @ R ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_165_abelian__groupI,axiom,
! [R: partia4934656038542163276t_unit] :
( ! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
=> ! [Y4: set_int] :
( ( member_set_int @ Y4 @ ( partia966996272515721803t_unit @ R ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R @ X @ Y4 ) @ ( partia966996272515721803t_unit @ R ) ) ) )
=> ( ( member_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ ( partia966996272515721803t_unit @ R ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
=> ! [Y4: set_int] :
( ( member_set_int @ Y4 @ ( partia966996272515721803t_unit @ R ) )
=> ! [Z2: set_int] :
( ( member_set_int @ Z2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ ( add_se5859248395121729892t_unit @ R @ X @ Y4 ) @ Z2 )
= ( add_se5859248395121729892t_unit @ R @ X @ ( add_se5859248395121729892t_unit @ R @ Y4 @ Z2 ) ) ) ) ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
=> ! [Y4: set_int] :
( ( member_set_int @ Y4 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ X @ Y4 )
= ( add_se5859248395121729892t_unit @ R @ Y4 @ X ) ) ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ ( zero_s6269048424454532197t_unit @ R ) @ X )
= X ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
=> ? [Xa: set_int] :
( ( member_set_int @ Xa @ ( partia966996272515721803t_unit @ R ) )
& ( ( add_se5859248395121729892t_unit @ R @ Xa @ X )
= ( zero_s6269048424454532197t_unit @ R ) ) ) )
=> ( abelia23968383328945916t_unit @ R ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_166_abelian__groupE_I5_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia23968383328945916t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ ( zero_s6269048424454532197t_unit @ R ) @ X2 )
= X2 ) ) ) ).
% abelian_groupE(5)
thf(fact_167_abelian__groupE_I6_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia23968383328945916t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ? [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
& ( ( add_se5859248395121729892t_unit @ R @ X @ X2 )
= ( zero_s6269048424454532197t_unit @ R ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_168_abelian__monoidI,axiom,
! [R: partia4934656038542163276t_unit] :
( ! [X: set_int,Y4: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y4 @ ( partia966996272515721803t_unit @ R ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R @ X @ Y4 ) @ ( partia966996272515721803t_unit @ R ) ) ) )
=> ( ( member_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ ( partia966996272515721803t_unit @ R ) )
=> ( ! [X: set_int,Y4: set_int,Z2: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y4 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Z2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ ( add_se5859248395121729892t_unit @ R @ X @ Y4 ) @ Z2 )
= ( add_se5859248395121729892t_unit @ R @ X @ ( add_se5859248395121729892t_unit @ R @ Y4 @ Z2 ) ) ) ) ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ ( zero_s6269048424454532197t_unit @ R ) @ X )
= X ) )
=> ( ! [X: set_int,Y4: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y4 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ X @ Y4 )
= ( add_se5859248395121729892t_unit @ R @ Y4 @ X ) ) ) )
=> ( abelia3815030880812984441t_unit @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_169_abelian__monoid_Ominus__unique,axiom,
! [G: partia4934656038542163276t_unit,Y: set_int,X2: set_int,Y2: set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( ( add_se5859248395121729892t_unit @ G @ Y @ X2 )
= ( zero_s6269048424454532197t_unit @ G ) )
=> ( ( ( add_se5859248395121729892t_unit @ G @ X2 @ Y2 )
= ( zero_s6269048424454532197t_unit @ G ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y2 @ ( partia966996272515721803t_unit @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_170_abelian__monoid_Or__zero,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ X2 @ ( zero_s6269048424454532197t_unit @ G ) )
= X2 ) ) ) ).
% abelian_monoid.r_zero
thf(fact_171_abelian__monoid_Ol__zero,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ ( zero_s6269048424454532197t_unit @ G ) @ X2 )
= X2 ) ) ) ).
% abelian_monoid.l_zero
thf(fact_172_abelian__monoidE_I4_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia3815030880812984441t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ ( zero_s6269048424454532197t_unit @ R ) @ X2 )
= X2 ) ) ) ).
% abelian_monoidE(4)
thf(fact_173_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ ( zero_s6269048424454532197t_unit @ R ) @ X2 )
= X2 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_174_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ X2 @ ( zero_s6269048424454532197t_unit @ R ) )
= X2 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_175_semiring_Or__null,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ X2 @ ( zero_s6269048424454532197t_unit @ R ) )
= ( zero_s6269048424454532197t_unit @ R ) ) ) ) ).
% semiring.r_null
thf(fact_176_semiring_Ol__null,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ ( zero_s6269048424454532197t_unit @ R ) @ X2 )
= ( zero_s6269048424454532197t_unit @ R ) ) ) ) ).
% semiring.l_null
thf(fact_177_ring_Oring__simprules_I2_J,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ ( partia966996272515721803t_unit @ R ) ) ) ).
% ring.ring_simprules(2)
thf(fact_178_abelian__groupE_I2_J,axiom,
! [R: partia4934656038542163276t_unit] :
( ( abelia23968383328945916t_unit @ R )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ ( partia966996272515721803t_unit @ R ) ) ) ).
% abelian_groupE(2)
thf(fact_179_abelian__monoid_Ozero__closed,axiom,
! [G: partia4934656038542163276t_unit] :
( ( abelia3815030880812984441t_unit @ G )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ G ) @ ( partia966996272515721803t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_180_abelian__monoidE_I2_J,axiom,
! [R: partia4934656038542163276t_unit] :
( ( abelia3815030880812984441t_unit @ R )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ ( partia966996272515721803t_unit @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_181_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ ( partia966996272515721803t_unit @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_182_abelian__monoid_Ofinsum__empty,axiom,
! [G: partia4934656038542163276t_unit,F: int > set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( finsum5140864651718467066it_int @ G @ F @ bot_bot_set_int )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_183_abelian__monoid_Ofinsum__empty,axiom,
! [G: partia4934656038542163276t_unit,F: set_set_int > set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( finsum1798061883449901670et_int @ G @ F @ bot_bo2384636101374064866et_int )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_184_abelian__monoid_Ofinsum__empty,axiom,
! [G: partia4934656038542163276t_unit,F: nat > set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( finsum5143355122227517342it_nat @ G @ F @ bot_bot_set_nat )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_185_abelian__monoid_Ofinsum__empty,axiom,
! [G: partia4934656038542163276t_unit,F: ( nat > set_int ) > set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( finsum1765041520989711135et_int @ G @ F @ bot_bo8417611410066262939et_int )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_186_abelian__monoid_Ofinsum__empty,axiom,
! [G: partia4934656038542163276t_unit,F: set_int > set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( finsum2830217253030306736et_int @ G @ F @ bot_bot_set_set_int )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_187_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_188_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_189_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_190_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_191_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_192_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_193_ring_Ois__abelian__group,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( abelia23968383328945916t_unit @ R ) ) ).
% ring.is_abelian_group
thf(fact_194_abelian__group_Oaxioms_I1_J,axiom,
! [G: partia4934656038542163276t_unit] :
( ( abelia23968383328945916t_unit @ G )
=> ( abelia3815030880812984441t_unit @ G ) ) ).
% abelian_group.axioms(1)
thf(fact_195_semiring_Oaxioms_I1_J,axiom,
! [R: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R )
=> ( abelia3815030880812984441t_unit @ R ) ) ).
% semiring.axioms(1)
thf(fact_196_s_Oonepideal,axiom,
princi8860937869964495385t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% s.onepideal
thf(fact_197_s_Oisgcd__divides__l,axiom,
! [A: set_int,B: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( isgcd_4636411027072998995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ A @ B ) ) ) ) ).
% s.isgcd_divides_l
thf(fact_198_s_Oisgcd__divides__r,axiom,
! [B: set_int,A: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ A )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( isgcd_4636411027072998995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ A @ B ) ) ) ) ).
% s.isgcd_divides_r
thf(fact_199_s_OboundD__carrier,axiom,
! [N: nat,F: nat > set_int,M: nat] :
( ( bound_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_set_int @ ( F @ M ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% s.boundD_carrier
thf(fact_200_s_Ochar__bound_I2_J,axiom,
! [X2: nat] :
( ( ord_less_nat @ zero_zero_nat @ X2 )
=> ( ( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( semiri1314217659103216013at_int @ X2 ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% s.char_bound(2)
thf(fact_201_s_Omonoid__cancelI,axiom,
( ! [A3: set_int,B2: set_int,C2: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C2 @ A3 )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C2 @ B2 ) )
=> ( ( member_set_int @ A3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: set_int,B2: set_int,C2: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A3 @ C2 )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B2 @ C2 ) )
=> ( ( member_set_int @ A3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid497721730651901107t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.monoid_cancelI
thf(fact_202_nat__pow__0,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int] :
( ( pow_se2518650051167492506it_nat @ G @ X2 @ zero_zero_nat )
= ( one_se8065767436706823081t_unit @ G ) ) ).
% nat_pow_0
thf(fact_203_dividesI_H,axiom,
! [B: set_int,G: partia4934656038542163276t_unit,A: set_int,C: set_int] :
( ( B
= ( mult_s3864001451298473021t_unit @ G @ A @ C ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G ) )
=> ( factor5186451337065598620t_unit @ G @ A @ B ) ) ) ).
% dividesI'
thf(fact_204_s_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 ) ) ) ) ).
% s.embed_char_eq_0
thf(fact_205_s_Ofinite__dimension__imp__subalgebra,axiom,
! [K: set_set_int,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ E )
=> ( embedd2743979684206749024t_unit @ K @ E @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.finite_dimension_imp_subalgebra
thf(fact_206_s_Odimension__zero,axiom,
! [K: set_set_int,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ zero_zero_nat @ K @ E )
=> ( E
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) ) ) ).
% s.dimension_zero
thf(fact_207_s_Oone__zeroI,axiom,
( ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
=> ( ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.one_zeroI
thf(fact_208_s_Oone__zeroD,axiom,
( ( ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) ) ).
% s.one_zeroD
thf(fact_209_s_Ocarrier__one__zero,axiom,
( ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
= ( ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.carrier_one_zero
thf(fact_210_s_Ocarrier__one__not__zero,axiom,
( ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
= ( ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.carrier_one_not_zero
thf(fact_211_s_Ozero__dim,axiom,
! [K: set_set_int] : ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ zero_zero_nat @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) ).
% s.zero_dim
thf(fact_212_s_Ozeropideal,axiom,
princi8860937869964495385t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% s.zeropideal
thf(fact_213_monoid__cancel_Ois__monoid__cancel,axiom,
! [G: partia4934656038542163276t_unit] :
( ( monoid497721730651901107t_unit @ G )
=> ( monoid497721730651901107t_unit @ G ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_214_monoid__cancel_Ol__cancel,axiom,
! [G: partia4934656038542163276t_unit,C: set_int,A: set_int,B: set_int] :
( ( monoid497721730651901107t_unit @ G )
=> ( ( ( mult_s3864001451298473021t_unit @ G @ C @ A )
= ( mult_s3864001451298473021t_unit @ G @ C @ B ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_215_monoid__cancel_Or__cancel,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,C: set_int,B: set_int] :
( ( monoid497721730651901107t_unit @ G )
=> ( ( ( mult_s3864001451298473021t_unit @ G @ A @ C )
= ( mult_s3864001451298473021t_unit @ G @ B @ C ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_216_isgcd__def,axiom,
( isgcd_4636411027072998995t_unit
= ( ^ [G2: partia4934656038542163276t_unit,X3: set_int,A4: set_int,B3: set_int] :
( ( factor5186451337065598620t_unit @ G2 @ X3 @ A4 )
& ( factor5186451337065598620t_unit @ G2 @ X3 @ B3 )
& ! [Y5: set_int] :
( ( member_set_int @ Y5 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( ( factor5186451337065598620t_unit @ G2 @ Y5 @ A4 )
& ( factor5186451337065598620t_unit @ G2 @ Y5 @ B3 ) )
=> ( factor5186451337065598620t_unit @ G2 @ Y5 @ X3 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_217_monoid__cancel_Odivides__mult__l,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int,C: set_int] :
( ( monoid497721730651901107t_unit @ G )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G ) )
=> ( ( factor5186451337065598620t_unit @ G @ ( mult_s3864001451298473021t_unit @ G @ C @ A ) @ ( mult_s3864001451298473021t_unit @ G @ C @ B ) )
= ( factor5186451337065598620t_unit @ G @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_218_monoid__cancel_Oproperfactor__mult__l,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int,C: set_int] :
( ( monoid497721730651901107t_unit @ G )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G ) )
=> ( ( proper1002977052347345036t_unit @ G @ ( mult_s3864001451298473021t_unit @ G @ C @ A ) @ ( mult_s3864001451298473021t_unit @ G @ C @ B ) )
= ( proper1002977052347345036t_unit @ G @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_l
thf(fact_219_monoid__cancel_Oproperfactor__mult__lI,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int,C: set_int] :
( ( monoid497721730651901107t_unit @ G )
=> ( ( proper1002977052347345036t_unit @ G @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G ) )
=> ( proper1002977052347345036t_unit @ G @ ( mult_s3864001451298473021t_unit @ G @ C @ A ) @ ( mult_s3864001451298473021t_unit @ G @ C @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_lI
thf(fact_220_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( ( partia966996272515721803t_unit @ R )
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) )
= ( ( one_se8065767436706823081t_unit @ R )
!= ( zero_s6269048424454532197t_unit @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_221_semiring_Ocarrier__one__zero,axiom,
! [R: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( ( partia966996272515721803t_unit @ R )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) )
= ( ( one_se8065767436706823081t_unit @ R )
= ( zero_s6269048424454532197t_unit @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_222_semiring_Oone__zeroI,axiom,
! [R: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( ( partia966996272515721803t_unit @ R )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) )
=> ( ( one_se8065767436706823081t_unit @ R )
= ( zero_s6269048424454532197t_unit @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_223_semiring_Oone__zeroD,axiom,
! [R: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R )
=> ( ( ( one_se8065767436706823081t_unit @ R )
= ( zero_s6269048424454532197t_unit @ R ) )
=> ( ( partia966996272515721803t_unit @ R )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) ) ) ) ).
% semiring.one_zeroD
thf(fact_224_properfactorE,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( proper1002977052347345036t_unit @ G @ A @ B )
=> ~ ( ( factor5186451337065598620t_unit @ G @ A @ B )
=> ( factor5186451337065598620t_unit @ G @ B @ A ) ) ) ).
% properfactorE
thf(fact_225_properfactorI,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( factor5186451337065598620t_unit @ G @ A @ B )
=> ( ~ ( factor5186451337065598620t_unit @ G @ B @ A )
=> ( proper1002977052347345036t_unit @ G @ A @ B ) ) ) ).
% properfactorI
thf(fact_226_properfactor__def,axiom,
( proper1002977052347345036t_unit
= ( ^ [G2: partia4934656038542163276t_unit,A4: set_int,B3: set_int] :
( ( factor5186451337065598620t_unit @ G2 @ A4 @ B3 )
& ~ ( factor5186451337065598620t_unit @ G2 @ B3 @ A4 ) ) ) ) ).
% properfactor_def
thf(fact_227_factor__def,axiom,
( factor5186451337065598620t_unit
= ( ^ [G2: partia4934656038542163276t_unit,A4: set_int,B3: set_int] :
? [X3: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ G2 ) )
& ( B3
= ( mult_s3864001451298473021t_unit @ G2 @ A4 @ X3 ) ) ) ) ) ).
% factor_def
thf(fact_228_dividesI,axiom,
! [C: set_int,G: partia4934656038542163276t_unit,B: set_int,A: set_int] :
( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G ) )
=> ( ( B
= ( mult_s3864001451298473021t_unit @ G @ A @ C ) )
=> ( factor5186451337065598620t_unit @ G @ A @ B ) ) ) ).
% dividesI
thf(fact_229_dividesE,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( factor5186451337065598620t_unit @ G @ A @ B )
=> ~ ! [C2: set_int] :
( ( B
= ( mult_s3864001451298473021t_unit @ G @ A @ C2 ) )
=> ~ ( member_set_int @ C2 @ ( partia966996272515721803t_unit @ G ) ) ) ) ).
% dividesE
thf(fact_230_dividesD,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( factor5186451337065598620t_unit @ G @ A @ B )
=> ? [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G ) )
& ( B
= ( mult_s3864001451298473021t_unit @ G @ A @ X ) ) ) ) ).
% dividesD
thf(fact_231_s_Ogenideal__one,axiom,
( ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
= ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% s.genideal_one
thf(fact_232_s_Ospace__subgroup__props_I6_J,axiom,
! [K: set_set_int,N: nat,E: set_set_int,K2: set_int,A: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( ( member_set_int @ K2 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ A ) @ E )
=> ( member_set_int @ A @ E ) ) ) ) ) ) ).
% s.space_subgroup_props(6)
thf(fact_233_s_Osubfield__m__inv__simprule,axiom,
! [K: set_set_int,K2: set_int,A: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ K2 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ A ) @ K )
=> ( member_set_int @ A @ K ) ) ) ) ) ).
% s.subfield_m_inv_simprule
thf(fact_234_s_Ochar__bound_I1_J,axiom,
! [X2: nat] :
( ( ord_less_nat @ zero_zero_nat @ X2 )
=> ( ( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( semiri1314217659103216013at_int @ X2 ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_less_eq_nat @ ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X2 ) ) ) ).
% s.char_bound(1)
thf(fact_235_s_Ogenideal__zero,axiom,
( ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) ).
% s.genideal_zero
thf(fact_236_s_Ogenideal__self_H,axiom,
! [I: set_int] :
( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ I @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ I @ bot_bot_set_set_int ) ) ) ) ).
% s.genideal_self'
thf(fact_237_ring_Ochar__bound_I2_J,axiom,
! [R: partia4934656038542163276t_unit,X2: nat] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ X2 )
=> ( ( ( ring_i2743490682209504680t_unit @ R @ ( semiri1314217659103216013at_int @ X2 ) )
= ( zero_s6269048424454532197t_unit @ R ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c6147214092195050492t_unit @ R ) ) ) ) ) ).
% ring.char_bound(2)
thf(fact_238_ring_Odimension__zero,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( embedd646006463343340164t_unit @ R @ zero_zero_nat @ K @ E )
=> ( E
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) ) ) ) ) ).
% ring.dimension_zero
thf(fact_239_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_240_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_241_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_242_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_243_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_244_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_245_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_246_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_247_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_248_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_249_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_250_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_251_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_252_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_253_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_254_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_255_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_256_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_257_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_258_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_259_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_260_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_261_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_262_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_263_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_264_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_265_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_266_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_267_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_268_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_269_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_270_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_271_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_272_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_273_eq__iff__diff__eq__0,axiom,
( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( minus_minus_int @ A4 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_274_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_275_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_276_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_277_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_278_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_279_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_280_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_281_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_282_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_283_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_284_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_285_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_286_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_287_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_288_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_289_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_290_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_291_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_292_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_293_ring_Odimension_Ocong,axiom,
embedd646006463343340164t_unit = embedd646006463343340164t_unit ).
% ring.dimension.cong
thf(fact_294_ring_Ofinite__dimension_Ocong,axiom,
embedd8246663962306818995t_unit = embedd8246663962306818995t_unit ).
% ring.finite_dimension.cong
thf(fact_295_ring_Ochar__bound_I1_J,axiom,
! [R: partia4934656038542163276t_unit,X2: nat] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ X2 )
=> ( ( ( ring_i2743490682209504680t_unit @ R @ ( semiri1314217659103216013at_int @ X2 ) )
= ( zero_s6269048424454532197t_unit @ R ) )
=> ( ord_less_eq_nat @ ( ring_c6147214092195050492t_unit @ R ) @ X2 ) ) ) ) ).
% ring.char_bound(1)
thf(fact_296_ring_Ospace__subgroup__props_I6_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,N: nat,E: set_set_int,K2: set_int,A: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( embedd646006463343340164t_unit @ R @ N @ K @ E )
=> ( ( member_set_int @ K2 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ ( mult_s3864001451298473021t_unit @ R @ K2 @ A ) @ E )
=> ( member_set_int @ A @ E ) ) ) ) ) ) ) ).
% ring.space_subgroup_props(6)
thf(fact_297_subalgebra_Osmult__closed,axiom,
! [K: set_set_int,V3: set_set_int,R: partia4934656038542163276t_unit,K2: set_int,V: set_int] :
( ( embedd2743979684206749024t_unit @ K @ V3 @ R )
=> ( ( member_set_int @ K2 @ K )
=> ( ( member_set_int @ V @ V3 )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R @ K2 @ V ) @ V3 ) ) ) ) ).
% subalgebra.smult_closed
thf(fact_298_ring_Osubring__props_I2_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ K ) ) ) ).
% ring.subring_props(2)
thf(fact_299_ring_Osubring__props_I7_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,H1: set_int,H2: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( member_set_int @ H1 @ K )
=> ( ( member_set_int @ H2 @ K )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R @ H1 @ H2 ) @ K ) ) ) ) ) ).
% ring.subring_props(7)
thf(fact_300_ring_Osubring__props_I4_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( K != bot_bot_set_set_int ) ) ) ).
% ring.subring_props(4)
thf(fact_301_ring_Osubring__props_I6_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,H1: set_int,H2: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( member_set_int @ H1 @ K )
=> ( ( member_set_int @ H2 @ K )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R @ H1 @ H2 ) @ K ) ) ) ) ) ).
% ring.subring_props(6)
thf(fact_302_ring_Oint__embed__closed,axiom,
! [R: partia4934656038542163276t_unit,K2: int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( member_set_int @ ( ring_i2743490682209504680t_unit @ R @ K2 ) @ ( partia966996272515721803t_unit @ R ) ) ) ).
% ring.int_embed_closed
thf(fact_303_ring_Osubring__props_I3_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ R ) @ K ) ) ) ).
% ring.subring_props(3)
thf(fact_304_ring_Odimension__is__inj,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,N: nat,E: set_set_int,M: nat] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( embedd646006463343340164t_unit @ R @ N @ K @ E )
=> ( ( embedd646006463343340164t_unit @ R @ M @ K @ E )
=> ( N = M ) ) ) ) ) ).
% ring.dimension_is_inj
thf(fact_305_ring_Otelescopic__base__dim_I1_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,F2: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( subfie3888952257595785920t_unit @ F2 @ R )
=> ( ( embedd8246663962306818995t_unit @ R @ K @ F2 )
=> ( ( embedd8246663962306818995t_unit @ R @ F2 @ E )
=> ( embedd8246663962306818995t_unit @ R @ K @ E ) ) ) ) ) ) ).
% ring.telescopic_base_dim(1)
thf(fact_306_ring_Ofinite__dimensionI,axiom,
! [R: partia4934656038542163276t_unit,N: nat,K: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( embedd646006463343340164t_unit @ R @ N @ K @ E )
=> ( embedd8246663962306818995t_unit @ R @ K @ E ) ) ) ).
% ring.finite_dimensionI
thf(fact_307_ring_Ofinite__dimensionE_H,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( embedd8246663962306818995t_unit @ R @ K @ E )
=> ~ ! [N2: nat] :
~ ( embedd646006463343340164t_unit @ R @ N2 @ K @ E ) ) ) ).
% ring.finite_dimensionE'
thf(fact_308_ring_Ofinite__dimension__def,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( embedd8246663962306818995t_unit @ R @ K @ E )
= ( ? [N3: nat] : ( embedd646006463343340164t_unit @ R @ N3 @ K @ E ) ) ) ) ).
% ring.finite_dimension_def
thf(fact_309_ring_Oint__embed__zero,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ring_i2743490682209504680t_unit @ R @ zero_zero_int )
= ( zero_s6269048424454532197t_unit @ R ) ) ) ).
% ring.int_embed_zero
thf(fact_310_ring_Ospace__subgroup__props_I2_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,N: nat,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( embedd646006463343340164t_unit @ R @ N @ K @ E )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ E ) ) ) ) ).
% ring.space_subgroup_props(2)
thf(fact_311_ring_Ospace__subgroup__props_I3_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,N: nat,E: set_set_int,V1: set_int,V2: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( embedd646006463343340164t_unit @ R @ N @ K @ E )
=> ( ( member_set_int @ V1 @ E )
=> ( ( member_set_int @ V2 @ E )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R @ V1 @ V2 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(3)
thf(fact_312_ring_Ospace__subgroup__props_I5_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,N: nat,E: set_set_int,K2: set_int,V: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( embedd646006463343340164t_unit @ R @ N @ K @ E )
=> ( ( member_set_int @ K2 @ K )
=> ( ( member_set_int @ V @ E )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R @ K2 @ V ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(5)
thf(fact_313_ring_Ounique__dimension,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( embedd8246663962306818995t_unit @ R @ K @ E )
=> ? [X: nat] :
( ( embedd646006463343340164t_unit @ R @ X @ K @ E )
& ! [Y3: nat] :
( ( embedd646006463343340164t_unit @ R @ Y3 @ K @ E )
=> ( Y3 = X ) ) ) ) ) ) ).
% ring.unique_dimension
thf(fact_314_ring_Ofinite__dimension__imp__subalgebra,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( embedd8246663962306818995t_unit @ R @ K @ E )
=> ( embedd2743979684206749024t_unit @ K @ E @ R ) ) ) ) ).
% ring.finite_dimension_imp_subalgebra
thf(fact_315_ring_Oembed__char__eq__0,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ring_i2743490682209504680t_unit @ R @ ( semiri1314217659103216013at_int @ ( ring_c6147214092195050492t_unit @ R ) ) )
= ( zero_s6269048424454532197t_unit @ R ) ) ) ).
% ring.embed_char_eq_0
thf(fact_316_ring_Ozero__dim,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( embedd646006463343340164t_unit @ R @ zero_zero_nat @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) ) ) ).
% ring.zero_dim
thf(fact_317_ring_Osubfield__m__inv__simprule,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,K2: set_int,A: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( member_set_int @ K2 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ ( mult_s3864001451298473021t_unit @ R @ K2 @ A ) @ K )
=> ( member_set_int @ A @ K ) ) ) ) ) ) ).
% ring.subfield_m_inv_simprule
thf(fact_318_insert__Diff__single,axiom,
! [A: int,A2: set_int] :
( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( insert_int @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_319_insert__Diff__single,axiom,
! [A: set_set_int,A2: set_set_set_int] :
( ( insert_set_set_int @ A @ ( minus_6857623457997529383et_int @ A2 @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) )
= ( insert_set_set_int @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_320_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_321_insert__Diff__single,axiom,
! [A: nat > set_int,A2: set_nat_set_int] :
( ( insert_nat_set_int @ A @ ( minus_3247115583872269408et_int @ A2 @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ) )
= ( insert_nat_set_int @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_322_insert__Diff__single,axiom,
! [A: set_int,A2: set_set_int] :
( ( insert_set_int @ A @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) )
= ( insert_set_int @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_323_s_Osubfield__m__inv_I3_J,axiom,
! [K: set_set_int,K2: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ K2 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 ) @ K2 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% s.subfield_m_inv(3)
thf(fact_324_s_Osubfield__m__inv_I2_J,axiom,
! [K: set_set_int,K2: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ K2 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 ) )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% s.subfield_m_inv(2)
thf(fact_325_s_OIdl__subset__ideal_H,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 ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) )
= ( member_set_int @ A @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ) ) ).
% s.Idl_subset_ideal'
thf(fact_326_s_Obound__upD,axiom,
! [F: nat > set_int] :
( ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [N2: nat] : ( bound_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ N2 @ F ) ) ).
% s.bound_upD
thf(fact_327_s_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 ) ) ) ).
% s.cgenideal_self
thf(fact_328_s_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 ) ) ) ) ) ).
% s.finite_carr_imp_char_ge_0
thf(fact_329_subset__antisym,axiom,
! [A2: set_nat_set_int,B4: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ B4 )
=> ( ( ord_le5995675665013768039et_int @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% subset_antisym
thf(fact_330_subset__antisym,axiom,
! [A2: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A2 @ B4 )
=> ( ( ord_less_eq_set_int @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% subset_antisym
thf(fact_331_subset__antisym,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ( ord_le4403425263959731960et_int @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% subset_antisym
thf(fact_332_subsetI,axiom,
! [A2: set_int,B4: set_int] :
( ! [X: int] :
( ( member_int @ X @ A2 )
=> ( member_int @ X @ B4 ) )
=> ( ord_less_eq_set_int @ A2 @ B4 ) ) ).
% subsetI
thf(fact_333_subsetI,axiom,
! [A2: set_nat_set_int,B4: set_nat_set_int] :
( ! [X: nat > set_int] :
( ( member_nat_set_int @ X @ A2 )
=> ( member_nat_set_int @ X @ B4 ) )
=> ( ord_le5995675665013768039et_int @ A2 @ B4 ) ) ).
% subsetI
thf(fact_334_subsetI,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ! [X: set_int] :
( ( member_set_int @ X @ A2 )
=> ( member_set_int @ X @ B4 ) )
=> ( ord_le4403425263959731960et_int @ A2 @ B4 ) ) ).
% subsetI
thf(fact_335_psubsetI,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ( A2 != B4 )
=> ( ord_less_set_set_int @ A2 @ B4 ) ) ) ).
% psubsetI
thf(fact_336_empty__iff,axiom,
! [C: nat > set_int] :
~ ( member_nat_set_int @ C @ bot_bo8417611410066262939et_int ) ).
% empty_iff
thf(fact_337_empty__iff,axiom,
! [C: set_int] :
~ ( member_set_int @ C @ bot_bot_set_set_int ) ).
% empty_iff
thf(fact_338_all__not__in__conv,axiom,
! [A2: set_nat_set_int] :
( ( ! [X3: nat > set_int] :
~ ( member_nat_set_int @ X3 @ A2 ) )
= ( A2 = bot_bo8417611410066262939et_int ) ) ).
% all_not_in_conv
thf(fact_339_all__not__in__conv,axiom,
! [A2: set_set_int] :
( ( ! [X3: set_int] :
~ ( member_set_int @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_int ) ) ).
% all_not_in_conv
thf(fact_340_Collect__empty__eq,axiom,
! [P: set_int > $o] :
( ( ( collect_set_int @ P )
= bot_bot_set_set_int )
= ( ! [X3: set_int] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_341_empty__Collect__eq,axiom,
! [P: set_int > $o] :
( ( bot_bot_set_set_int
= ( collect_set_int @ P ) )
= ( ! [X3: set_int] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_342_insertCI,axiom,
! [A: set_int,B4: set_set_int,B: set_int] :
( ( ~ ( member_set_int @ A @ B4 )
=> ( A = B ) )
=> ( member_set_int @ A @ ( insert_set_int @ B @ B4 ) ) ) ).
% insertCI
thf(fact_343_insertCI,axiom,
! [A: nat > set_int,B4: set_nat_set_int,B: nat > set_int] :
( ( ~ ( member_nat_set_int @ A @ B4 )
=> ( A = B ) )
=> ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ B4 ) ) ) ).
% insertCI
thf(fact_344_insert__iff,axiom,
! [A: set_int,B: set_int,A2: set_set_int] :
( ( member_set_int @ A @ ( insert_set_int @ B @ A2 ) )
= ( ( A = B )
| ( member_set_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_345_insert__iff,axiom,
! [A: nat > set_int,B: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_set_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_346_insert__absorb2,axiom,
! [X2: set_int,A2: set_set_int] :
( ( insert_set_int @ X2 @ ( insert_set_int @ X2 @ A2 ) )
= ( insert_set_int @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_347_DiffI,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ C @ A2 )
=> ( ~ ( member_nat_set_int @ C @ B4 )
=> ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B4 ) ) ) ) ).
% DiffI
thf(fact_348_DiffI,axiom,
! [C: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ A2 )
=> ( ~ ( member_set_int @ C @ B4 )
=> ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A2 @ B4 ) ) ) ) ).
% DiffI
thf(fact_349_Diff__iff,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B4 ) )
= ( ( member_nat_set_int @ C @ A2 )
& ~ ( member_nat_set_int @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_350_Diff__iff,axiom,
! [C: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A2 @ B4 ) )
= ( ( member_set_int @ C @ A2 )
& ~ ( member_set_int @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_351_Diff__idemp,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( minus_8897228262479074673et_int @ ( minus_8897228262479074673et_int @ A2 @ B4 ) @ B4 )
= ( minus_8897228262479074673et_int @ A2 @ B4 ) ) ).
% Diff_idemp
thf(fact_352_s_Osubring__props_I1_J,axiom,
! [K: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.subring_props(1)
thf(fact_353_s_Osubalgebra__in__carrier,axiom,
! [K: set_set_int,V3: set_set_int] :
( ( embedd2743979684206749024t_unit @ K @ V3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ord_le4403425263959731960et_int @ V3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.subalgebra_in_carrier
thf(fact_354_s_Ocarrier__is__subalgebra,axiom,
! [K: set_set_int] :
( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( embedd2743979684206749024t_unit @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% s.carrier_is_subalgebra
thf(fact_355_s_Osubset__Idl__subset,axiom,
! [I4: set_set_int,H: set_set_int] :
( ( ord_le4403425263959731960et_int @ I4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ H @ I4 )
=> ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H ) @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I4 ) ) ) ) ).
% s.subset_Idl_subset
thf(fact_356_s_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 ) ) ) ).
% s.genideal_self
thf(fact_357_s_Ospace__subgroup__props_I1_J,axiom,
! [K: set_set_int,N: nat,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% s.space_subgroup_props(1)
thf(fact_358_s_Oinv__unique_H,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( Y
= ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) ) ) ) ) ) ).
% s.inv_unique'
thf(fact_359_s_Oinv__char,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 )
= Y ) ) ) ) ) ).
% s.inv_char
thf(fact_360_s_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K: set_set_int,E: set_set_int,V3: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ E )
=> ( ( embedd2743979684206749024t_unit @ K @ V3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ord_le4403425263959731960et_int @ V3 @ E )
=> ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ V3 ) ) ) ) ) ).
% s.subalbegra_incl_imp_finite_dimension
thf(fact_361_empty__subsetI,axiom,
! [A2: set_set_int] : ( ord_le4403425263959731960et_int @ bot_bot_set_set_int @ A2 ) ).
% empty_subsetI
thf(fact_362_subset__empty,axiom,
! [A2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ bot_bot_set_set_int )
= ( A2 = bot_bot_set_set_int ) ) ).
% subset_empty
thf(fact_363_insert__subset,axiom,
! [X2: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ ( insert_nat_set_int @ X2 @ A2 ) @ B4 )
= ( ( member_nat_set_int @ X2 @ B4 )
& ( ord_le5995675665013768039et_int @ A2 @ B4 ) ) ) ).
% insert_subset
thf(fact_364_insert__subset,axiom,
! [X2: set_int,A2: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( insert_set_int @ X2 @ A2 ) @ B4 )
= ( ( member_set_int @ X2 @ B4 )
& ( ord_le4403425263959731960et_int @ A2 @ B4 ) ) ) ).
% insert_subset
thf(fact_365_singletonI,axiom,
! [A: nat > set_int] : ( member_nat_set_int @ A @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ) ).
% singletonI
thf(fact_366_singletonI,axiom,
! [A: set_int] : ( member_set_int @ A @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ).
% singletonI
thf(fact_367_Diff__cancel,axiom,
! [A2: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ A2 )
= bot_bot_set_set_int ) ).
% Diff_cancel
thf(fact_368_empty__Diff,axiom,
! [A2: set_set_int] :
( ( minus_8897228262479074673et_int @ bot_bot_set_set_int @ A2 )
= bot_bot_set_set_int ) ).
% empty_Diff
thf(fact_369_Diff__empty,axiom,
! [A2: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ bot_bot_set_set_int )
= A2 ) ).
% Diff_empty
thf(fact_370_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_371_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_372_insert__Diff1,axiom,
! [X2: nat > set_int,B4: set_nat_set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ X2 @ B4 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X2 @ A2 ) @ B4 )
= ( minus_3247115583872269408et_int @ A2 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_373_insert__Diff1,axiom,
! [X2: set_int,B4: set_set_int,A2: set_set_int] :
( ( member_set_int @ X2 @ B4 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X2 @ A2 ) @ B4 )
= ( minus_8897228262479074673et_int @ A2 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_374_Diff__insert0,axiom,
! [X2: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ~ ( member_nat_set_int @ X2 @ A2 )
=> ( ( minus_3247115583872269408et_int @ A2 @ ( insert_nat_set_int @ X2 @ B4 ) )
= ( minus_3247115583872269408et_int @ A2 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_375_Diff__insert0,axiom,
! [X2: set_int,A2: set_set_int,B4: set_set_int] :
( ~ ( member_set_int @ X2 @ A2 )
=> ( ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ X2 @ B4 ) )
= ( minus_8897228262479074673et_int @ A2 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_376_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_377_s_Osubfield__m__inv_I1_J,axiom,
! [K: set_set_int,K2: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ K2 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
=> ( member_set_int @ ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 ) @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) ) ) ) ).
% s.subfield_m_inv(1)
thf(fact_378_singleton__insert__inj__eq_H,axiom,
! [A: set_int,A2: set_set_int,B: set_int] :
( ( ( insert_set_int @ A @ A2 )
= ( insert_set_int @ B @ bot_bot_set_set_int ) )
= ( ( A = B )
& ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_379_singleton__insert__inj__eq,axiom,
! [B: set_int,A: set_int,A2: set_set_int] :
( ( ( insert_set_int @ B @ bot_bot_set_set_int )
= ( insert_set_int @ A @ A2 ) )
= ( ( A = B )
& ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_380_Diff__eq__empty__iff,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ( minus_8897228262479074673et_int @ A2 @ B4 )
= bot_bot_set_set_int )
= ( ord_le4403425263959731960et_int @ A2 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_381_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_382_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_383_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_384_s_Oinv__one,axiom,
( ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% s.inv_one
thf(fact_385_s_Ofinsum__infinite,axiom,
! [A2: set_nat_set_int,F: ( nat > set_int ) > set_int] :
( ~ ( finite7455725759970522984et_int @ A2 )
=> ( ( finsum1765041520989711135et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.finsum_infinite
thf(fact_386_s_Ofinsum__infinite,axiom,
! [A2: set_set_int,F: set_int > set_int] :
( ~ ( finite6197958912794628473et_int @ A2 )
=> ( ( finsum2830217253030306736et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.finsum_infinite
thf(fact_387_insert__mono,axiom,
! [C3: set_set_int,D2: set_set_int,A: set_int] :
( ( ord_le4403425263959731960et_int @ C3 @ D2 )
=> ( ord_le4403425263959731960et_int @ ( insert_set_int @ A @ C3 ) @ ( insert_set_int @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_388_subset__insert,axiom,
! [X2: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ~ ( member_nat_set_int @ X2 @ A2 )
=> ( ( ord_le5995675665013768039et_int @ A2 @ ( insert_nat_set_int @ X2 @ B4 ) )
= ( ord_le5995675665013768039et_int @ A2 @ B4 ) ) ) ).
% subset_insert
thf(fact_389_subset__insert,axiom,
! [X2: set_int,A2: set_set_int,B4: set_set_int] :
( ~ ( member_set_int @ X2 @ A2 )
=> ( ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X2 @ B4 ) )
= ( ord_le4403425263959731960et_int @ A2 @ B4 ) ) ) ).
% subset_insert
thf(fact_390_subset__insertI,axiom,
! [B4: set_set_int,A: set_int] : ( ord_le4403425263959731960et_int @ B4 @ ( insert_set_int @ A @ B4 ) ) ).
% subset_insertI
thf(fact_391_subset__insertI2,axiom,
! [A2: set_set_int,B4: set_set_int,B: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_392_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_393_Collect__mono__iff,axiom,
! [P: set_int > $o,Q: set_int > $o] :
( ( ord_le4403425263959731960et_int @ ( collect_set_int @ P ) @ ( collect_set_int @ Q ) )
= ( ! [X3: set_int] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_394_set__eq__subset,axiom,
( ( ^ [Y6: set_set_int,Z3: set_set_int] : ( Y6 = Z3 ) )
= ( ^ [A5: set_set_int,B5: set_set_int] :
( ( ord_le4403425263959731960et_int @ A5 @ B5 )
& ( ord_le4403425263959731960et_int @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_395_subset__trans,axiom,
! [A2: set_set_int,B4: set_set_int,C3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ( ord_le4403425263959731960et_int @ B4 @ C3 )
=> ( ord_le4403425263959731960et_int @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_396_Collect__mono,axiom,
! [P: set_int > $o,Q: set_int > $o] :
( ! [X: set_int] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le4403425263959731960et_int @ ( collect_set_int @ P ) @ ( collect_set_int @ Q ) ) ) ).
% Collect_mono
thf(fact_397_subset__refl,axiom,
! [A2: set_set_int] : ( ord_le4403425263959731960et_int @ A2 @ A2 ) ).
% subset_refl
thf(fact_398_double__diff,axiom,
! [A2: set_set_int,B4: set_set_int,C3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ( ord_le4403425263959731960et_int @ B4 @ C3 )
=> ( ( minus_8897228262479074673et_int @ B4 @ ( minus_8897228262479074673et_int @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_399_Diff__subset,axiom,
! [A2: set_set_int,B4: set_set_int] : ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A2 @ B4 ) @ A2 ) ).
% Diff_subset
thf(fact_400_subset__iff,axiom,
( ord_le5995675665013768039et_int
= ( ^ [A5: set_nat_set_int,B5: set_nat_set_int] :
! [T2: nat > set_int] :
( ( member_nat_set_int @ T2 @ A5 )
=> ( member_nat_set_int @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_401_subset__iff,axiom,
( ord_le4403425263959731960et_int
= ( ^ [A5: set_set_int,B5: set_set_int] :
! [T2: set_int] :
( ( member_set_int @ T2 @ A5 )
=> ( member_set_int @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_402_equalityD2,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( A2 = B4 )
=> ( ord_le4403425263959731960et_int @ B4 @ A2 ) ) ).
% equalityD2
thf(fact_403_equalityD1,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( A2 = B4 )
=> ( ord_le4403425263959731960et_int @ A2 @ B4 ) ) ).
% equalityD1
thf(fact_404_subset__eq,axiom,
( ord_le5995675665013768039et_int
= ( ^ [A5: set_nat_set_int,B5: set_nat_set_int] :
! [X3: nat > set_int] :
( ( member_nat_set_int @ X3 @ A5 )
=> ( member_nat_set_int @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_405_subset__eq,axiom,
( ord_le4403425263959731960et_int
= ( ^ [A5: set_set_int,B5: set_set_int] :
! [X3: set_int] :
( ( member_set_int @ X3 @ A5 )
=> ( member_set_int @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_406_equalityE,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( A2 = B4 )
=> ~ ( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ~ ( ord_le4403425263959731960et_int @ B4 @ A2 ) ) ) ).
% equalityE
thf(fact_407_Diff__mono,axiom,
! [A2: set_set_int,C3: set_set_int,D2: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ C3 )
=> ( ( ord_le4403425263959731960et_int @ D2 @ B4 )
=> ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A2 @ B4 ) @ ( minus_8897228262479074673et_int @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_408_subsetD,axiom,
! [A2: set_nat_set_int,B4: set_nat_set_int,C: nat > set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ B4 )
=> ( ( member_nat_set_int @ C @ A2 )
=> ( member_nat_set_int @ C @ B4 ) ) ) ).
% subsetD
thf(fact_409_subsetD,axiom,
! [A2: set_set_int,B4: set_set_int,C: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ( member_set_int @ C @ A2 )
=> ( member_set_int @ C @ B4 ) ) ) ).
% subsetD
thf(fact_410_in__mono,axiom,
! [A2: set_nat_set_int,B4: set_nat_set_int,X2: nat > set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ B4 )
=> ( ( member_nat_set_int @ X2 @ A2 )
=> ( member_nat_set_int @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_411_in__mono,axiom,
! [A2: set_set_int,B4: set_set_int,X2: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ( member_set_int @ X2 @ A2 )
=> ( member_set_int @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_412_subset__iff__psubset__eq,axiom,
( ord_le4403425263959731960et_int
= ( ^ [A5: set_set_int,B5: set_set_int] :
( ( ord_less_set_set_int @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_413_subset__psubset__trans,axiom,
! [A2: set_set_int,B4: set_set_int,C3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ( ord_less_set_set_int @ B4 @ C3 )
=> ( ord_less_set_set_int @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_414_subset__not__subset__eq,axiom,
( ord_less_set_set_int
= ( ^ [A5: set_set_int,B5: set_set_int] :
( ( ord_le4403425263959731960et_int @ A5 @ B5 )
& ~ ( ord_le4403425263959731960et_int @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_415_psubset__subset__trans,axiom,
! [A2: set_set_int,B4: set_set_int,C3: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B4 )
=> ( ( ord_le4403425263959731960et_int @ B4 @ C3 )
=> ( ord_less_set_set_int @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_416_psubset__imp__subset,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B4 )
=> ( ord_le4403425263959731960et_int @ A2 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_417_psubset__eq,axiom,
( ord_less_set_set_int
= ( ^ [A5: set_set_int,B5: set_set_int] :
( ( ord_le4403425263959731960et_int @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_418_psubsetE,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B4 )
=> ~ ( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ord_le4403425263959731960et_int @ B4 @ A2 ) ) ) ).
% psubsetE
thf(fact_419_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_420_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_421_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_422_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_423_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_424_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_425_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_426_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_427_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_428_le__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_429_eq__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_430_subset__singleton__iff,axiom,
! [X4: set_set_int,A: set_int] :
( ( ord_le4403425263959731960et_int @ X4 @ ( insert_set_int @ A @ bot_bot_set_set_int ) )
= ( ( X4 = bot_bot_set_set_int )
| ( X4
= ( insert_set_int @ A @ bot_bot_set_set_int ) ) ) ) ).
% subset_singleton_iff
thf(fact_431_subset__singletonD,axiom,
! [A2: set_set_int,X2: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X2 @ bot_bot_set_set_int ) )
=> ( ( A2 = bot_bot_set_set_int )
| ( A2
= ( insert_set_int @ X2 @ bot_bot_set_set_int ) ) ) ) ).
% subset_singletonD
thf(fact_432_subfieldE_I3_J,axiom,
! [K: set_set_int,R: partia4934656038542163276t_unit] :
( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ R ) ) ) ).
% subfieldE(3)
thf(fact_433_subset__Diff__insert,axiom,
! [A2: set_nat_set_int,B4: set_nat_set_int,X2: nat > set_int,C3: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ ( minus_3247115583872269408et_int @ B4 @ ( insert_nat_set_int @ X2 @ C3 ) ) )
= ( ( ord_le5995675665013768039et_int @ A2 @ ( minus_3247115583872269408et_int @ B4 @ C3 ) )
& ~ ( member_nat_set_int @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_434_subset__Diff__insert,axiom,
! [A2: set_set_int,B4: set_set_int,X2: set_int,C3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( minus_8897228262479074673et_int @ B4 @ ( insert_set_int @ X2 @ C3 ) ) )
= ( ( ord_le4403425263959731960et_int @ A2 @ ( minus_8897228262479074673et_int @ B4 @ C3 ) )
& ~ ( member_set_int @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_435_subset__insert__iff,axiom,
! [A2: set_nat_set_int,X2: nat > set_int,B4: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ ( insert_nat_set_int @ X2 @ B4 ) )
= ( ( ( member_nat_set_int @ X2 @ A2 )
=> ( ord_le5995675665013768039et_int @ ( minus_3247115583872269408et_int @ A2 @ ( insert_nat_set_int @ X2 @ bot_bo8417611410066262939et_int ) ) @ B4 ) )
& ( ~ ( member_nat_set_int @ X2 @ A2 )
=> ( ord_le5995675665013768039et_int @ A2 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_436_subset__insert__iff,axiom,
! [A2: set_set_int,X2: set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X2 @ B4 ) )
= ( ( ( member_set_int @ X2 @ A2 )
=> ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ X2 @ bot_bot_set_set_int ) ) @ B4 ) )
& ( ~ ( member_set_int @ X2 @ A2 )
=> ( ord_le4403425263959731960et_int @ A2 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_437_Diff__single__insert,axiom,
! [A2: set_set_int,X2: set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ X2 @ bot_bot_set_set_int ) ) @ B4 )
=> ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X2 @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_438_psubset__insert__iff,axiom,
! [A2: set_nat_set_int,X2: nat > set_int,B4: set_nat_set_int] :
( ( ord_le2931775347370382171et_int @ A2 @ ( insert_nat_set_int @ X2 @ B4 ) )
= ( ( ( member_nat_set_int @ X2 @ B4 )
=> ( ord_le2931775347370382171et_int @ A2 @ B4 ) )
& ( ~ ( member_nat_set_int @ X2 @ B4 )
=> ( ( ( member_nat_set_int @ X2 @ A2 )
=> ( ord_le2931775347370382171et_int @ ( minus_3247115583872269408et_int @ A2 @ ( insert_nat_set_int @ X2 @ bot_bo8417611410066262939et_int ) ) @ B4 ) )
& ( ~ ( member_nat_set_int @ X2 @ A2 )
=> ( ord_le5995675665013768039et_int @ A2 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_439_psubset__insert__iff,axiom,
! [A2: set_set_int,X2: set_int,B4: set_set_int] :
( ( ord_less_set_set_int @ A2 @ ( insert_set_int @ X2 @ B4 ) )
= ( ( ( member_set_int @ X2 @ B4 )
=> ( ord_less_set_set_int @ A2 @ B4 ) )
& ( ~ ( member_set_int @ X2 @ B4 )
=> ( ( ( member_set_int @ X2 @ A2 )
=> ( ord_less_set_set_int @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ X2 @ bot_bot_set_set_int ) ) @ B4 ) )
& ( ~ ( member_set_int @ X2 @ A2 )
=> ( ord_le4403425263959731960et_int @ A2 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_440_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_441_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_442_less__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_443_emptyE,axiom,
! [A: nat > set_int] :
~ ( member_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ).
% emptyE
thf(fact_444_emptyE,axiom,
! [A: set_int] :
~ ( member_set_int @ A @ bot_bot_set_set_int ) ).
% emptyE
thf(fact_445_equals0D,axiom,
! [A2: set_nat_set_int,A: nat > set_int] :
( ( A2 = bot_bo8417611410066262939et_int )
=> ~ ( member_nat_set_int @ A @ A2 ) ) ).
% equals0D
thf(fact_446_equals0D,axiom,
! [A2: set_set_int,A: set_int] :
( ( A2 = bot_bot_set_set_int )
=> ~ ( member_set_int @ A @ A2 ) ) ).
% equals0D
thf(fact_447_equals0I,axiom,
! [A2: set_nat_set_int] :
( ! [Y4: nat > set_int] :
~ ( member_nat_set_int @ Y4 @ A2 )
=> ( A2 = bot_bo8417611410066262939et_int ) ) ).
% equals0I
thf(fact_448_equals0I,axiom,
! [A2: set_set_int] :
( ! [Y4: set_int] :
~ ( member_set_int @ Y4 @ A2 )
=> ( A2 = bot_bot_set_set_int ) ) ).
% equals0I
thf(fact_449_ex__in__conv,axiom,
! [A2: set_nat_set_int] :
( ( ? [X3: nat > set_int] : ( member_nat_set_int @ X3 @ A2 ) )
= ( A2 != bot_bo8417611410066262939et_int ) ) ).
% ex_in_conv
thf(fact_450_ex__in__conv,axiom,
! [A2: set_set_int] :
( ( ? [X3: set_int] : ( member_set_int @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_int ) ) ).
% ex_in_conv
thf(fact_451_not__psubset__empty,axiom,
! [A2: set_set_int] :
~ ( ord_less_set_set_int @ A2 @ bot_bot_set_set_int ) ).
% not_psubset_empty
thf(fact_452_ring_Osubring__props_I1_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ R ) ) ) ) ).
% ring.subring_props(1)
thf(fact_453_insertE,axiom,
! [A: set_int,B: set_int,A2: set_set_int] :
( ( member_set_int @ A @ ( insert_set_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_454_insertE,axiom,
! [A: nat > set_int,B: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat_set_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_455_insertI1,axiom,
! [A: set_int,B4: set_set_int] : ( member_set_int @ A @ ( insert_set_int @ A @ B4 ) ) ).
% insertI1
thf(fact_456_insertI1,axiom,
! [A: nat > set_int,B4: set_nat_set_int] : ( member_nat_set_int @ A @ ( insert_nat_set_int @ A @ B4 ) ) ).
% insertI1
thf(fact_457_insertI2,axiom,
! [A: set_int,B4: set_set_int,B: set_int] :
( ( member_set_int @ A @ B4 )
=> ( member_set_int @ A @ ( insert_set_int @ B @ B4 ) ) ) ).
% insertI2
thf(fact_458_insertI2,axiom,
! [A: nat > set_int,B4: set_nat_set_int,B: nat > set_int] :
( ( member_nat_set_int @ A @ B4 )
=> ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ B4 ) ) ) ).
% insertI2
thf(fact_459_Set_Oset__insert,axiom,
! [X2: set_int,A2: set_set_int] :
( ( member_set_int @ X2 @ A2 )
=> ~ ! [B6: set_set_int] :
( ( A2
= ( insert_set_int @ X2 @ B6 ) )
=> ( member_set_int @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_460_Set_Oset__insert,axiom,
! [X2: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ X2 @ A2 )
=> ~ ! [B6: set_nat_set_int] :
( ( A2
= ( insert_nat_set_int @ X2 @ B6 ) )
=> ( member_nat_set_int @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_461_insert__ident,axiom,
! [X2: set_int,A2: set_set_int,B4: set_set_int] :
( ~ ( member_set_int @ X2 @ A2 )
=> ( ~ ( member_set_int @ X2 @ B4 )
=> ( ( ( insert_set_int @ X2 @ A2 )
= ( insert_set_int @ X2 @ B4 ) )
= ( A2 = B4 ) ) ) ) ).
% insert_ident
thf(fact_462_insert__ident,axiom,
! [X2: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ~ ( member_nat_set_int @ X2 @ A2 )
=> ( ~ ( member_nat_set_int @ X2 @ B4 )
=> ( ( ( insert_nat_set_int @ X2 @ A2 )
= ( insert_nat_set_int @ X2 @ B4 ) )
= ( A2 = B4 ) ) ) ) ).
% insert_ident
thf(fact_463_insert__absorb,axiom,
! [A: set_int,A2: set_set_int] :
( ( member_set_int @ A @ A2 )
=> ( ( insert_set_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_464_insert__absorb,axiom,
! [A: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ A2 )
=> ( ( insert_nat_set_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_465_insert__eq__iff,axiom,
! [A: set_int,A2: set_set_int,B: set_int,B4: set_set_int] :
( ~ ( member_set_int @ A @ A2 )
=> ( ~ ( member_set_int @ B @ B4 )
=> ( ( ( insert_set_int @ A @ A2 )
= ( insert_set_int @ B @ B4 ) )
= ( ( ( A = B )
=> ( A2 = B4 ) )
& ( ( A != B )
=> ? [C4: set_set_int] :
( ( A2
= ( insert_set_int @ B @ C4 ) )
& ~ ( member_set_int @ B @ C4 )
& ( B4
= ( insert_set_int @ A @ C4 ) )
& ~ ( member_set_int @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_466_insert__eq__iff,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B: nat > set_int,B4: set_nat_set_int] :
( ~ ( member_nat_set_int @ A @ A2 )
=> ( ~ ( member_nat_set_int @ B @ B4 )
=> ( ( ( insert_nat_set_int @ A @ A2 )
= ( insert_nat_set_int @ B @ B4 ) )
= ( ( ( A = B )
=> ( A2 = B4 ) )
& ( ( A != B )
=> ? [C4: set_nat_set_int] :
( ( A2
= ( insert_nat_set_int @ B @ C4 ) )
& ~ ( member_nat_set_int @ B @ C4 )
& ( B4
= ( insert_nat_set_int @ A @ C4 ) )
& ~ ( member_nat_set_int @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_467_insert__commute,axiom,
! [X2: set_int,Y: set_int,A2: set_set_int] :
( ( insert_set_int @ X2 @ ( insert_set_int @ Y @ A2 ) )
= ( insert_set_int @ Y @ ( insert_set_int @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_468_mk__disjoint__insert,axiom,
! [A: set_int,A2: set_set_int] :
( ( member_set_int @ A @ A2 )
=> ? [B6: set_set_int] :
( ( A2
= ( insert_set_int @ A @ B6 ) )
& ~ ( member_set_int @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_469_mk__disjoint__insert,axiom,
! [A: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ A2 )
=> ? [B6: set_nat_set_int] :
( ( A2
= ( insert_nat_set_int @ A @ B6 ) )
& ~ ( member_nat_set_int @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_470_abelian__monoid_Ofinsum__infinite,axiom,
! [G: partia4934656038542163276t_unit,A2: set_nat_set_int,F: ( nat > set_int ) > set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ~ ( finite7455725759970522984et_int @ A2 )
=> ( ( finsum1765041520989711135et_int @ G @ F @ A2 )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ) ).
% abelian_monoid.finsum_infinite
thf(fact_471_abelian__monoid_Ofinsum__infinite,axiom,
! [G: partia4934656038542163276t_unit,A2: set_set_int,F: set_int > set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ~ ( finite6197958912794628473et_int @ A2 )
=> ( ( finsum2830217253030306736et_int @ G @ F @ A2 )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ) ).
% abelian_monoid.finsum_infinite
thf(fact_472_ring_Osubalgebra__in__carrier,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,V3: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( embedd2743979684206749024t_unit @ K @ V3 @ R )
=> ( ord_le4403425263959731960et_int @ V3 @ ( partia966996272515721803t_unit @ R ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_473_ring_Ocarrier__is__subalgebra,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ R ) )
=> ( embedd2743979684206749024t_unit @ K @ ( partia966996272515721803t_unit @ R ) @ R ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_474_psubset__imp__ex__mem,axiom,
! [A2: set_nat_set_int,B4: set_nat_set_int] :
( ( ord_le2931775347370382171et_int @ A2 @ B4 )
=> ? [B2: nat > set_int] : ( member_nat_set_int @ B2 @ ( minus_3247115583872269408et_int @ B4 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_475_psubset__imp__ex__mem,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B4 )
=> ? [B2: set_int] : ( member_set_int @ B2 @ ( minus_8897228262479074673et_int @ B4 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_476_DiffE,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B4 ) )
=> ~ ( ( member_nat_set_int @ C @ A2 )
=> ( member_nat_set_int @ C @ B4 ) ) ) ).
% DiffE
thf(fact_477_DiffE,axiom,
! [C: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A2 @ B4 ) )
=> ~ ( ( member_set_int @ C @ A2 )
=> ( member_set_int @ C @ B4 ) ) ) ).
% DiffE
thf(fact_478_DiffD1,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B4 ) )
=> ( member_nat_set_int @ C @ A2 ) ) ).
% DiffD1
thf(fact_479_DiffD1,axiom,
! [C: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A2 @ B4 ) )
=> ( member_set_int @ C @ A2 ) ) ).
% DiffD1
thf(fact_480_DiffD2,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B4 ) )
=> ~ ( member_nat_set_int @ C @ B4 ) ) ).
% DiffD2
thf(fact_481_DiffD2,axiom,
! [C: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A2 @ B4 ) )
=> ~ ( member_set_int @ C @ B4 ) ) ).
% DiffD2
thf(fact_482_ring_Osubfield__m__inv_I1_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,K2: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( member_set_int @ K2 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) ) )
=> ( member_set_int @ ( m_inv_4894562657074299959t_unit @ R @ K2 ) @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) ) ) ) ) ) ).
% ring.subfield_m_inv(1)
thf(fact_483_ring_Ospace__subgroup__props_I1_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,N: nat,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( embedd646006463343340164t_unit @ R @ N @ K @ E )
=> ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ R ) ) ) ) ) ).
% ring.space_subgroup_props(1)
thf(fact_484_ring_Osubalbegra__incl__imp__finite__dimension,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,E: set_set_int,V3: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( embedd8246663962306818995t_unit @ R @ K @ E )
=> ( ( embedd2743979684206749024t_unit @ K @ V3 @ R )
=> ( ( ord_le4403425263959731960et_int @ V3 @ E )
=> ( embedd8246663962306818995t_unit @ R @ K @ V3 ) ) ) ) ) ) ).
% ring.subalbegra_incl_imp_finite_dimension
thf(fact_485_fin__zfact,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% fin_zfact
thf(fact_486_ring_Ofinite__carr__imp__char__ge__0,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ R ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c6147214092195050492t_unit @ R ) ) ) ) ).
% ring.finite_carr_imp_char_ge_0
thf(fact_487_ring_Osubfield__m__inv_I2_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,K2: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( member_set_int @ K2 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ K2 @ ( m_inv_4894562657074299959t_unit @ R @ K2 ) )
= ( one_se8065767436706823081t_unit @ R ) ) ) ) ) ).
% ring.subfield_m_inv(2)
thf(fact_488_ring_Osubfield__m__inv_I3_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,K2: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( member_set_int @ K2 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ ( m_inv_4894562657074299959t_unit @ R @ K2 ) @ K2 )
= ( one_se8065767436706823081t_unit @ R ) ) ) ) ) ).
% ring.subfield_m_inv(3)
thf(fact_489_singletonD,axiom,
! [B: nat > set_int,A: nat > set_int] :
( ( member_nat_set_int @ B @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_490_singletonD,axiom,
! [B: set_int,A: set_int] :
( ( member_set_int @ B @ ( insert_set_int @ A @ bot_bot_set_set_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_491_singleton__iff,axiom,
! [B: nat > set_int,A: nat > set_int] :
( ( member_nat_set_int @ B @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_492_singleton__iff,axiom,
! [B: set_int,A: set_int] :
( ( member_set_int @ B @ ( insert_set_int @ A @ bot_bot_set_set_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_493_doubleton__eq__iff,axiom,
! [A: set_int,B: set_int,C: set_int,D: set_int] :
( ( ( insert_set_int @ A @ ( insert_set_int @ B @ bot_bot_set_set_int ) )
= ( insert_set_int @ C @ ( insert_set_int @ D @ bot_bot_set_set_int ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_494_insert__not__empty,axiom,
! [A: set_int,A2: set_set_int] :
( ( insert_set_int @ A @ A2 )
!= bot_bot_set_set_int ) ).
% insert_not_empty
thf(fact_495_singleton__inject,axiom,
! [A: set_int,B: set_int] :
( ( ( insert_set_int @ A @ bot_bot_set_set_int )
= ( insert_set_int @ B @ bot_bot_set_set_int ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_496_insert__Diff__if,axiom,
! [X2: nat > set_int,B4: set_nat_set_int,A2: set_nat_set_int] :
( ( ( member_nat_set_int @ X2 @ B4 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X2 @ A2 ) @ B4 )
= ( minus_3247115583872269408et_int @ A2 @ B4 ) ) )
& ( ~ ( member_nat_set_int @ X2 @ B4 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X2 @ A2 ) @ B4 )
= ( insert_nat_set_int @ X2 @ ( minus_3247115583872269408et_int @ A2 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_497_insert__Diff__if,axiom,
! [X2: set_int,B4: set_set_int,A2: set_set_int] :
( ( ( member_set_int @ X2 @ B4 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X2 @ A2 ) @ B4 )
= ( minus_8897228262479074673et_int @ A2 @ B4 ) ) )
& ( ~ ( member_set_int @ X2 @ B4 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X2 @ A2 ) @ B4 )
= ( insert_set_int @ X2 @ ( minus_8897228262479074673et_int @ A2 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_498_subfieldE_I4_J,axiom,
! [K: set_set_int,R: partia4934656038542163276t_unit,K1: set_int,K22: set_int] :
( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( member_set_int @ K1 @ K )
=> ( ( member_set_int @ K22 @ K )
=> ( ( mult_s3864001451298473021t_unit @ R @ K1 @ K22 )
= ( mult_s3864001451298473021t_unit @ R @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_499_Diff__insert__absorb,axiom,
! [X2: nat > set_int,A2: set_nat_set_int] :
( ~ ( member_nat_set_int @ X2 @ A2 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X2 @ A2 ) @ ( insert_nat_set_int @ X2 @ bot_bo8417611410066262939et_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_500_Diff__insert__absorb,axiom,
! [X2: set_int,A2: set_set_int] :
( ~ ( member_set_int @ X2 @ A2 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X2 @ A2 ) @ ( insert_set_int @ X2 @ bot_bot_set_set_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_501_Diff__insert2,axiom,
! [A2: set_set_int,A: set_int,B4: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ B4 ) )
= ( minus_8897228262479074673et_int @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_502_insert__Diff,axiom,
! [A: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ A2 )
=> ( ( insert_nat_set_int @ A @ ( minus_3247115583872269408et_int @ A2 @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_503_insert__Diff,axiom,
! [A: set_int,A2: set_set_int] :
( ( member_set_int @ A @ A2 )
=> ( ( insert_set_int @ A @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_504_Diff__insert,axiom,
! [A2: set_set_int,A: set_int,B4: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ B4 ) )
= ( minus_8897228262479074673et_int @ ( minus_8897228262479074673et_int @ A2 @ B4 ) @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ) ).
% Diff_insert
thf(fact_505_subfieldE_I5_J,axiom,
! [K: set_set_int,R: partia4934656038542163276t_unit,K1: set_int,K22: set_int] :
( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( member_set_int @ K1 @ K )
=> ( ( member_set_int @ K22 @ K )
=> ( ( ( mult_s3864001451298473021t_unit @ R @ K1 @ K22 )
= ( zero_s6269048424454532197t_unit @ R ) )
=> ( ( K1
= ( zero_s6269048424454532197t_unit @ R ) )
| ( K22
= ( zero_s6269048424454532197t_unit @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_506_subfieldE_I6_J,axiom,
! [K: set_set_int,R: partia4934656038542163276t_unit] :
( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( one_se8065767436706823081t_unit @ R )
!= ( zero_s6269048424454532197t_unit @ R ) ) ) ).
% subfieldE(6)
thf(fact_507_s_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 ) ) ) ) ) ).
% s.order_gt_0_iff_finite
thf(fact_508_mem__upI,axiom,
! [F: nat > set_int,R: partia4934656038542163276t_unit] :
( ! [N2: nat] : ( member_set_int @ ( F @ N2 ) @ ( partia966996272515721803t_unit @ R ) )
=> ( ? [N4: nat] : ( bound_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ N4 @ F )
=> ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_509_s_Oint__embed__diff,axiom,
! [X2: int,Y: int] :
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( minus_minus_int @ X2 @ Y ) )
= ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ).
% s.int_embed_diff
thf(fact_510_s_Oline__extension__smult__closed,axiom,
! [K: set_set_int,E: set_set_int,A: set_int,K2: set_int,U: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ! [K3: set_int,V4: set_int] :
( ( member_set_int @ K3 @ K )
=> ( ( member_set_int @ V4 @ E )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K3 @ V4 ) @ E ) ) )
=> ( ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ K2 @ K )
=> ( ( member_set_int @ U @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ A @ E ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ U ) @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ A @ E ) ) ) ) ) ) ) ) ).
% s.line_extension_smult_closed
thf(fact_511_finite__Diff__insert,axiom,
! [A2: set_set_int,A: set_int,B4: set_set_int] :
( ( finite6197958912794628473et_int @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ B4 ) ) )
= ( finite6197958912794628473et_int @ ( minus_8897228262479074673et_int @ A2 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_512_s_Oa__lcos__m__assoc,axiom,
! [M4: set_set_int,G3: set_int,H3: set_int] :
( ( ord_le4403425263959731960et_int @ M4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ G3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ H3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ G3 @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H3 @ M4 ) )
= ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ G3 @ H3 ) @ M4 ) ) ) ) ) ).
% s.a_lcos_m_assoc
thf(fact_513_s_Oline__extension__in__carrier,axiom,
! [K: set_set_int,A: set_int,E: set_set_int] :
( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ A @ E ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% s.line_extension_in_carrier
thf(fact_514_s_Oa__l__coset__subset__G,axiom,
! [H: set_set_int,X2: set_int] :
( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ H ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% s.a_l_coset_subset_G
thf(fact_515_s_Oline__extension__mem__iff,axiom,
! [U: set_int,K: set_set_int,A: set_int,E: set_set_int] :
( ( member_set_int @ U @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ A @ E ) )
= ( ? [X3: set_int] :
( ( member_set_int @ X3 @ K )
& ? [Y5: set_int] :
( ( member_set_int @ Y5 @ E )
& ( U
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X3 @ A ) @ Y5 ) ) ) ) ) ) ).
% s.line_extension_mem_iff
thf(fact_516_s_Oa__lcos__mult__one,axiom,
! [M4: set_set_int] :
( ( ord_le4403425263959731960et_int @ M4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ M4 )
= M4 ) ) ).
% s.a_lcos_mult_one
thf(fact_517_finite__insert,axiom,
! [A: set_int,A2: set_set_int] :
( ( finite6197958912794628473et_int @ ( insert_set_int @ A @ A2 ) )
= ( finite6197958912794628473et_int @ A2 ) ) ).
% finite_insert
thf(fact_518_finite__Diff,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( finite6197958912794628473et_int @ A2 )
=> ( finite6197958912794628473et_int @ ( minus_8897228262479074673et_int @ A2 @ B4 ) ) ) ).
% finite_Diff
thf(fact_519_finite__Diff2,axiom,
! [B4: set_set_int,A2: set_set_int] :
( ( finite6197958912794628473et_int @ B4 )
=> ( ( finite6197958912794628473et_int @ ( minus_8897228262479074673et_int @ A2 @ B4 ) )
= ( finite6197958912794628473et_int @ A2 ) ) ) ).
% finite_Diff2
thf(fact_520_bound_Ointro,axiom,
! [N: nat,F: nat > set_int,Z: set_int] :
( ! [M5: nat] :
( ( ord_less_nat @ N @ M5 )
=> ( ( F @ M5 )
= Z ) )
=> ( bound_set_int @ Z @ N @ F ) ) ).
% bound.intro
thf(fact_521_s_Ominus__closed,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( 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 ) ) @ X2 @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% s.minus_closed
thf(fact_522_s_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 ) ) ) ) ).
% s.r_right_minus_eq
thf(fact_523_psubsetD,axiom,
! [A2: set_set_int,B4: set_set_int,C: set_int] :
( ( ord_less_set_set_int @ A2 @ B4 )
=> ( ( member_set_int @ C @ A2 )
=> ( member_set_int @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_524_psubsetD,axiom,
! [A2: set_nat_set_int,B4: set_nat_set_int,C: nat > set_int] :
( ( ord_le2931775347370382171et_int @ A2 @ B4 )
=> ( ( member_nat_set_int @ C @ A2 )
=> ( member_nat_set_int @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_525_ring_Oline__extension_Ocong,axiom,
embedd4283282269743769663t_unit = embedd4283282269743769663t_unit ).
% ring.line_extension.cong
thf(fact_526_ring_Oring__simprules_I4_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( member_set_int @ ( a_minu5974516859897376926t_unit @ R @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_527_abelian__group_Ominus__closed,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( member_set_int @ ( a_minu5974516859897376926t_unit @ G @ X2 @ Y ) @ ( partia966996272515721803t_unit @ G ) ) ) ) ) ).
% abelian_group.minus_closed
thf(fact_528_ring_Oint__embed__diff,axiom,
! [R: partia4934656038542163276t_unit,X2: int,Y: int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ring_i2743490682209504680t_unit @ R @ ( minus_minus_int @ X2 @ Y ) )
= ( a_minu5974516859897376926t_unit @ R @ ( ring_i2743490682209504680t_unit @ R @ X2 ) @ ( ring_i2743490682209504680t_unit @ R @ Y ) ) ) ) ).
% ring.int_embed_diff
thf(fact_529_finite__psubset__induct,axiom,
! [A2: set_set_int,P: set_set_int > $o] :
( ( finite6197958912794628473et_int @ A2 )
=> ( ! [A6: set_set_int] :
( ( finite6197958912794628473et_int @ A6 )
=> ( ! [B7: set_set_int] :
( ( ord_less_set_set_int @ B7 @ A6 )
=> ( P @ B7 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_530_ring_Oline__extension__in__carrier,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,A: set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R ) )
=> ( ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ R ) )
=> ( ord_le4403425263959731960et_int @ ( embedd4283282269743769663t_unit @ R @ K @ A @ E ) @ ( partia966996272515721803t_unit @ R ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_531_ring_Oline__extension__mem__iff,axiom,
! [R: partia4934656038542163276t_unit,U: set_int,K: set_set_int,A: set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ U @ ( embedd4283282269743769663t_unit @ R @ K @ A @ E ) )
= ( ? [X3: set_int] :
( ( member_set_int @ X3 @ K )
& ? [Y5: set_int] :
( ( member_set_int @ Y5 @ E )
& ( U
= ( add_se5859248395121729892t_unit @ R @ ( mult_s3864001451298473021t_unit @ R @ X3 @ A ) @ Y5 ) ) ) ) ) ) ) ).
% ring.line_extension_mem_iff
thf(fact_532_finite__has__maximal2,axiom,
! [A2: set_nat_set_int,A: nat > set_int] :
( ( finite7455725759970522984et_int @ A2 )
=> ( ( member_nat_set_int @ A @ A2 )
=> ? [X: nat > set_int] :
( ( member_nat_set_int @ X @ A2 )
& ( ord_le3704955753469811889et_int @ A @ X )
& ! [Xa: nat > set_int] :
( ( member_nat_set_int @ Xa @ A2 )
=> ( ( ord_le3704955753469811889et_int @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_533_finite__has__maximal2,axiom,
! [A2: set_set_int,A: set_int] :
( ( finite6197958912794628473et_int @ A2 )
=> ( ( member_set_int @ A @ A2 )
=> ? [X: set_int] :
( ( member_set_int @ X @ A2 )
& ( ord_less_eq_set_int @ A @ X )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A2 )
=> ( ( ord_less_eq_set_int @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_534_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( ord_less_eq_nat @ A @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_535_finite__has__maximal2,axiom,
! [A2: set_set_set_int,A: set_set_int] :
( ( finite4249678464180374575et_int @ A2 )
=> ( ( member_set_set_int @ A @ A2 )
=> ? [X: set_set_int] :
( ( member_set_set_int @ X @ A2 )
& ( ord_le4403425263959731960et_int @ A @ X )
& ! [Xa: set_set_int] :
( ( member_set_set_int @ Xa @ A2 )
=> ( ( ord_le4403425263959731960et_int @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_536_finite__has__maximal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X: int] :
( ( member_int @ X @ A2 )
& ( ord_less_eq_int @ A @ X )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_537_finite__has__minimal2,axiom,
! [A2: set_nat_set_int,A: nat > set_int] :
( ( finite7455725759970522984et_int @ A2 )
=> ( ( member_nat_set_int @ A @ A2 )
=> ? [X: nat > set_int] :
( ( member_nat_set_int @ X @ A2 )
& ( ord_le3704955753469811889et_int @ X @ A )
& ! [Xa: nat > set_int] :
( ( member_nat_set_int @ Xa @ A2 )
=> ( ( ord_le3704955753469811889et_int @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_538_finite__has__minimal2,axiom,
! [A2: set_set_int,A: set_int] :
( ( finite6197958912794628473et_int @ A2 )
=> ( ( member_set_int @ A @ A2 )
=> ? [X: set_int] :
( ( member_set_int @ X @ A2 )
& ( ord_less_eq_set_int @ X @ A )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A2 )
=> ( ( ord_less_eq_set_int @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_539_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( ord_less_eq_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_540_finite__has__minimal2,axiom,
! [A2: set_set_set_int,A: set_set_int] :
( ( finite4249678464180374575et_int @ A2 )
=> ( ( member_set_set_int @ A @ A2 )
=> ? [X: set_set_int] :
( ( member_set_set_int @ X @ A2 )
& ( ord_le4403425263959731960et_int @ X @ A )
& ! [Xa: set_set_int] :
( ( member_set_set_int @ Xa @ A2 )
=> ( ( ord_le4403425263959731960et_int @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_541_finite__has__minimal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X: int] :
( ( member_int @ X @ A2 )
& ( ord_less_eq_int @ X @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_542_finite__subset,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ( finite6197958912794628473et_int @ B4 )
=> ( finite6197958912794628473et_int @ A2 ) ) ) ).
% finite_subset
thf(fact_543_infinite__super,axiom,
! [S2: set_set_int,T3: set_set_int] :
( ( ord_le4403425263959731960et_int @ S2 @ T3 )
=> ( ~ ( finite6197958912794628473et_int @ S2 )
=> ~ ( finite6197958912794628473et_int @ T3 ) ) ) ).
% infinite_super
thf(fact_544_rev__finite__subset,axiom,
! [B4: set_set_int,A2: set_set_int] :
( ( finite6197958912794628473et_int @ B4 )
=> ( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( finite6197958912794628473et_int @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_545_finite_OemptyI,axiom,
finite6197958912794628473et_int @ bot_bot_set_set_int ).
% finite.emptyI
thf(fact_546_infinite__imp__nonempty,axiom,
! [S2: set_set_int] :
( ~ ( finite6197958912794628473et_int @ S2 )
=> ( S2 != bot_bot_set_set_int ) ) ).
% infinite_imp_nonempty
thf(fact_547_finite_OinsertI,axiom,
! [A2: set_set_int,A: set_int] :
( ( finite6197958912794628473et_int @ A2 )
=> ( finite6197958912794628473et_int @ ( insert_set_int @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_548_ring_Oline__extension__smult__closed,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,E: set_set_int,A: set_int,K2: set_int,U: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ! [K3: set_int,V4: set_int] :
( ( member_set_int @ K3 @ K )
=> ( ( member_set_int @ V4 @ E )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R @ K3 @ V4 ) @ E ) ) )
=> ( ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ K2 @ K )
=> ( ( member_set_int @ U @ ( embedd4283282269743769663t_unit @ R @ K @ A @ E ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R @ K2 @ U ) @ ( embedd4283282269743769663t_unit @ R @ K @ A @ E ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_549_Diff__infinite__finite,axiom,
! [T3: set_set_int,S2: set_set_int] :
( ( finite6197958912794628473et_int @ T3 )
=> ( ~ ( finite6197958912794628473et_int @ S2 )
=> ~ ( finite6197958912794628473et_int @ ( minus_8897228262479074673et_int @ S2 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_550_mem__upD,axiom,
! [F: nat > set_int,R: partia4934656038542163276t_unit,N: nat] :
( ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ R ) )
=> ( member_set_int @ ( F @ N ) @ ( partia966996272515721803t_unit @ R ) ) ) ).
% mem_upD
thf(fact_551_bound__def,axiom,
( bound_set_int
= ( ^ [Z4: set_int,N3: nat,F3: nat > set_int] :
! [M3: nat] :
( ( ord_less_nat @ N3 @ M3 )
=> ( ( F3 @ M3 )
= Z4 ) ) ) ) ).
% bound_def
thf(fact_552_bound_Obound,axiom,
! [Z: set_int,N: nat,F: nat > set_int,M: nat] :
( ( bound_set_int @ Z @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( ( F @ M )
= Z ) ) ) ).
% bound.bound
thf(fact_553_bound__below,axiom,
! [Z: set_int,M: nat,F: nat > set_int,N: nat] :
( ( bound_set_int @ Z @ M @ F )
=> ( ( ( F @ N )
!= Z )
=> ( ord_less_eq_nat @ N @ M ) ) ) ).
% bound_below
thf(fact_554_finite__has__minimal,axiom,
! [A2: set_set_int] :
( ( finite6197958912794628473et_int @ A2 )
=> ( ( A2 != bot_bot_set_set_int )
=> ? [X: set_int] :
( ( member_set_int @ X @ A2 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A2 )
=> ( ( ord_less_eq_set_int @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_555_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_556_finite__has__minimal,axiom,
! [A2: set_set_set_int] :
( ( finite4249678464180374575et_int @ A2 )
=> ( ( A2 != bot_bo2384636101374064866et_int )
=> ? [X: set_set_int] :
( ( member_set_set_int @ X @ A2 )
& ! [Xa: set_set_int] :
( ( member_set_set_int @ Xa @ A2 )
=> ( ( ord_le4403425263959731960et_int @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_557_finite__has__minimal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X: int] :
( ( member_int @ X @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_558_finite__has__maximal,axiom,
! [A2: set_set_int] :
( ( finite6197958912794628473et_int @ A2 )
=> ( ( A2 != bot_bot_set_set_int )
=> ? [X: set_int] :
( ( member_set_int @ X @ A2 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A2 )
=> ( ( ord_less_eq_set_int @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_559_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_560_finite__has__maximal,axiom,
! [A2: set_set_set_int] :
( ( finite4249678464180374575et_int @ A2 )
=> ( ( A2 != bot_bo2384636101374064866et_int )
=> ? [X: set_set_int] :
( ( member_set_set_int @ X @ A2 )
& ! [Xa: set_set_int] :
( ( member_set_set_int @ Xa @ A2 )
=> ( ( ord_le4403425263959731960et_int @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_561_finite__has__maximal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X: int] :
( ( member_int @ X @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_562_finite_Ocases,axiom,
! [A: set_set_int] :
( ( finite6197958912794628473et_int @ A )
=> ( ( A != bot_bot_set_set_int )
=> ~ ! [A6: set_set_int] :
( ? [A3: set_int] :
( A
= ( insert_set_int @ A3 @ A6 ) )
=> ~ ( finite6197958912794628473et_int @ A6 ) ) ) ) ).
% finite.cases
thf(fact_563_finite_Osimps,axiom,
( finite6197958912794628473et_int
= ( ^ [A4: set_set_int] :
( ( A4 = bot_bot_set_set_int )
| ? [A5: set_set_int,B3: set_int] :
( ( A4
= ( insert_set_int @ B3 @ A5 ) )
& ( finite6197958912794628473et_int @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_564_finite__induct,axiom,
! [F2: set_nat_set_int,P: set_nat_set_int > $o] :
( ( finite7455725759970522984et_int @ F2 )
=> ( ( P @ bot_bo8417611410066262939et_int )
=> ( ! [X: nat > set_int,F4: set_nat_set_int] :
( ( finite7455725759970522984et_int @ F4 )
=> ( ~ ( member_nat_set_int @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat_set_int @ X @ F4 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_565_finite__induct,axiom,
! [F2: set_set_int,P: set_set_int > $o] :
( ( finite6197958912794628473et_int @ F2 )
=> ( ( P @ bot_bot_set_set_int )
=> ( ! [X: set_int,F4: set_set_int] :
( ( finite6197958912794628473et_int @ F4 )
=> ( ~ ( member_set_int @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_int @ X @ F4 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_566_finite__ne__induct,axiom,
! [F2: set_nat_set_int,P: set_nat_set_int > $o] :
( ( finite7455725759970522984et_int @ F2 )
=> ( ( F2 != bot_bo8417611410066262939et_int )
=> ( ! [X: nat > set_int] : ( P @ ( insert_nat_set_int @ X @ bot_bo8417611410066262939et_int ) )
=> ( ! [X: nat > set_int,F4: set_nat_set_int] :
( ( finite7455725759970522984et_int @ F4 )
=> ( ( F4 != bot_bo8417611410066262939et_int )
=> ( ~ ( member_nat_set_int @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat_set_int @ X @ F4 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_567_finite__ne__induct,axiom,
! [F2: set_set_int,P: set_set_int > $o] :
( ( finite6197958912794628473et_int @ F2 )
=> ( ( F2 != bot_bot_set_set_int )
=> ( ! [X: set_int] : ( P @ ( insert_set_int @ X @ bot_bot_set_set_int ) )
=> ( ! [X: set_int,F4: set_set_int] :
( ( finite6197958912794628473et_int @ F4 )
=> ( ( F4 != bot_bot_set_set_int )
=> ( ~ ( member_set_int @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_int @ X @ F4 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_568_infinite__finite__induct,axiom,
! [P: set_nat_set_int > $o,A2: set_nat_set_int] :
( ! [A6: set_nat_set_int] :
( ~ ( finite7455725759970522984et_int @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bo8417611410066262939et_int )
=> ( ! [X: nat > set_int,F4: set_nat_set_int] :
( ( finite7455725759970522984et_int @ F4 )
=> ( ~ ( member_nat_set_int @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat_set_int @ X @ F4 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_569_infinite__finite__induct,axiom,
! [P: set_set_int > $o,A2: set_set_int] :
( ! [A6: set_set_int] :
( ~ ( finite6197958912794628473et_int @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_set_int )
=> ( ! [X: set_int,F4: set_set_int] :
( ( finite6197958912794628473et_int @ F4 )
=> ( ~ ( member_set_int @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_int @ X @ F4 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_570_finite__subset__induct,axiom,
! [F2: set_nat_set_int,A2: set_nat_set_int,P: set_nat_set_int > $o] :
( ( finite7455725759970522984et_int @ F2 )
=> ( ( ord_le5995675665013768039et_int @ F2 @ A2 )
=> ( ( P @ bot_bo8417611410066262939et_int )
=> ( ! [A3: nat > set_int,F4: set_nat_set_int] :
( ( finite7455725759970522984et_int @ F4 )
=> ( ( member_nat_set_int @ A3 @ A2 )
=> ( ~ ( member_nat_set_int @ A3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat_set_int @ A3 @ F4 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_571_finite__subset__induct,axiom,
! [F2: set_set_int,A2: set_set_int,P: set_set_int > $o] :
( ( finite6197958912794628473et_int @ F2 )
=> ( ( ord_le4403425263959731960et_int @ F2 @ A2 )
=> ( ( P @ bot_bot_set_set_int )
=> ( ! [A3: set_int,F4: set_set_int] :
( ( finite6197958912794628473et_int @ F4 )
=> ( ( member_set_int @ A3 @ A2 )
=> ( ~ ( member_set_int @ A3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_int @ A3 @ F4 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_572_finite__subset__induct_H,axiom,
! [F2: set_nat_set_int,A2: set_nat_set_int,P: set_nat_set_int > $o] :
( ( finite7455725759970522984et_int @ F2 )
=> ( ( ord_le5995675665013768039et_int @ F2 @ A2 )
=> ( ( P @ bot_bo8417611410066262939et_int )
=> ( ! [A3: nat > set_int,F4: set_nat_set_int] :
( ( finite7455725759970522984et_int @ F4 )
=> ( ( member_nat_set_int @ A3 @ A2 )
=> ( ( ord_le5995675665013768039et_int @ F4 @ A2 )
=> ( ~ ( member_nat_set_int @ A3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat_set_int @ A3 @ F4 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_573_finite__subset__induct_H,axiom,
! [F2: set_set_int,A2: set_set_int,P: set_set_int > $o] :
( ( finite6197958912794628473et_int @ F2 )
=> ( ( ord_le4403425263959731960et_int @ F2 @ A2 )
=> ( ( P @ bot_bot_set_set_int )
=> ( ! [A3: set_int,F4: set_set_int] :
( ( finite6197958912794628473et_int @ F4 )
=> ( ( member_set_int @ A3 @ A2 )
=> ( ( ord_le4403425263959731960et_int @ F4 @ A2 )
=> ( ~ ( member_set_int @ A3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_int @ A3 @ F4 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_574_finite__induct__select,axiom,
! [S2: set_set_int,P: set_set_int > $o] :
( ( finite6197958912794628473et_int @ S2 )
=> ( ( P @ bot_bot_set_set_int )
=> ( ! [T4: set_set_int] :
( ( ord_less_set_set_int @ T4 @ S2 )
=> ( ( P @ T4 )
=> ? [X5: set_int] :
( ( member_set_int @ X5 @ ( minus_8897228262479074673et_int @ S2 @ T4 ) )
& ( P @ ( insert_set_int @ X5 @ T4 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_575_finite__empty__induct,axiom,
! [A2: set_nat_set_int,P: set_nat_set_int > $o] :
( ( finite7455725759970522984et_int @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: nat > set_int,A6: set_nat_set_int] :
( ( finite7455725759970522984et_int @ A6 )
=> ( ( member_nat_set_int @ A3 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_3247115583872269408et_int @ A6 @ ( insert_nat_set_int @ A3 @ bot_bo8417611410066262939et_int ) ) ) ) ) )
=> ( P @ bot_bo8417611410066262939et_int ) ) ) ) ).
% finite_empty_induct
thf(fact_576_finite__empty__induct,axiom,
! [A2: set_set_int,P: set_set_int > $o] :
( ( finite6197958912794628473et_int @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: set_int,A6: set_set_int] :
( ( finite6197958912794628473et_int @ A6 )
=> ( ( member_set_int @ A3 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_8897228262479074673et_int @ A6 @ ( insert_set_int @ A3 @ bot_bot_set_set_int ) ) ) ) ) )
=> ( P @ bot_bot_set_set_int ) ) ) ) ).
% finite_empty_induct
thf(fact_577_infinite__coinduct,axiom,
! [X4: set_set_int > $o,A2: set_set_int] :
( ( X4 @ A2 )
=> ( ! [A6: set_set_int] :
( ( X4 @ A6 )
=> ? [X5: set_int] :
( ( member_set_int @ X5 @ A6 )
& ( ( X4 @ ( minus_8897228262479074673et_int @ A6 @ ( insert_set_int @ X5 @ bot_bot_set_set_int ) ) )
| ~ ( finite6197958912794628473et_int @ ( minus_8897228262479074673et_int @ A6 @ ( insert_set_int @ X5 @ bot_bot_set_set_int ) ) ) ) ) )
=> ~ ( finite6197958912794628473et_int @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_578_infinite__remove,axiom,
! [S2: set_set_int,A: set_int] :
( ~ ( finite6197958912794628473et_int @ S2 )
=> ~ ( finite6197958912794628473et_int @ ( minus_8897228262479074673et_int @ S2 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ) ) ).
% infinite_remove
thf(fact_579_ring_Obound__upD,axiom,
! [R: partia4934656038542163276t_unit,F: nat > set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ R ) )
=> ? [N2: nat] : ( bound_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ N2 @ F ) ) ) ).
% ring.bound_upD
thf(fact_580_finite__remove__induct,axiom,
! [B4: set_nat_set_int,P: set_nat_set_int > $o] :
( ( finite7455725759970522984et_int @ B4 )
=> ( ( P @ bot_bo8417611410066262939et_int )
=> ( ! [A6: set_nat_set_int] :
( ( finite7455725759970522984et_int @ A6 )
=> ( ( A6 != bot_bo8417611410066262939et_int )
=> ( ( ord_le5995675665013768039et_int @ A6 @ B4 )
=> ( ! [X5: nat > set_int] :
( ( member_nat_set_int @ X5 @ A6 )
=> ( P @ ( minus_3247115583872269408et_int @ A6 @ ( insert_nat_set_int @ X5 @ bot_bo8417611410066262939et_int ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_581_finite__remove__induct,axiom,
! [B4: set_set_int,P: set_set_int > $o] :
( ( finite6197958912794628473et_int @ B4 )
=> ( ( P @ bot_bot_set_set_int )
=> ( ! [A6: set_set_int] :
( ( finite6197958912794628473et_int @ A6 )
=> ( ( A6 != bot_bot_set_set_int )
=> ( ( ord_le4403425263959731960et_int @ A6 @ B4 )
=> ( ! [X5: set_int] :
( ( member_set_int @ X5 @ A6 )
=> ( P @ ( minus_8897228262479074673et_int @ A6 @ ( insert_set_int @ X5 @ bot_bot_set_set_int ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_582_remove__induct,axiom,
! [P: set_nat_set_int > $o,B4: set_nat_set_int] :
( ( P @ bot_bo8417611410066262939et_int )
=> ( ( ~ ( finite7455725759970522984et_int @ B4 )
=> ( P @ B4 ) )
=> ( ! [A6: set_nat_set_int] :
( ( finite7455725759970522984et_int @ A6 )
=> ( ( A6 != bot_bo8417611410066262939et_int )
=> ( ( ord_le5995675665013768039et_int @ A6 @ B4 )
=> ( ! [X5: nat > set_int] :
( ( member_nat_set_int @ X5 @ A6 )
=> ( P @ ( minus_3247115583872269408et_int @ A6 @ ( insert_nat_set_int @ X5 @ bot_bo8417611410066262939et_int ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_583_remove__induct,axiom,
! [P: set_set_int > $o,B4: set_set_int] :
( ( P @ bot_bot_set_set_int )
=> ( ( ~ ( finite6197958912794628473et_int @ B4 )
=> ( P @ B4 ) )
=> ( ! [A6: set_set_int] :
( ( finite6197958912794628473et_int @ A6 )
=> ( ( A6 != bot_bot_set_set_int )
=> ( ( ord_le4403425263959731960et_int @ A6 @ B4 )
=> ( ! [X5: set_int] :
( ( member_set_int @ X5 @ A6 )
=> ( P @ ( minus_8897228262479074673et_int @ A6 @ ( insert_set_int @ X5 @ bot_bot_set_set_int ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_584_abelian__monoid_OboundD__carrier,axiom,
! [G: partia4934656038542163276t_unit,N: nat,F: nat > set_int,M: nat] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( bound_set_int @ ( zero_s6269048424454532197t_unit @ G ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_set_int @ ( F @ M ) @ ( partia966996272515721803t_unit @ G ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_585_s_Odimension_Ocases,axiom,
! [A1: nat,A22: set_set_int,A32: set_set_int] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
=> ~ ! [V4: set_int,E2: set_set_int,N2: nat] :
( ( A1
= ( suc @ N2 ) )
=> ( ( A32
= ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A22 @ V4 @ E2 ) )
=> ( ( member_set_int @ V4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ~ ( member_set_int @ V4 @ E2 )
=> ~ ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ A22 @ E2 ) ) ) ) ) ) ) ).
% s.dimension.cases
thf(fact_586_s_Odimension_Osimps,axiom,
! [A1: nat,A22: set_set_int,A32: set_set_int] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A1 @ A22 @ A32 )
= ( ? [K4: set_set_int] :
( ( A1 = zero_zero_nat )
& ( A22 = K4 )
& ( A32
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
| ? [V5: set_int,E3: set_set_int,N3: nat,K4: set_set_int] :
( ( A1
= ( suc @ N3 ) )
& ( A22 = K4 )
& ( A32
= ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K4 @ V5 @ E3 ) )
& ( member_set_int @ V5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ~ ( member_set_int @ V5 @ E3 )
& ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N3 @ K4 @ E3 ) ) ) ) ).
% s.dimension.simps
thf(fact_587_s_Oset__add__zero,axiom,
! [A2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ A2 )
= A2 ) ) ).
% s.set_add_zero
thf(fact_588_s_Odimension__backwards,axiom,
! [K: set_set_int,N: nat,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( suc @ N ) @ K @ E )
=> ? [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ? [E4: set_set_int] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E4 )
& ~ ( member_set_int @ X @ E4 )
& ( E
= ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ X @ E4 ) ) ) ) ) ) ).
% s.dimension_backwards
thf(fact_589_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_590_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_591_s_Osetadd__subset__G,axiom,
! [H: set_set_int,K: set_set_int] :
( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H @ K ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% s.setadd_subset_G
thf(fact_592_s_Oset__add__comm,axiom,
! [I4: set_set_int,J3: set_set_int] :
( ( ord_le4403425263959731960et_int @ I4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ J3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I4 @ J3 )
= ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ J3 @ I4 ) ) ) ) ).
% s.set_add_comm
thf(fact_593_s_Oset__add__closed,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ B4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A2 @ B4 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% s.set_add_closed
thf(fact_594_s_Osum__space__dim_I1_J,axiom,
! [K: set_set_int,E: set_set_int,F2: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ E )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ F2 )
=> ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ E @ F2 ) ) ) ) ) ).
% s.sum_space_dim(1)
thf(fact_595_s_Onat__pow__Suc2,axiom,
! [X2: set_int,N: nat] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( suc @ N ) )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ N ) ) ) ) ).
% s.nat_pow_Suc2
thf(fact_596_s_OSuc__dim,axiom,
! [V: set_int,E: set_set_int,N: nat,K: set_set_int] :
( ( member_set_int @ V @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ~ ( member_set_int @ V @ E )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( suc @ N ) @ K @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ V @ E ) ) ) ) ) ).
% s.Suc_dim
thf(fact_597_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_598_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_599_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_600_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_601_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_602_Suc__diff__diff,axiom,
! [M: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_603_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_604_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_605_nat__pow__Suc,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,N: nat] :
( ( pow_se2518650051167492506it_nat @ G @ X2 @ ( suc @ N ) )
= ( mult_s3864001451298473021t_unit @ G @ ( pow_se2518650051167492506it_nat @ G @ X2 @ N ) @ X2 ) ) ).
% nat_pow_Suc
thf(fact_606_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_607_s_Onat__pow__Suc,axiom,
! [X2: set_int,N: nat] :
( ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( suc @ N ) )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ N ) @ X2 ) ) ).
% s.nat_pow_Suc
thf(fact_608_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_609_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_610_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_611_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_612_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_613_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_614_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X: nat] : ( P @ X @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X: nat,Y4: nat] :
( ( P @ X @ Y4 )
=> ( P @ ( suc @ X ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_615_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_616_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_617_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_618_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_619_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_620_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_621_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_622_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_623_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_624_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_625_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_626_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_627_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
=> ( P @ I5 ) ) )
= ( ( P @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( P @ I5 ) ) ) ) ).
% All_less_Suc
thf(fact_628_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_629_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_630_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
& ( P @ I5 ) ) )
= ( ( P @ N )
| ? [I5: nat] :
( ( ord_less_nat @ I5 @ N )
& ( P @ I5 ) ) ) ) ).
% Ex_less_Suc
thf(fact_631_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_632_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_633_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_634_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_635_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_636_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_637_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_638_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_639_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_640_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M5: nat] :
( M7
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_641_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_642_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_643_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_644_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_645_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_646_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X: nat] : ( R @ X @ X )
=> ( ! [X: nat,Y4: nat,Z2: nat] :
( ( R @ X @ Y4 )
=> ( ( R @ Y4 @ Z2 )
=> ( R @ X @ Z2 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_647_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_648_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_649_Suc__inject,axiom,
! [X2: nat,Y: nat] :
( ( ( suc @ X2 )
= ( suc @ Y ) )
=> ( X2 = Y ) ) ).
% Suc_inject
thf(fact_650_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_651_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_652_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_653_lift__Suc__antimono__le,axiom,
! [F: nat > set_set_int,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_le4403425263959731960et_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_le4403425263959731960et_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_654_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_655_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_656_lift__Suc__mono__le,axiom,
! [F: nat > set_set_int,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_le4403425263959731960et_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_le4403425263959731960et_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_657_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_658_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_659_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_660_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_661_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_662_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
& ( P @ I5 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I5: nat] :
( ( ord_less_nat @ I5 @ N )
& ( P @ ( suc @ I5 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_663_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_664_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
=> ( P @ I5 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( P @ ( suc @ I5 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_665_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_666_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J4: nat] :
( ( M
= ( suc @ J4 ) )
& ( ord_less_nat @ J4 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_667_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_668_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_669_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_670_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_671_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_672_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_673_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_674_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_675_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_676_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_677_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_678_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_679_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_680_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_681_ring_Osum__space__dim_I1_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,E: set_set_int,F2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( embedd8246663962306818995t_unit @ R @ K @ E )
=> ( ( embedd8246663962306818995t_unit @ R @ K @ F2 )
=> ( embedd8246663962306818995t_unit @ R @ K @ ( set_ad273131178244904872t_unit @ R @ E @ F2 ) ) ) ) ) ) ).
% ring.sum_space_dim(1)
thf(fact_682_ring_OSuc__dim,axiom,
! [R: partia4934656038542163276t_unit,V: set_int,E: set_set_int,N: nat,K: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ V @ ( partia966996272515721803t_unit @ R ) )
=> ( ~ ( member_set_int @ V @ E )
=> ( ( embedd646006463343340164t_unit @ R @ N @ K @ E )
=> ( embedd646006463343340164t_unit @ R @ ( suc @ N ) @ K @ ( embedd4283282269743769663t_unit @ R @ K @ V @ E ) ) ) ) ) ) ).
% ring.Suc_dim
thf(fact_683_ring_Odimension__backwards,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,N: nat,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( embedd646006463343340164t_unit @ R @ ( suc @ N ) @ K @ E )
=> ? [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
& ? [E4: set_set_int] :
( ( embedd646006463343340164t_unit @ R @ N @ K @ E4 )
& ~ ( member_set_int @ X @ E4 )
& ( E
= ( embedd4283282269743769663t_unit @ R @ K @ X @ E4 ) ) ) ) ) ) ) ).
% ring.dimension_backwards
thf(fact_684_ring_Odimension_Osimps,axiom,
! [R: partia4934656038542163276t_unit,A1: nat,A22: set_set_int,A32: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( embedd646006463343340164t_unit @ R @ A1 @ A22 @ A32 )
= ( ? [K4: set_set_int] :
( ( A1 = zero_zero_nat )
& ( A22 = K4 )
& ( A32
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) ) )
| ? [V5: set_int,E3: set_set_int,N3: nat,K4: set_set_int] :
( ( A1
= ( suc @ N3 ) )
& ( A22 = K4 )
& ( A32
= ( embedd4283282269743769663t_unit @ R @ K4 @ V5 @ E3 ) )
& ( member_set_int @ V5 @ ( partia966996272515721803t_unit @ R ) )
& ~ ( member_set_int @ V5 @ E3 )
& ( embedd646006463343340164t_unit @ R @ N3 @ K4 @ E3 ) ) ) ) ) ).
% ring.dimension.simps
thf(fact_685_ring_Odimension_Ocases,axiom,
! [R: partia4934656038542163276t_unit,A1: nat,A22: set_set_int,A32: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( embedd646006463343340164t_unit @ R @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) ) )
=> ~ ! [V4: set_int,E2: set_set_int,N2: nat] :
( ( A1
= ( suc @ N2 ) )
=> ( ( A32
= ( embedd4283282269743769663t_unit @ R @ A22 @ V4 @ E2 ) )
=> ( ( member_set_int @ V4 @ ( partia966996272515721803t_unit @ R ) )
=> ( ~ ( member_set_int @ V4 @ E2 )
=> ~ ( embedd646006463343340164t_unit @ R @ N2 @ A22 @ E2 ) ) ) ) ) ) ) ) ).
% ring.dimension.cases
thf(fact_686_ring_Oset__add__zero,axiom,
! [R: partia4934656038542163276t_unit,A2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ord_le4403425263959731960et_int @ A2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( set_ad273131178244904872t_unit @ R @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) @ A2 )
= A2 ) ) ) ).
% ring.set_add_zero
thf(fact_687_s_Oadd__additive__subgroups,axiom,
! [H: set_set_int,K: set_set_int] :
( ( additi7073586575563672860t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( additi7073586575563672860t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( additi7073586575563672860t_unit @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H @ K ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.add_additive_subgroups
thf(fact_688_ring_Ogenideal__one,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( genide1545711809618862555t_unit @ R @ ( insert_set_int @ ( one_se8065767436706823081t_unit @ R ) @ bot_bot_set_set_int ) )
= ( partia966996272515721803t_unit @ R ) ) ) ).
% ring.genideal_one
thf(fact_689_zdiff__int__split,axiom,
! [P: int > $o,X2: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X2 @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_690_imp__le__cong,axiom,
! [X2: int,X6: int,P: $o,P2: $o] :
( ( X2 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_691_conj__le__cong,axiom,
! [X2: int,X6: int,P: $o,P2: $o] :
( ( X2 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_692_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ Z5 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_693_pinf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ Z5 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_694_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ Z5 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_695_pinf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ Z5 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_696_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_697_pinf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_698_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_699_pinf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_700_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_701_pinf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_702_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_703_pinf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_704_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z5 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_705_minf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ X @ Z5 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_706_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z5 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_707_minf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ X @ Z5 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_708_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_709_minf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_710_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_711_minf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_712_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_713_minf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_714_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_715_minf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_716_principalideal_Ois__principalideal,axiom,
! [I4: set_set_int,R: partia4934656038542163276t_unit] :
( ( princi8860937869964495385t_unit @ I4 @ R )
=> ( princi8860937869964495385t_unit @ I4 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_717_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_718_minf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% minf(8)
thf(fact_719_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_720_minf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ord_less_eq_int @ X5 @ T ) ) ).
% minf(6)
thf(fact_721_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_722_pinf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ord_less_eq_int @ T @ X5 ) ) ).
% pinf(8)
thf(fact_723_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_724_pinf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% pinf(6)
thf(fact_725_ring_Ocgenideal__self,axiom,
! [R: partia4934656038542163276t_unit,I: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ I @ ( partia966996272515721803t_unit @ R ) )
=> ( member_set_int @ I @ ( cgenid8502489213727343375t_unit @ R @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_726_ring_Oonepideal,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( princi8860937869964495385t_unit @ ( partia966996272515721803t_unit @ R ) @ R ) ) ).
% ring.onepideal
thf(fact_727_ring_Ogenideal__self,axiom,
! [R: partia4934656038542163276t_unit,S2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ord_le4403425263959731960et_int @ S2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ord_le4403425263959731960et_int @ S2 @ ( genide1545711809618862555t_unit @ R @ S2 ) ) ) ) ).
% ring.genideal_self
thf(fact_728_ring_Osubset__Idl__subset,axiom,
! [R: partia4934656038542163276t_unit,I4: set_set_int,H: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ord_le4403425263959731960et_int @ I4 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( ord_le4403425263959731960et_int @ H @ I4 )
=> ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ R @ H ) @ ( genide1545711809618862555t_unit @ R @ I4 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_729_ring_Oset__add__comm,axiom,
! [R: partia4934656038542163276t_unit,I4: set_set_int,J3: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ord_le4403425263959731960et_int @ I4 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( ord_le4403425263959731960et_int @ J3 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( set_ad273131178244904872t_unit @ R @ I4 @ J3 )
= ( set_ad273131178244904872t_unit @ R @ J3 @ I4 ) ) ) ) ) ).
% ring.set_add_comm
thf(fact_730_ring_Ogenideal__self_H,axiom,
! [R: partia4934656038542163276t_unit,I: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ I @ ( partia966996272515721803t_unit @ R ) )
=> ( member_set_int @ I @ ( genide1545711809618862555t_unit @ R @ ( insert_set_int @ I @ bot_bot_set_set_int ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_731_ring_Ogenideal__zero,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( genide1545711809618862555t_unit @ R @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) ) ) ).
% ring.genideal_zero
thf(fact_732_ring_Ozeropideal,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( princi8860937869964495385t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) @ R ) ) ).
% ring.zeropideal
thf(fact_733_principalideal_Ogenerate,axiom,
! [I4: set_set_int,R: partia4934656038542163276t_unit] :
( ( princi8860937869964495385t_unit @ I4 @ R )
=> ? [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
& ( I4
= ( genide1545711809618862555t_unit @ R @ ( insert_set_int @ X @ bot_bot_set_set_int ) ) ) ) ) ).
% principalideal.generate
thf(fact_734_ring_OIdl__subset__ideal_H,axiom,
! [R: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R ) )
=> ( ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ R @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) @ ( genide1545711809618862555t_unit @ R @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) )
= ( member_set_int @ A @ ( genide1545711809618862555t_unit @ R @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_735_s_Oint__embed__one,axiom,
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ one_one_int )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% s.int_embed_one
thf(fact_736_s_Oadd_Oone__in__subset,axiom,
! [H: set_set_int] :
( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( H != bot_bot_set_set_int )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ H )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ H ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ H )
=> ! [Xa2: set_int] :
( ( member_set_int @ Xa2 @ H )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Xa2 ) @ H ) ) )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ H ) ) ) ) ) ).
% s.add.one_in_subset
thf(fact_737_s_Oembed__char__eq__0__iff,axiom,
! [N: int] :
( ( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) @ N ) ) ).
% s.embed_char_eq_0_iff
thf(fact_738_s_Osubring__props_I5_J,axiom,
! [K: set_set_int,H3: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ H3 @ K )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H3 ) @ K ) ) ) ).
% s.subring_props(5)
thf(fact_739_s_Or__neg2,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ Y ) )
= Y ) ) ) ).
% s.r_neg2
thf(fact_740_s_Or__neg1,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( 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 ) ) @ X2 ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) )
= Y ) ) ) ).
% s.r_neg1
thf(fact_741_s_Ominus__add,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( 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 ) ) @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ) ) ).
% s.minus_add
thf(fact_742_s_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 ) ) ) ) ) ) ).
% s.add.inv_solve_right'
thf(fact_743_s_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 ) ) ) ) ) ) ).
% s.add.inv_solve_right
thf(fact_744_s_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 ) ) ) ) ) ) ).
% s.add.inv_solve_left'
thf(fact_745_s_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 ) ) ) ) ) ) ).
% s.add.inv_solve_left
thf(fact_746_s_Oadd_Oinv__mult__group,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( 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 ) ) @ X2 @ 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 ) ) @ X2 ) ) ) ) ) ).
% s.add.inv_mult_group
thf(fact_747_s_Oa__transpose__inv,axiom,
! [X2: set_int,Y: set_int,Z: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= Z )
=> ( ( member_set_int @ X2 @ ( 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 ) ) @ X2 ) @ Z )
= Y ) ) ) ) ) ).
% s.a_transpose_inv
thf(fact_748_s_Or__minus,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) ) ) ) ) ).
% s.r_minus
thf(fact_749_s_Ol__minus,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( 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 ) ) @ X2 ) @ Y )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) ) ) ) ) ).
% s.l_minus
thf(fact_750_s_Ospace__subgroup__props_I4_J,axiom,
! [K: set_set_int,N: nat,E: set_set_int,V: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( ( member_set_int @ V @ E )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ V ) @ E ) ) ) ) ).
% s.space_subgroup_props(4)
thf(fact_751_s_Ominus__eq,axiom,
! [X2: set_int,Y: set_int] :
( ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ).
% s.minus_eq
thf(fact_752_s_Or__neg,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.r_neg
thf(fact_753_s_Ominus__equality,axiom,
! [Y: set_int,X2: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 )
= Y ) ) ) ) ).
% s.minus_equality
thf(fact_754_s_Ol__neg,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ X2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.l_neg
thf(fact_755_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_756_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_757_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_758_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_759_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_760_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_761_s_Ominus__minus,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) )
= X2 ) ) ).
% s.minus_minus
thf(fact_762_s_Oadd_Oinv__closed,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.add.inv_closed
thf(fact_763_s_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 ) ) ) ) ).
% s.minus_zero
thf(fact_764_s_Oadd_Oinv__eq__1__iff,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( X2
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% s.add.inv_eq_1_iff
thf(fact_765_s_Oinv__neg__one,axiom,
( ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.inv_neg_one
thf(fact_766_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_767_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_768_ring_Oring__simprules_I3_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R @ X2 ) @ ( partia966996272515721803t_unit @ R ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_769_ring_Oring__simprules_I20_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( a_inv_5951419416477254493t_unit @ R @ ( a_inv_5951419416477254493t_unit @ R @ X2 ) )
= X2 ) ) ) ).
% ring.ring_simprules(20)
thf(fact_770_ring_Ominus__zero,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( a_inv_5951419416477254493t_unit @ R @ ( zero_s6269048424454532197t_unit @ R ) )
= ( zero_s6269048424454532197t_unit @ R ) ) ) ).
% ring.minus_zero
thf(fact_771_ring_Osubring__props_I5_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,H3: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( member_set_int @ H3 @ K )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R @ H3 ) @ K ) ) ) ) ).
% ring.subring_props(5)
thf(fact_772_abelian__group_Oa__inv__closed,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ G @ X2 ) @ ( partia966996272515721803t_unit @ G ) ) ) ) ).
% abelian_group.a_inv_closed
thf(fact_773_abelian__group_Ominus__minus,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( a_inv_5951419416477254493t_unit @ G @ ( a_inv_5951419416477254493t_unit @ G @ X2 ) )
= X2 ) ) ) ).
% abelian_group.minus_minus
thf(fact_774_a__minus__def,axiom,
( a_minu5974516859897376926t_unit
= ( ^ [R2: partia4934656038542163276t_unit,X3: set_int,Y5: set_int] : ( add_se5859248395121729892t_unit @ R2 @ X3 @ ( a_inv_5951419416477254493t_unit @ R2 @ Y5 ) ) ) ) ).
% a_minus_def
thf(fact_775_ring_Oring__simprules_I17_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ X2 @ ( add_se5859248395121729892t_unit @ R @ ( a_inv_5951419416477254493t_unit @ R @ X2 ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_776_ring_Oring__simprules_I18_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ ( a_inv_5951419416477254493t_unit @ R @ X2 ) @ ( add_se5859248395121729892t_unit @ R @ X2 @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_777_ring_Oring__simprules_I19_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( a_inv_5951419416477254493t_unit @ R @ ( add_se5859248395121729892t_unit @ R @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ R @ ( a_inv_5951419416477254493t_unit @ R @ X2 ) @ ( a_inv_5951419416477254493t_unit @ R @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_778_ring_Or__minus,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ X2 @ ( a_inv_5951419416477254493t_unit @ R @ Y ) )
= ( a_inv_5951419416477254493t_unit @ R @ ( mult_s3864001451298473021t_unit @ R @ X2 @ Y ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_779_ring_Ol__minus,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ ( a_inv_5951419416477254493t_unit @ R @ X2 ) @ Y )
= ( a_inv_5951419416477254493t_unit @ R @ ( mult_s3864001451298473021t_unit @ R @ X2 @ Y ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_780_abelian__group_Ominus__add,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( a_inv_5951419416477254493t_unit @ G @ ( add_se5859248395121729892t_unit @ G @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ G @ ( a_inv_5951419416477254493t_unit @ G @ X2 ) @ ( a_inv_5951419416477254493t_unit @ G @ Y ) ) ) ) ) ) ).
% abelian_group.minus_add
thf(fact_781_abelian__group_Or__neg2,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ X2 @ ( add_se5859248395121729892t_unit @ G @ ( a_inv_5951419416477254493t_unit @ G @ X2 ) @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg2
thf(fact_782_abelian__group_Or__neg1,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ ( a_inv_5951419416477254493t_unit @ G @ X2 ) @ ( add_se5859248395121729892t_unit @ G @ X2 @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg1
thf(fact_783_ring_Ospace__subgroup__props_I4_J,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,N: nat,E: set_set_int,V: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( embedd646006463343340164t_unit @ R @ N @ K @ E )
=> ( ( member_set_int @ V @ E )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R @ V ) @ E ) ) ) ) ) ).
% ring.space_subgroup_props(4)
thf(fact_784_ring_Oring__simprules_I14_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( a_minu5974516859897376926t_unit @ R @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ R @ X2 @ ( a_inv_5951419416477254493t_unit @ R @ Y ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_785_ring_Oinv__neg__one,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( m_inv_4894562657074299959t_unit @ R @ ( a_inv_5951419416477254493t_unit @ R @ ( one_se8065767436706823081t_unit @ R ) ) )
= ( a_inv_5951419416477254493t_unit @ R @ ( one_se8065767436706823081t_unit @ R ) ) ) ) ).
% ring.inv_neg_one
thf(fact_786_abelian__group_Ominus__eq,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( a_minu5974516859897376926t_unit @ G @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ G @ X2 @ ( a_inv_5951419416477254493t_unit @ G @ Y ) ) ) ) ).
% abelian_group.minus_eq
thf(fact_787_ring_Oring__simprules_I9_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ ( a_inv_5951419416477254493t_unit @ R @ X2 ) @ X2 )
= ( zero_s6269048424454532197t_unit @ R ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_788_ring_Oring__simprules_I16_J,axiom,
! [R: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( add_se5859248395121729892t_unit @ R @ X2 @ ( a_inv_5951419416477254493t_unit @ R @ X2 ) )
= ( zero_s6269048424454532197t_unit @ R ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_789_ring_Oint__embed__one,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ring_i2743490682209504680t_unit @ R @ one_one_int )
= ( one_se8065767436706823081t_unit @ R ) ) ) ).
% ring.int_embed_one
thf(fact_790_abelian__group_Ol__neg,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ ( a_inv_5951419416477254493t_unit @ G @ X2 ) @ X2 )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ) ).
% abelian_group.l_neg
thf(fact_791_abelian__group_Or__neg,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ X2 @ ( a_inv_5951419416477254493t_unit @ G @ X2 ) )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ) ).
% abelian_group.r_neg
thf(fact_792_abelian__group_Ominus__equality,axiom,
! [G: partia4934656038542163276t_unit,Y: set_int,X2: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( ( add_se5859248395121729892t_unit @ G @ Y @ X2 )
= ( zero_s6269048424454532197t_unit @ G ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( a_inv_5951419416477254493t_unit @ G @ X2 )
= Y ) ) ) ) ) ).
% abelian_group.minus_equality
thf(fact_793_ring_Oembed__char__eq__0__iff,axiom,
! [R: partia4934656038542163276t_unit,N: int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ( ring_i2743490682209504680t_unit @ R @ N )
= ( zero_s6269048424454532197t_unit @ R ) )
= ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ ( ring_c6147214092195050492t_unit @ R ) ) @ N ) ) ) ).
% ring.embed_char_eq_0_iff
thf(fact_794_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_795_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_796_s_Oadd_Oint__pow__diff,axiom,
! [X2: set_int,N: int,M: int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( minus_minus_int @ N @ M ) @ X2 )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ X2 ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ X2 ) ) ) ) ) ).
% s.add.int_pow_diff
thf(fact_797_s_Oadd_Oint__pow__distrib,axiom,
! [X2: set_int,Y: set_int,I: int] :
( ( member_set_int @ X2 @ ( 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 ) ) @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X2 ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ Y ) ) ) ) ) ).
% s.add.int_pow_distrib
thf(fact_798_s_Oadd_Oint__pow__mult__distrib,axiom,
! [X2: set_int,Y: set_int,I: int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 ) )
=> ( ( member_set_int @ X2 @ ( 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 ) ) @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X2 ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ Y ) ) ) ) ) ) ).
% s.add.int_pow_mult_distrib
thf(fact_799_s_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 ) ) ) ) ) ).
% s.add_pow_ldistr_int
thf(fact_800_s_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 ) ) ) ) ) ).
% s.add_pow_rdistr_int
thf(fact_801_s_Otelescopic__base__aux,axiom,
! [K: set_set_int,F2: set_set_int,N: nat,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( subfie3888952257595785920t_unit @ F2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ F2 )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ one_one_nat @ F2 @ E )
=> ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E ) ) ) ) ) ).
% s.telescopic_base_aux
thf(fact_802_s_Oadd_Oint__pow__inv,axiom,
! [X2: set_int,I: int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X2 ) ) ) ) ).
% s.add.int_pow_inv
thf(fact_803_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_804_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_805_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_806_s_Oadd_Oint__pow__closed,axiom,
! [X2: set_int,I: int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.add.int_pow_closed
thf(fact_807_s_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 ) ) ) ) ).
% s.add.int_pow_one
thf(fact_808_s_Oadd_Oint__pow__1,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ one_one_int @ X2 )
= X2 ) ) ).
% s.add.int_pow_1
thf(fact_809_s_Onat__pow__eone,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ one_one_nat )
= X2 ) ) ).
% s.nat_pow_eone
thf(fact_810_s_Odimension__one,axiom,
! [K: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ one_one_nat @ K @ K ) ) ).
% s.dimension_one
thf(fact_811_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_812_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_813_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_814_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_815_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_816_ring_Otelescopic__base__aux,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,F2: set_set_int,N: nat,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( subfie3888952257595785920t_unit @ F2 @ R )
=> ( ( embedd646006463343340164t_unit @ R @ N @ K @ F2 )
=> ( ( embedd646006463343340164t_unit @ R @ one_one_nat @ F2 @ E )
=> ( embedd646006463343340164t_unit @ R @ N @ K @ E ) ) ) ) ) ) ).
% ring.telescopic_base_aux
thf(fact_817_ring_Odimension__one,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( embedd646006463343340164t_unit @ R @ one_one_nat @ K @ K ) ) ) ).
% ring.dimension_one
thf(fact_818_int__embed__def,axiom,
( ring_i2743490682209504680t_unit
= ( ^ [R2: partia4934656038542163276t_unit,K5: int] : ( add_po7581009264371422883it_int @ R2 @ K5 @ ( one_se8065767436706823081t_unit @ R2 ) ) ) ) ).
% int_embed_def
thf(fact_819_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_820_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_821_ring_Oadd__pow__ldistr__int,axiom,
! [R: partia4934656038542163276t_unit,A: set_int,B: set_int,K2: int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ ( add_po7581009264371422883it_int @ R @ K2 @ A ) @ B )
= ( add_po7581009264371422883it_int @ R @ K2 @ ( mult_s3864001451298473021t_unit @ R @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_ldistr_int
thf(fact_822_ring_Oadd__pow__rdistr__int,axiom,
! [R: partia4934656038542163276t_unit,A: set_int,B: set_int,K2: int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R ) )
=> ( ( mult_s3864001451298473021t_unit @ R @ A @ ( add_po7581009264371422883it_int @ R @ K2 @ B ) )
= ( add_po7581009264371422883it_int @ R @ K2 @ ( mult_s3864001451298473021t_unit @ R @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_rdistr_int
thf(fact_823_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_824_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_825_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_826_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_827_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_828_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_829_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_830_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_831_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_832_int__diff__cases,axiom,
! [Z: int] :
~ ! [M5: nat,N2: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_833_minus__int__code_I1_J,axiom,
! [K2: int] :
( ( minus_minus_int @ K2 @ zero_zero_int )
= K2 ) ).
% minus_int_code(1)
thf(fact_834_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_835_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N )
=> ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_836_zdvd__zdiffD,axiom,
! [K2: int,M: int,N: int] :
( ( dvd_dvd_int @ K2 @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K2 @ N )
=> ( dvd_dvd_int @ K2 @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_837_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_838_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_839_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ? [N2: nat] :
( K2
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_840_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ~ ! [N2: nat] :
( K2
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_841_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_842_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_843_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_844_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( ( dvd_dvd_int @ Z @ N )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ Z @ N ) ) ) ).
% zdvd_imp_le
thf(fact_845_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M @ N )
=> ( ( dvd_dvd_int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_846_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N2: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_847_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K2
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_848_s_OsubringI,axiom,
! [H: set_set_int] :
( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ H )
=> ( ! [H4: set_int] :
( ( member_set_int @ H4 @ H )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H4 ) @ H ) )
=> ( ! [H12: set_int,H22: set_int] :
( ( member_set_int @ H12 @ H )
=> ( ( member_set_int @ H22 @ H )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H12 @ H22 ) @ H ) ) )
=> ( ! [H12: set_int,H22: set_int] :
( ( member_set_int @ H12 @ H )
=> ( ( member_set_int @ H22 @ H )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H12 @ H22 ) @ H ) ) )
=> ( subrin7689096310803670856t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ) ).
% s.subringI
thf(fact_849_s_Oadd_Oint__pow__mult,axiom,
! [X2: set_int,I: int,J: int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( plus_plus_int @ I @ J ) @ X2 )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X2 ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ J @ X2 ) ) ) ) ).
% s.add.int_pow_mult
thf(fact_850_s_Onat__pow__mult,axiom,
! [X2: set_int,N: nat,M: nat] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ N ) @ ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ M ) )
= ( pow_se2518650051167492506it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( plus_plus_nat @ N @ M ) ) ) ) ).
% s.nat_pow_mult
thf(fact_851_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_852_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_853_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_854_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_855_s_Ocarrier__is__subring,axiom,
subrin7689096310803670856t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% s.carrier_is_subring
thf(fact_856_s_Oint__embed__range,axiom,
! [K: set_set_int,K2: int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( member_set_int @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 ) @ K ) ) ).
% s.int_embed_range
thf(fact_857_s_Oint__embed__add,axiom,
! [X2: int,Y: int] :
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( plus_plus_int @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ).
% s.int_embed_add
thf(fact_858_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_859_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_860_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_861_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_862_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_863_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_864_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_865_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_866_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_867_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_868_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_869_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_870_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_871_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y ) )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_872_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_873_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_874_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_875_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_876_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_877_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_878_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_879_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_880_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_881_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_882_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_883_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_884_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_885_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_886_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_887_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_888_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_889_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_890_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_891_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_892_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_893_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_894_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_895_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_896_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_897_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_898_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_899_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_900_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_901_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_902_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_903_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_904_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_905_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_906_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_907_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_908_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_909_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_910_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_911_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_912_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_913_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_914_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_915_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_916_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_917_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_918_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_919_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_920_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_921_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_922_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_923_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_924_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_925_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_926_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_927_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_928_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_929_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_930_subringE_I5_J,axiom,
! [H: set_set_int,R: partia4934656038542163276t_unit,H3: set_int] :
( ( subrin7689096310803670856t_unit @ H @ R )
=> ( ( member_set_int @ H3 @ H )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R @ H3 ) @ H ) ) ) ).
% subringE(5)
thf(fact_931_plus__int__code_I1_J,axiom,
! [K2: int] :
( ( plus_plus_int @ K2 @ zero_zero_int )
= K2 ) ).
% plus_int_code(1)
thf(fact_932_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_933_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_934_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_935_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_936_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_937_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_938_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_939_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_940_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_941_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_942_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_943_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_944_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_945_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_946_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_947_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_948_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_949_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_950_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_951_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_952_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_953_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_954_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_955_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_956_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_957_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_958_subringE_I1_J,axiom,
! [H: set_set_int,R: partia4934656038542163276t_unit] :
( ( subrin7689096310803670856t_unit @ H @ R )
=> ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ R ) ) ) ).
% subringE(1)
thf(fact_959_ring_Ocarrier__is__subring,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( subrin7689096310803670856t_unit @ ( partia966996272515721803t_unit @ R ) @ R ) ) ).
% ring.carrier_is_subring
thf(fact_960_subringE_I2_J,axiom,
! [H: set_set_int,R: partia4934656038542163276t_unit] :
( ( subrin7689096310803670856t_unit @ H @ R )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ H ) ) ).
% subringE(2)
thf(fact_961_subringE_I7_J,axiom,
! [H: set_set_int,R: partia4934656038542163276t_unit,H1: set_int,H2: set_int] :
( ( subrin7689096310803670856t_unit @ H @ R )
=> ( ( member_set_int @ H1 @ H )
=> ( ( member_set_int @ H2 @ H )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R @ H1 @ H2 ) @ H ) ) ) ) ).
% subringE(7)
thf(fact_962_subringE_I4_J,axiom,
! [H: set_set_int,R: partia4934656038542163276t_unit] :
( ( subrin7689096310803670856t_unit @ H @ R )
=> ( H != bot_bot_set_set_int ) ) ).
% subringE(4)
thf(fact_963_subringE_I6_J,axiom,
! [H: set_set_int,R: partia4934656038542163276t_unit,H1: set_int,H2: set_int] :
( ( subrin7689096310803670856t_unit @ H @ R )
=> ( ( member_set_int @ H1 @ H )
=> ( ( member_set_int @ H2 @ H )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R @ H1 @ H2 ) @ H ) ) ) ) ).
% subringE(6)
thf(fact_964_subfieldE_I1_J,axiom,
! [K: set_set_int,R: partia4934656038542163276t_unit] :
( ( subfie3888952257595785920t_unit @ K @ R )
=> ( subrin7689096310803670856t_unit @ K @ R ) ) ).
% subfieldE(1)
thf(fact_965_subringE_I3_J,axiom,
! [H: set_set_int,R: partia4934656038542163276t_unit] :
( ( subrin7689096310803670856t_unit @ H @ R )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ R ) @ H ) ) ).
% subringE(3)
thf(fact_966_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_967_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_968_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_969_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_970_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_971_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_972_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_973_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_974_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_975_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_976_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_977_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_978_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_979_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_980_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_981_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_982_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_983_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_984_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_985_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_986_diff__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_987_Nat_Odiff__cancel,axiom,
! [K2: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_988_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_989_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_990_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_991_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_992_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_993_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_994_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_995_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_996_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_997_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_998_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_999_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_1000_group__cancel_Oadd2,axiom,
! [B4: int,K2: int,B: int,A: int] :
( ( B4
= ( plus_plus_int @ K2 @ B ) )
=> ( ( plus_plus_int @ A @ B4 )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1001_group__cancel_Oadd2,axiom,
! [B4: nat,K2: nat,B: nat,A: nat] :
( ( B4
= ( plus_plus_nat @ K2 @ B ) )
=> ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1002_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_1003_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_1004_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_1005_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_1006_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_1007_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_1008_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_1009_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1010_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1011_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K5: nat] :
( N3
= ( plus_plus_nat @ M3 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1012_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1013_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1014_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_1015_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_1016_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K2 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1017_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_1018_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1019_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1020_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1021_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_1022_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_1023_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_1024_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_1025_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_1026_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
? [C5: nat] :
( B3
= ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_1027_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_1028_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_1029_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_1030_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_1031_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_1032_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_1033_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_1034_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_1035_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_1036_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_1037_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_1038_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_1039_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_1040_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_1041_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_1042_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_1043_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_1044_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_1045_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_1046_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_1047_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_1048_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_1049_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_1050_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_1051_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_1052_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1053_add__nonneg__eq__0__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X2 @ Y )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1054_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1055_add__nonpos__eq__0__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X2 @ Y )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1056_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_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_ring_Oint__embed__range,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,K2: int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subrin7689096310803670856t_unit @ K @ R )
=> ( member_set_int @ ( ring_i2743490682209504680t_unit @ R @ K2 ) @ K ) ) ) ).
% ring.int_embed_range
thf(fact_1064_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_1065_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_1066_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_1067_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_1068_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_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_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_1079_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_1080_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_1081_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_1082_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_1083_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_1084_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_1085_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_1086_minf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_1087_minf_I10_J,axiom,
! [D: int,S: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_1088_minf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% minf(9)
thf(fact_1089_minf_I9_J,axiom,
! [D: int,S: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% minf(9)
thf(fact_1090_pinf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_1091_pinf_I10_J,axiom,
! [D: int,S: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_1092_pinf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% pinf(9)
thf(fact_1093_pinf_I9_J,axiom,
! [D: int,S: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% pinf(9)
thf(fact_1094_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1095_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1096_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_1097_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1098_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1099_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1100_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1101_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
? [K5: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M3 @ K5 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1102_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1103_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1104_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K2: nat] :
( ! [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K2 ) @ ( F @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1105_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_1106_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1107_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1108_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1109_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1110_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_1111_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_1112_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_1113_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_1114_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_1115_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z4: int] :
? [N3: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1116_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_1117_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_1118_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_1119_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_1120_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_1121_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_1122_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_1123_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_1124_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_1125_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_1126_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_1127_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_1128_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_1129_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 ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1130_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 ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1131_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_1132_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_1133_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z4: int] :
? [N3: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1134_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_1135_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_1136_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_1137_ring_Oint__embed__add,axiom,
! [R: partia4934656038542163276t_unit,X2: int,Y: int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ring_i2743490682209504680t_unit @ R @ ( plus_plus_int @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ R @ ( ring_i2743490682209504680t_unit @ R @ X2 ) @ ( ring_i2743490682209504680t_unit @ R @ Y ) ) ) ) ).
% ring.int_embed_add
thf(fact_1138_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1139_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_1140_ring_OsubringI,axiom,
! [R: partia4934656038542163276t_unit,H: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ ( one_se8065767436706823081t_unit @ R ) @ H )
=> ( ! [H4: set_int] :
( ( member_set_int @ H4 @ H )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R @ H4 ) @ H ) )
=> ( ! [H12: set_int,H22: set_int] :
( ( member_set_int @ H12 @ H )
=> ( ( member_set_int @ H22 @ H )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R @ H12 @ H22 ) @ H ) ) )
=> ( ! [H12: set_int,H22: set_int] :
( ( member_set_int @ H12 @ H )
=> ( ( member_set_int @ H22 @ H )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R @ H12 @ H22 ) @ H ) ) )
=> ( subrin7689096310803670856t_unit @ H @ R ) ) ) ) ) ) ) ).
% ring.subringI
thf(fact_1141_s_Odimension__direct__sum__space,axiom,
! [K: set_set_int,N: nat,E: set_set_int,M: nat,F2: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ K @ F2 )
=> ( ( ( inf_inf_set_set_int @ E @ F2 )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
=> ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( plus_plus_nat @ N @ M ) @ K @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ E @ F2 ) ) ) ) ) ) ).
% s.dimension_direct_sum_space
thf(fact_1142_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_1143_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_1144_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_1145_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_1146_IntI,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ C @ A2 )
=> ( ( member_nat_set_int @ C @ B4 )
=> ( member_nat_set_int @ C @ ( inf_in1752217752563533465et_int @ A2 @ B4 ) ) ) ) ).
% IntI
thf(fact_1147_IntI,axiom,
! [C: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ A2 )
=> ( ( member_set_int @ C @ B4 )
=> ( member_set_int @ C @ ( inf_inf_set_set_int @ A2 @ B4 ) ) ) ) ).
% IntI
thf(fact_1148_Int__iff,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( inf_in1752217752563533465et_int @ A2 @ B4 ) )
= ( ( member_nat_set_int @ C @ A2 )
& ( member_nat_set_int @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_1149_Int__iff,axiom,
! [C: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ ( inf_inf_set_set_int @ A2 @ B4 ) )
= ( ( member_set_int @ C @ A2 )
& ( member_set_int @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_1150_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_1151_s_Osubring__inter,axiom,
! [I4: set_set_int,J3: set_set_int] :
( ( subrin7689096310803670856t_unit @ I4 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( subrin7689096310803670856t_unit @ J3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( subrin7689096310803670856t_unit @ ( inf_inf_set_set_int @ I4 @ J3 ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.subring_inter
thf(fact_1152_s_Osubalgebra__inter,axiom,
! [K: set_set_int,V3: set_set_int,V6: set_set_int] :
( ( embedd2743979684206749024t_unit @ K @ V3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd2743979684206749024t_unit @ K @ V6 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( embedd2743979684206749024t_unit @ K @ ( inf_inf_set_set_int @ V3 @ V6 ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.subalgebra_inter
thf(fact_1153_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_1154_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_1155_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_1156_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_1157_dvd__1__left,axiom,
! [K2: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K2 ) ).
% dvd_1_left
thf(fact_1158_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( M
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_1159_Int__subset__iff,axiom,
! [C3: set_set_int,A2: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ C3 @ ( inf_inf_set_set_int @ A2 @ B4 ) )
= ( ( ord_le4403425263959731960et_int @ C3 @ A2 )
& ( ord_le4403425263959731960et_int @ C3 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_1160_Int__insert__left__if0,axiom,
! [A: nat > set_int,C3: set_nat_set_int,B4: set_nat_set_int] :
( ~ ( member_nat_set_int @ A @ C3 )
=> ( ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ B4 ) @ C3 )
= ( inf_in1752217752563533465et_int @ B4 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_1161_Int__insert__left__if0,axiom,
! [A: set_int,C3: set_set_int,B4: set_set_int] :
( ~ ( member_set_int @ A @ C3 )
=> ( ( inf_inf_set_set_int @ ( insert_set_int @ A @ B4 ) @ C3 )
= ( inf_inf_set_set_int @ B4 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_1162_Int__insert__left__if1,axiom,
! [A: nat > set_int,C3: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ A @ C3 )
=> ( ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ B4 ) @ C3 )
= ( insert_nat_set_int @ A @ ( inf_in1752217752563533465et_int @ B4 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1163_Int__insert__left__if1,axiom,
! [A: set_int,C3: set_set_int,B4: set_set_int] :
( ( member_set_int @ A @ C3 )
=> ( ( inf_inf_set_set_int @ ( insert_set_int @ A @ B4 ) @ C3 )
= ( insert_set_int @ A @ ( inf_inf_set_set_int @ B4 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1164_insert__inter__insert,axiom,
! [A: set_int,A2: set_set_int,B4: set_set_int] :
( ( inf_inf_set_set_int @ ( insert_set_int @ A @ A2 ) @ ( insert_set_int @ A @ B4 ) )
= ( insert_set_int @ A @ ( inf_inf_set_set_int @ A2 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_1165_Int__insert__right__if0,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ~ ( member_nat_set_int @ A @ A2 )
=> ( ( inf_in1752217752563533465et_int @ A2 @ ( insert_nat_set_int @ A @ B4 ) )
= ( inf_in1752217752563533465et_int @ A2 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_1166_Int__insert__right__if0,axiom,
! [A: set_int,A2: set_set_int,B4: set_set_int] :
( ~ ( member_set_int @ A @ A2 )
=> ( ( inf_inf_set_set_int @ A2 @ ( insert_set_int @ A @ B4 ) )
= ( inf_inf_set_set_int @ A2 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_1167_Int__insert__right__if1,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ A @ A2 )
=> ( ( inf_in1752217752563533465et_int @ A2 @ ( insert_nat_set_int @ A @ B4 ) )
= ( insert_nat_set_int @ A @ ( inf_in1752217752563533465et_int @ A2 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1168_Int__insert__right__if1,axiom,
! [A: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ A @ A2 )
=> ( ( inf_inf_set_set_int @ A2 @ ( insert_set_int @ A @ B4 ) )
= ( insert_set_int @ A @ ( inf_inf_set_set_int @ A2 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1169_finite__Int,axiom,
! [F2: set_set_int,G: set_set_int] :
( ( ( finite6197958912794628473et_int @ F2 )
| ( finite6197958912794628473et_int @ G ) )
=> ( finite6197958912794628473et_int @ ( inf_inf_set_set_int @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_1170_int__dvd__int__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% int_dvd_int_iff
thf(fact_1171_s_Odimension__sum__space,axiom,
! [K: set_set_int,N: nat,E: set_set_int,M: nat,F2: set_set_int,K2: nat] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ K @ F2 )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ K @ ( inf_inf_set_set_int @ E @ F2 ) )
=> ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K2 ) @ K @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ E @ F2 ) ) ) ) ) ) ).
% s.dimension_sum_space
thf(fact_1172_disjoint__insert_I2_J,axiom,
! [A2: set_nat_set_int,B: nat > set_int,B4: set_nat_set_int] :
( ( bot_bo8417611410066262939et_int
= ( inf_in1752217752563533465et_int @ A2 @ ( insert_nat_set_int @ B @ B4 ) ) )
= ( ~ ( member_nat_set_int @ B @ A2 )
& ( bot_bo8417611410066262939et_int
= ( inf_in1752217752563533465et_int @ A2 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1173_disjoint__insert_I2_J,axiom,
! [A2: set_set_int,B: set_int,B4: set_set_int] :
( ( bot_bot_set_set_int
= ( inf_inf_set_set_int @ A2 @ ( insert_set_int @ B @ B4 ) ) )
= ( ~ ( member_set_int @ B @ A2 )
& ( bot_bot_set_set_int
= ( inf_inf_set_set_int @ A2 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1174_disjoint__insert_I1_J,axiom,
! [B4: set_nat_set_int,A: nat > set_int,A2: set_nat_set_int] :
( ( ( inf_in1752217752563533465et_int @ B4 @ ( insert_nat_set_int @ A @ A2 ) )
= bot_bo8417611410066262939et_int )
= ( ~ ( member_nat_set_int @ A @ B4 )
& ( ( inf_in1752217752563533465et_int @ B4 @ A2 )
= bot_bo8417611410066262939et_int ) ) ) ).
% disjoint_insert(1)
thf(fact_1175_disjoint__insert_I1_J,axiom,
! [B4: set_set_int,A: set_int,A2: set_set_int] :
( ( ( inf_inf_set_set_int @ B4 @ ( insert_set_int @ A @ A2 ) )
= bot_bot_set_set_int )
= ( ~ ( member_set_int @ A @ B4 )
& ( ( inf_inf_set_set_int @ B4 @ A2 )
= bot_bot_set_set_int ) ) ) ).
% disjoint_insert(1)
thf(fact_1176_insert__disjoint_I2_J,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( bot_bo8417611410066262939et_int
= ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ A2 ) @ B4 ) )
= ( ~ ( member_nat_set_int @ A @ B4 )
& ( bot_bo8417611410066262939et_int
= ( inf_in1752217752563533465et_int @ A2 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1177_insert__disjoint_I2_J,axiom,
! [A: set_int,A2: set_set_int,B4: set_set_int] :
( ( bot_bot_set_set_int
= ( inf_inf_set_set_int @ ( insert_set_int @ A @ A2 ) @ B4 ) )
= ( ~ ( member_set_int @ A @ B4 )
& ( bot_bot_set_set_int
= ( inf_inf_set_set_int @ A2 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1178_insert__disjoint_I1_J,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ A2 ) @ B4 )
= bot_bo8417611410066262939et_int )
= ( ~ ( member_nat_set_int @ A @ B4 )
& ( ( inf_in1752217752563533465et_int @ A2 @ B4 )
= bot_bo8417611410066262939et_int ) ) ) ).
% insert_disjoint(1)
thf(fact_1179_insert__disjoint_I1_J,axiom,
! [A: set_int,A2: set_set_int,B4: set_set_int] :
( ( ( inf_inf_set_set_int @ ( insert_set_int @ A @ A2 ) @ B4 )
= bot_bot_set_set_int )
= ( ~ ( member_set_int @ A @ B4 )
& ( ( inf_inf_set_set_int @ A2 @ B4 )
= bot_bot_set_set_int ) ) ) ).
% insert_disjoint(1)
thf(fact_1180_Diff__disjoint,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( inf_inf_set_set_int @ A2 @ ( minus_8897228262479074673et_int @ B4 @ A2 ) )
= bot_bot_set_set_int ) ).
% Diff_disjoint
thf(fact_1181_Diff__triv,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ( inf_inf_set_set_int @ A2 @ B4 )
= bot_bot_set_set_int )
=> ( ( minus_8897228262479074673et_int @ A2 @ B4 )
= A2 ) ) ).
% Diff_triv
thf(fact_1182_Int__Diff__disjoint,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( inf_inf_set_set_int @ ( inf_inf_set_set_int @ A2 @ B4 ) @ ( minus_8897228262479074673et_int @ A2 @ B4 ) )
= bot_bot_set_set_int ) ).
% Int_Diff_disjoint
thf(fact_1183_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_1184_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_1185_dvd__diffD,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K2 @ M ) ) ) ) ).
% dvd_diffD
thf(fact_1186_dvd__diffD1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K2 @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_1187_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( dvd_dvd_nat @ M @ N )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1188_Int__mono,axiom,
! [A2: set_set_int,C3: set_set_int,B4: set_set_int,D2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ C3 )
=> ( ( ord_le4403425263959731960et_int @ B4 @ D2 )
=> ( ord_le4403425263959731960et_int @ ( inf_inf_set_set_int @ A2 @ B4 ) @ ( inf_inf_set_set_int @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1189_Int__lower1,axiom,
! [A2: set_set_int,B4: set_set_int] : ( ord_le4403425263959731960et_int @ ( inf_inf_set_set_int @ A2 @ B4 ) @ A2 ) ).
% Int_lower1
thf(fact_1190_Int__lower2,axiom,
! [A2: set_set_int,B4: set_set_int] : ( ord_le4403425263959731960et_int @ ( inf_inf_set_set_int @ A2 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_1191_Int__absorb1,axiom,
! [B4: set_set_int,A2: set_set_int] :
( ( ord_le4403425263959731960et_int @ B4 @ A2 )
=> ( ( inf_inf_set_set_int @ A2 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_1192_Int__absorb2,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ( inf_inf_set_set_int @ A2 @ B4 )
= A2 ) ) ).
% Int_absorb2
thf(fact_1193_Int__greatest,axiom,
! [C3: set_set_int,A2: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ C3 @ A2 )
=> ( ( ord_le4403425263959731960et_int @ C3 @ B4 )
=> ( ord_le4403425263959731960et_int @ C3 @ ( inf_inf_set_set_int @ A2 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_1194_Int__Collect__mono,axiom,
! [A2: set_nat_set_int,B4: set_nat_set_int,P: ( nat > set_int ) > $o,Q: ( nat > set_int ) > $o] :
( ( ord_le5995675665013768039et_int @ A2 @ B4 )
=> ( ! [X: nat > set_int] :
( ( member_nat_set_int @ X @ A2 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le5995675665013768039et_int @ ( inf_in1752217752563533465et_int @ A2 @ ( collect_nat_set_int @ P ) ) @ ( inf_in1752217752563533465et_int @ B4 @ ( collect_nat_set_int @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1195_Int__Collect__mono,axiom,
! [A2: set_set_int,B4: set_set_int,P: set_int > $o,Q: set_int > $o] :
( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ A2 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le4403425263959731960et_int @ ( inf_inf_set_set_int @ A2 @ ( collect_set_int @ P ) ) @ ( inf_inf_set_set_int @ B4 @ ( collect_set_int @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1196_dvd__diff__nat,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ M )
=> ( ( dvd_dvd_nat @ K2 @ N )
=> ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_1197_IntE,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( inf_in1752217752563533465et_int @ A2 @ B4 ) )
=> ~ ( ( member_nat_set_int @ C @ A2 )
=> ~ ( member_nat_set_int @ C @ B4 ) ) ) ).
% IntE
thf(fact_1198_IntE,axiom,
! [C: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ ( inf_inf_set_set_int @ A2 @ B4 ) )
=> ~ ( ( member_set_int @ C @ A2 )
=> ~ ( member_set_int @ C @ B4 ) ) ) ).
% IntE
thf(fact_1199_IntD1,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( inf_in1752217752563533465et_int @ A2 @ B4 ) )
=> ( member_nat_set_int @ C @ A2 ) ) ).
% IntD1
thf(fact_1200_IntD1,axiom,
! [C: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ ( inf_inf_set_set_int @ A2 @ B4 ) )
=> ( member_set_int @ C @ A2 ) ) ).
% IntD1
thf(fact_1201_IntD2,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( inf_in1752217752563533465et_int @ A2 @ B4 ) )
=> ( member_nat_set_int @ C @ B4 ) ) ).
% IntD2
thf(fact_1202_IntD2,axiom,
! [C: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ ( inf_inf_set_set_int @ A2 @ B4 ) )
=> ( member_set_int @ C @ B4 ) ) ).
% IntD2
thf(fact_1203_Int__assoc,axiom,
! [A2: set_set_int,B4: set_set_int,C3: set_set_int] :
( ( inf_inf_set_set_int @ ( inf_inf_set_set_int @ A2 @ B4 ) @ C3 )
= ( inf_inf_set_set_int @ A2 @ ( inf_inf_set_set_int @ B4 @ C3 ) ) ) ).
% Int_assoc
thf(fact_1204_Int__absorb,axiom,
! [A2: set_set_int] :
( ( inf_inf_set_set_int @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_1205_Int__commute,axiom,
( inf_inf_set_set_int
= ( ^ [A5: set_set_int,B5: set_set_int] : ( inf_inf_set_set_int @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_1206_Int__left__absorb,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( inf_inf_set_set_int @ A2 @ ( inf_inf_set_set_int @ A2 @ B4 ) )
= ( inf_inf_set_set_int @ A2 @ B4 ) ) ).
% Int_left_absorb
thf(fact_1207_Int__left__commute,axiom,
! [A2: set_set_int,B4: set_set_int,C3: set_set_int] :
( ( inf_inf_set_set_int @ A2 @ ( inf_inf_set_set_int @ B4 @ C3 ) )
= ( inf_inf_set_set_int @ B4 @ ( inf_inf_set_set_int @ A2 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_1208_Diff__Int__distrib2,axiom,
! [A2: set_set_int,B4: set_set_int,C3: set_set_int] :
( ( inf_inf_set_set_int @ ( minus_8897228262479074673et_int @ A2 @ B4 ) @ C3 )
= ( minus_8897228262479074673et_int @ ( inf_inf_set_set_int @ A2 @ C3 ) @ ( inf_inf_set_set_int @ B4 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1209_Diff__Int__distrib,axiom,
! [C3: set_set_int,A2: set_set_int,B4: set_set_int] :
( ( inf_inf_set_set_int @ C3 @ ( minus_8897228262479074673et_int @ A2 @ B4 ) )
= ( minus_8897228262479074673et_int @ ( inf_inf_set_set_int @ C3 @ A2 ) @ ( inf_inf_set_set_int @ C3 @ B4 ) ) ) ).
% Diff_Int_distrib
thf(fact_1210_Diff__Diff__Int,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ ( minus_8897228262479074673et_int @ A2 @ B4 ) )
= ( inf_inf_set_set_int @ A2 @ B4 ) ) ).
% Diff_Diff_Int
thf(fact_1211_Diff__Int2,axiom,
! [A2: set_set_int,C3: set_set_int,B4: set_set_int] :
( ( minus_8897228262479074673et_int @ ( inf_inf_set_set_int @ A2 @ C3 ) @ ( inf_inf_set_set_int @ B4 @ C3 ) )
= ( minus_8897228262479074673et_int @ ( inf_inf_set_set_int @ A2 @ C3 ) @ B4 ) ) ).
% Diff_Int2
thf(fact_1212_Int__Diff,axiom,
! [A2: set_set_int,B4: set_set_int,C3: set_set_int] :
( ( minus_8897228262479074673et_int @ ( inf_inf_set_set_int @ A2 @ B4 ) @ C3 )
= ( inf_inf_set_set_int @ A2 @ ( minus_8897228262479074673et_int @ B4 @ C3 ) ) ) ).
% Int_Diff
thf(fact_1213_Int__insert__left,axiom,
! [A: nat > set_int,C3: set_nat_set_int,B4: set_nat_set_int] :
( ( ( member_nat_set_int @ A @ C3 )
=> ( ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ B4 ) @ C3 )
= ( insert_nat_set_int @ A @ ( inf_in1752217752563533465et_int @ B4 @ C3 ) ) ) )
& ( ~ ( member_nat_set_int @ A @ C3 )
=> ( ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ B4 ) @ C3 )
= ( inf_in1752217752563533465et_int @ B4 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_1214_Int__insert__left,axiom,
! [A: set_int,C3: set_set_int,B4: set_set_int] :
( ( ( member_set_int @ A @ C3 )
=> ( ( inf_inf_set_set_int @ ( insert_set_int @ A @ B4 ) @ C3 )
= ( insert_set_int @ A @ ( inf_inf_set_set_int @ B4 @ C3 ) ) ) )
& ( ~ ( member_set_int @ A @ C3 )
=> ( ( inf_inf_set_set_int @ ( insert_set_int @ A @ B4 ) @ C3 )
= ( inf_inf_set_set_int @ B4 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_1215_Int__insert__right,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( ( member_nat_set_int @ A @ A2 )
=> ( ( inf_in1752217752563533465et_int @ A2 @ ( insert_nat_set_int @ A @ B4 ) )
= ( insert_nat_set_int @ A @ ( inf_in1752217752563533465et_int @ A2 @ B4 ) ) ) )
& ( ~ ( member_nat_set_int @ A @ A2 )
=> ( ( inf_in1752217752563533465et_int @ A2 @ ( insert_nat_set_int @ A @ B4 ) )
= ( inf_in1752217752563533465et_int @ A2 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1216_Int__insert__right,axiom,
! [A: set_int,A2: set_set_int,B4: set_set_int] :
( ( ( member_set_int @ A @ A2 )
=> ( ( inf_inf_set_set_int @ A2 @ ( insert_set_int @ A @ B4 ) )
= ( insert_set_int @ A @ ( inf_inf_set_set_int @ A2 @ B4 ) ) ) )
& ( ~ ( member_set_int @ A @ A2 )
=> ( ( inf_inf_set_set_int @ A2 @ ( insert_set_int @ A @ B4 ) )
= ( inf_inf_set_set_int @ A2 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1217_disjoint__iff__not__equal,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ( inf_inf_set_set_int @ A2 @ B4 )
= bot_bot_set_set_int )
= ( ! [X3: set_int] :
( ( member_set_int @ X3 @ A2 )
=> ! [Y5: set_int] :
( ( member_set_int @ Y5 @ B4 )
=> ( X3 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1218_Int__empty__right,axiom,
! [A2: set_set_int] :
( ( inf_inf_set_set_int @ A2 @ bot_bot_set_set_int )
= bot_bot_set_set_int ) ).
% Int_empty_right
thf(fact_1219_Int__empty__left,axiom,
! [B4: set_set_int] :
( ( inf_inf_set_set_int @ bot_bot_set_set_int @ B4 )
= bot_bot_set_set_int ) ).
% Int_empty_left
thf(fact_1220_disjoint__iff,axiom,
! [A2: set_nat_set_int,B4: set_nat_set_int] :
( ( ( inf_in1752217752563533465et_int @ A2 @ B4 )
= bot_bo8417611410066262939et_int )
= ( ! [X3: nat > set_int] :
( ( member_nat_set_int @ X3 @ A2 )
=> ~ ( member_nat_set_int @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1221_disjoint__iff,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ( inf_inf_set_set_int @ A2 @ B4 )
= bot_bot_set_set_int )
= ( ! [X3: set_int] :
( ( member_set_int @ X3 @ A2 )
=> ~ ( member_set_int @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1222_Int__emptyI,axiom,
! [A2: set_nat_set_int,B4: set_nat_set_int] :
( ! [X: nat > set_int] :
( ( member_nat_set_int @ X @ A2 )
=> ~ ( member_nat_set_int @ X @ B4 ) )
=> ( ( inf_in1752217752563533465et_int @ A2 @ B4 )
= bot_bo8417611410066262939et_int ) ) ).
% Int_emptyI
thf(fact_1223_Int__emptyI,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ! [X: set_int] :
( ( member_set_int @ X @ A2 )
=> ~ ( member_set_int @ X @ B4 ) )
=> ( ( inf_inf_set_set_int @ A2 @ B4 )
= bot_bot_set_set_int ) ) ).
% Int_emptyI
thf(fact_1224_ring_Osubring__inter,axiom,
! [R: partia4934656038542163276t_unit,I4: set_set_int,J3: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subrin7689096310803670856t_unit @ I4 @ R )
=> ( ( subrin7689096310803670856t_unit @ J3 @ R )
=> ( subrin7689096310803670856t_unit @ ( inf_inf_set_set_int @ I4 @ J3 ) @ R ) ) ) ) ).
% ring.subring_inter
thf(fact_1225_ring_Osubalgebra__inter,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,V3: set_set_int,V6: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( embedd2743979684206749024t_unit @ K @ V3 @ R )
=> ( ( embedd2743979684206749024t_unit @ K @ V6 @ R )
=> ( embedd2743979684206749024t_unit @ K @ ( inf_inf_set_set_int @ V3 @ V6 ) @ R ) ) ) ) ).
% ring.subalgebra_inter
thf(fact_1226_dvd__imp__le,axiom,
! [K2: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% dvd_imp_le
thf(fact_1227_linorder__neqE__linordered__idom,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1228_ring_Odimension__sum__space,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,N: nat,E: set_set_int,M: nat,F2: set_set_int,K2: nat] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( embedd646006463343340164t_unit @ R @ N @ K @ E )
=> ( ( embedd646006463343340164t_unit @ R @ M @ K @ F2 )
=> ( ( embedd646006463343340164t_unit @ R @ K2 @ K @ ( inf_inf_set_set_int @ E @ F2 ) )
=> ( embedd646006463343340164t_unit @ R @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K2 ) @ K @ ( set_ad273131178244904872t_unit @ R @ E @ F2 ) ) ) ) ) ) ) ).
% ring.dimension_sum_space
thf(fact_1229_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1230_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_1231_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_1232_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_1233_dvd__diff,axiom,
! [X2: int,Y: int,Z: int] :
( ( dvd_dvd_int @ X2 @ Y )
=> ( ( dvd_dvd_int @ X2 @ Z )
=> ( dvd_dvd_int @ X2 @ ( minus_minus_int @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_1234_ring_Odimension__direct__sum__space,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,N: nat,E: set_set_int,M: nat,F2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ( embedd646006463343340164t_unit @ R @ N @ K @ E )
=> ( ( embedd646006463343340164t_unit @ R @ M @ K @ F2 )
=> ( ( ( inf_inf_set_set_int @ E @ F2 )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ bot_bot_set_set_int ) )
=> ( embedd646006463343340164t_unit @ R @ ( plus_plus_nat @ N @ M ) @ K @ ( set_ad273131178244904872t_unit @ R @ E @ F2 ) ) ) ) ) ) ) ).
% ring.dimension_direct_sum_space
thf(fact_1235_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_1236_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_1237_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_1238_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_1239_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1240_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1241_add__less__zeroD,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X2 @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X2 @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_1242_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1243_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_1244_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1245_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_1246_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1247_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_1248_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_1249_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_1250_add__le__add__imp__diff__le,axiom,
! [I: nat,K2: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1251_add__le__add__imp__diff__le,axiom,
! [I: int,K2: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1252_add__le__imp__le__diff,axiom,
! [I: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_1253_add__le__imp__le__diff,axiom,
! [I: int,K2: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_1254_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_1255_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_1256_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_1257_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_1258_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1259_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1260_s_OsubcringI,axiom,
! [H: set_set_int] :
( ( subrin7689096310803670856t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ! [H12: set_int,H22: set_int] :
( ( member_set_int @ H12 @ H )
=> ( ( member_set_int @ H22 @ H )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H12 @ H22 )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H22 @ H12 ) ) ) )
=> ( subcri1024317279029940167t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.subcringI
thf(fact_1261_s_Osubcring__inter,axiom,
! [I4: set_set_int,J3: set_set_int] :
( ( subcri1024317279029940167t_unit @ I4 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( subcri1024317279029940167t_unit @ J3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( subcri1024317279029940167t_unit @ ( inf_inf_set_set_int @ I4 @ J3 ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.subcring_inter
thf(fact_1262_subcring_Oaxioms_I1_J,axiom,
! [H: set_set_int,R: partia4934656038542163276t_unit] :
( ( subcri1024317279029940167t_unit @ H @ R )
=> ( subrin7689096310803670856t_unit @ H @ R ) ) ).
% subcring.axioms(1)
thf(fact_1263_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_1264_ring_Osubcring__inter,axiom,
! [R: partia4934656038542163276t_unit,I4: set_set_int,J3: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( subcri1024317279029940167t_unit @ I4 @ R )
=> ( ( subcri1024317279029940167t_unit @ J3 @ R )
=> ( subcri1024317279029940167t_unit @ ( inf_inf_set_set_int @ I4 @ J3 ) @ R ) ) ) ) ).
% ring.subcring_inter
thf(fact_1265_subcringE_I1_J,axiom,
! [H: set_set_int,R: partia4934656038542163276t_unit] :
( ( subcri1024317279029940167t_unit @ H @ R )
=> ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ R ) ) ) ).
% subcringE(1)
thf(fact_1266_subcringE_I5_J,axiom,
! [H: set_set_int,R: partia4934656038542163276t_unit,H3: set_int] :
( ( subcri1024317279029940167t_unit @ H @ R )
=> ( ( member_set_int @ H3 @ H )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R @ H3 ) @ H ) ) ) ).
% subcringE(5)
thf(fact_1267_subcringE_I2_J,axiom,
! [H: set_set_int,R: partia4934656038542163276t_unit] :
( ( subcri1024317279029940167t_unit @ H @ R )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ H ) ) ).
% subcringE(2)
% 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,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ring_zfact_iso @ n @ zero_zero_nat )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
%------------------------------------------------------------------------------