TPTP Problem File: SLH0398^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Pluennecke_Ruzsa_Inequality/0003_Pluennecke_Ruzsa_Inequality/prob_00135_004411__12061722_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1405 ( 529 unt; 131 typ; 0 def)
% Number of atoms : 3662 (1182 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 11597 ( 377 ~; 46 |; 252 &;9158 @)
% ( 0 <=>;1764 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 17 ( 16 usr)
% Number of type conns : 622 ( 622 >; 0 *; 0 +; 0 <<)
% Number of symbols : 116 ( 115 usr; 25 con; 0-5 aty)
% Number of variables : 3385 ( 170 ^;3130 !; 85 ?;3385 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:19:45.134
%------------------------------------------------------------------------------
% Could-be-implicit typings (16)
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% Explicit typings (115)
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
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thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
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thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Set_Oremove_001tf__a,type,
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thf(sy_c_Set_Othe__elem_001tf__a,type,
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thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_member_001tf__a,type,
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thf(sy_v_B,type,
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% Relevant facts (1272)
thf(fact_0_False,axiom,
( ( inf_inf_set_a @ b @ g )
!= bot_bot_set_a ) ).
% False
thf(fact_1_assms,axiom,
~ ( finite_finite_a @ ( inf_inf_set_a @ a2 @ g ) ) ).
% assms
thf(fact_2_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( addition @ X @ Y )
= ( addition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_3_unit__closed,axiom,
member_a @ zero @ g ).
% unit_closed
thf(fact_4_associative,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B @ g )
=> ( ( member_a @ C @ g )
=> ( ( addition @ ( addition @ A @ B ) @ C )
= ( addition @ A @ ( addition @ B @ C ) ) ) ) ) ) ).
% associative
thf(fact_5_composition__closed,axiom,
! [A: a,B: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B @ g )
=> ( member_a @ ( addition @ A @ B ) @ g ) ) ) ).
% composition_closed
thf(fact_6_sumsetp_Ocases,axiom,
! [A2: a > $o,B2: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B2 @ A )
=> ~ ! [A3: a,B3: a] :
( ( A
= ( addition @ A3 @ B3 ) )
=> ( ( A2 @ A3 )
=> ( ( member_a @ A3 @ g )
=> ( ( B2 @ B3 )
=> ~ ( member_a @ B3 @ g ) ) ) ) ) ) ).
% sumsetp.cases
thf(fact_7_sumsetp_Osimps,axiom,
! [A2: a > $o,B2: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B2 @ A )
= ( ? [A4: a,B4: a] :
( ( A
= ( addition @ A4 @ B4 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ g )
& ( B2 @ B4 )
& ( member_a @ B4 @ g ) ) ) ) ).
% sumsetp.simps
thf(fact_8_sumsetp_OsumsetI,axiom,
! [A2: a > $o,A: a,B2: a > $o,B: a] :
( ( A2 @ A )
=> ( ( member_a @ A @ g )
=> ( ( B2 @ B )
=> ( ( member_a @ B @ g )
=> ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B2 @ ( addition @ A @ B ) ) ) ) ) ) ).
% sumsetp.sumsetI
thf(fact_9_sumset_Ocases,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) )
=> ~ ! [A3: a,B3: a] :
( ( A
= ( addition @ A3 @ B3 ) )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ g )
=> ( ( member_a @ B3 @ B2 )
=> ~ ( member_a @ B3 @ g ) ) ) ) ) ) ).
% sumset.cases
thf(fact_10_sumset_Osimps,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) )
= ( ? [A4: a,B4: a] :
( ( A
= ( addition @ A4 @ B4 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ g )
& ( member_a @ B4 @ B2 )
& ( member_a @ B4 @ g ) ) ) ) ).
% sumset.simps
thf(fact_11_sumset__commute,axiom,
! [A2: set_a,B2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 )
= ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ A2 ) ) ).
% sumset_commute
thf(fact_12_sumset__assoc,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ C2 )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ C2 ) ) ) ).
% sumset_assoc
thf(fact_13_sumset_OsumsetI,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( ( member_a @ B @ B2 )
=> ( ( member_a @ B @ g )
=> ( member_a @ ( addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ) ) ).
% sumset.sumsetI
thf(fact_14_local_Oinverse__unique,axiom,
! [U: a,V: a,V2: a] :
( ( ( addition @ U @ V )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( member_a @ V @ g )
=> ( V2 = V ) ) ) ) ) ) ).
% local.inverse_unique
thf(fact_15_finite__sumset,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ).
% finite_sumset
thf(fact_16_sumset__empty_H_I1_J,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ A2 )
= bot_bot_set_a ) ) ).
% sumset_empty'(1)
thf(fact_17_sumset__empty_H_I2_J,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% sumset_empty'(2)
thf(fact_18_finite__sumset_H,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B2 @ g ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ).
% finite_sumset'
thf(fact_19_additive__abelian__group__axioms,axiom,
pluenn1164192988769422572roup_a @ g @ addition @ zero ).
% additive_abelian_group_axioms
thf(fact_20_right__unit,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( addition @ A @ zero )
= A ) ) ).
% right_unit
thf(fact_21_left__unit,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( addition @ zero @ A )
= A ) ) ).
% left_unit
thf(fact_22_sumset__empty_I1_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% sumset_empty(1)
thf(fact_23_sumset__empty_I2_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% sumset_empty(2)
thf(fact_24_sumset__Int__carrier,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ g )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ).
% sumset_Int_carrier
thf(fact_25_sumset__Int__carrier__eq_I1_J,axiom,
! [A2: set_a,B2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( inf_inf_set_a @ B2 @ g ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ).
% sumset_Int_carrier_eq(1)
thf(fact_26_sumset__Int__carrier__eq_I2_J,axiom,
! [A2: set_a,B2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( inf_inf_set_a @ A2 @ g ) @ B2 )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ).
% sumset_Int_carrier_eq(2)
thf(fact_27_sumset__is__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B2 @ g )
= bot_bot_set_a ) ) ) ).
% sumset_is_empty_iff
thf(fact_28_commutative__monoid__axioms,axiom,
group_4866109990395492029noid_a @ g @ addition @ zero ).
% commutative_monoid_axioms
thf(fact_29_additive__abelian__group_Osumset_Ocong,axiom,
pluenn3038260743871226533mset_a = pluenn3038260743871226533mset_a ).
% additive_abelian_group.sumset.cong
thf(fact_30_additive__abelian__group_Osumsetp_Ocong,axiom,
pluenn895083305082786853setp_a = pluenn895083305082786853setp_a ).
% additive_abelian_group.sumsetp.cong
thf(fact_31_abelian__group__axioms,axiom,
group_201663378560352916roup_a @ g @ addition @ zero ).
% abelian_group_axioms
thf(fact_32_finite__Int,axiom,
! [F: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F @ G ) ) ) ).
% finite_Int
thf(fact_33_finite__Int,axiom,
! [F: set_a,G: set_a] :
( ( ( finite_finite_a @ F )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F @ G ) ) ) ).
% finite_Int
thf(fact_34_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_35_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_36_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_37_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_38_group__axioms,axiom,
group_group_a @ g @ addition @ zero ).
% group_axioms
thf(fact_39_sumset__subset__carrier,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ g ) ).
% sumset_subset_carrier
thf(fact_40_sumset__mono,axiom,
! [A5: set_a,A2: set_a,B5: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ B5 @ B2 )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ).
% sumset_mono
thf(fact_41_invertibleE,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ! [V3: a] :
( ( ( ( addition @ U @ V3 )
= zero )
& ( ( addition @ V3 @ U )
= zero ) )
=> ~ ( member_a @ V3 @ g ) )
=> ~ ( member_a @ U @ g ) ) ) ).
% invertibleE
thf(fact_42_invertible__def,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
= ( ? [X2: a] :
( ( member_a @ X2 @ g )
& ( ( addition @ U @ X2 )
= zero )
& ( ( addition @ X2 @ U )
= zero ) ) ) ) ) ).
% invertible_def
thf(fact_43_unit__invertible,axiom,
group_invertible_a @ g @ addition @ zero @ zero ).
% unit_invertible
thf(fact_44_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_inf__right__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_47_inf_Oright__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ B )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.right_idem
thf(fact_48_inf__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_49_inf_Oleft__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.left_idem
thf(fact_50_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_51_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_52_inf_Obounded__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_53_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_54_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_55_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_56_invertible__right__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z @ g )
=> ( ( ( addition @ Y @ X )
= ( addition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% invertible_right_cancel
thf(fact_57_invertible__left__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z @ g )
=> ( ( ( addition @ X @ Y )
= ( addition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ).
% invertible_left_cancel
thf(fact_58_invertibleI,axiom,
! [U: a,V2: a] :
( ( ( addition @ U @ V2 )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ) ) ) ).
% invertibleI
thf(fact_59_invertible,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ).
% invertible
thf(fact_60_composition__invertible,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( group_invertible_a @ g @ addition @ zero @ Y )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( group_invertible_a @ g @ addition @ zero @ ( addition @ X @ Y ) ) ) ) ) ) ).
% composition_invertible
thf(fact_61_additive__abelian__group__def,axiom,
pluenn1164192988769422572roup_a = group_201663378560352916roup_a ).
% additive_abelian_group_def
thf(fact_62_additive__abelian__group_Oaxioms,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( group_201663378560352916roup_a @ G @ Addition @ Zero ) ) ).
% additive_abelian_group.axioms
thf(fact_63_additive__abelian__group_Ointro,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( group_201663378560352916roup_a @ G @ Addition @ Zero )
=> ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero ) ) ).
% additive_abelian_group.intro
thf(fact_64_additive__abelian__group_Osumset__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) @ G ) ) ).
% additive_abelian_group.sumset_subset_carrier
thf(fact_65_additive__abelian__group_Osumset__mono,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A5: set_a,A2: set_a,B5: set_a,B2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ B5 @ B2 )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) ) ) ) ) ).
% additive_abelian_group.sumset_mono
thf(fact_66_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_67_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_68_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_69_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_70_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_71_rev__finite__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_72_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_73_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_74_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_75_finite__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_76_inf_OcoboundedI2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_77_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_78_inf_OcoboundedI1,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_79_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_80_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_81_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_82_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_83_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_84_inf_Ocobounded2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_85_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_86_inf_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_87_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_88_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_89_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( A4
= ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_90_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_91_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_92_inf_OboundedI,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_93_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_94_inf_OboundedE,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_95_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_96_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_97_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_98_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_99_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_100_inf_Oabsorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_101_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_102_inf_Oabsorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_103_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_104_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_105_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( inf_inf_nat @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_106_inf__unique,axiom,
! [F2: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F2 @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F2 @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ X3 @ Z2 )
=> ( ord_less_eq_set_a @ X3 @ ( F2 @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_107_inf__unique,axiom,
! [F2: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F2 @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F2 @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ ( F2 @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_108_inf_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% inf.orderI
thf(fact_109_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_110_inf_OorderE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( A
= ( inf_inf_set_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_111_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_112_le__infI2,axiom,
! [B: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_113_le__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_114_le__infI1,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_115_le__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_116_inf__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_117_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_118_le__infI,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_119_le__infI,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_120_le__infE,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B ) ) ) ).
% le_infE
thf(fact_121_le__infE,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_122_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_123_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_124_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_125_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_126_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_127_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_128_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_129_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_130_additive__abelian__group_Osumset__commute,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ B2 @ A2 ) ) ) ).
% additive_abelian_group.sumset_commute
thf(fact_131_additive__abelian__group_Osumset_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B: a,B2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ G )
=> ( ( member_a @ B @ B2 )
=> ( ( member_a @ B @ G )
=> ( member_a @ ( Addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.sumsetI
thf(fact_132_additive__abelian__group_Osumset__assoc,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a,C2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) @ C2 )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B2 @ C2 ) ) ) ) ).
% additive_abelian_group.sumset_assoc
thf(fact_133_additive__abelian__group_Osumset_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) )
= ( ? [A4: a,B4: a] :
( ( A
= ( Addition @ A4 @ B4 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ G )
& ( member_a @ B4 @ B2 )
& ( member_a @ B4 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset.simps
thf(fact_134_additive__abelian__group_Osumset_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) )
=> ~ ! [A3: a,B3: a] :
( ( A
= ( Addition @ A3 @ B3 ) )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ G )
=> ( ( member_a @ B3 @ B2 )
=> ~ ( member_a @ B3 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.cases
thf(fact_135_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_136_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_137_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_138_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_139_additive__abelian__group_Osumsetp_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,A: a,B2: a > $o,B: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( A2 @ A )
=> ( ( member_a @ A @ G )
=> ( ( B2 @ B )
=> ( ( member_a @ B @ G )
=> ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B2 @ ( Addition @ A @ B ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.sumsetI
thf(fact_140_additive__abelian__group_Osumsetp_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,B2: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B2 @ A )
= ( ? [A4: a,B4: a] :
( ( A
= ( Addition @ A4 @ B4 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ G )
& ( B2 @ B4 )
& ( member_a @ B4 @ G ) ) ) ) ) ).
% additive_abelian_group.sumsetp.simps
thf(fact_141_additive__abelian__group_Osumsetp_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,B2: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B2 @ A )
=> ~ ! [A3: a,B3: a] :
( ( A
= ( Addition @ A3 @ B3 ) )
=> ( ( A2 @ A3 )
=> ( ( member_a @ A3 @ G )
=> ( ( B2 @ B3 )
=> ~ ( member_a @ B3 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.cases
thf(fact_142_additive__abelian__group_Osumset__empty_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ) ).
% additive_abelian_group.sumset_empty(1)
thf(fact_143_additive__abelian__group_Osumset__empty_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ) ).
% additive_abelian_group.sumset_empty(2)
thf(fact_144_additive__abelian__group_Ofinite__sumset,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B2: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B2 ) ) ) ) ) ).
% additive_abelian_group.finite_sumset
thf(fact_145_additive__abelian__group_Ofinite__sumset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) ) ) ) ) ).
% additive_abelian_group.finite_sumset
thf(fact_146_additive__abelian__group_Osumset__Int__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) @ G )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) ) ) ).
% additive_abelian_group.sumset_Int_carrier
thf(fact_147_additive__abelian__group_Osumset__Int__carrier__eq_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( inf_inf_set_a @ B2 @ G ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) ) ) ).
% additive_abelian_group.sumset_Int_carrier_eq(1)
thf(fact_148_additive__abelian__group_Osumset__Int__carrier__eq_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( inf_inf_set_a @ A2 @ G ) @ B2 )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) ) ) ).
% additive_abelian_group.sumset_Int_carrier_eq(2)
thf(fact_149_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_150_inf_Oleft__commute,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_151_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_a,K: set_a,B: set_a,A: set_a] :
( ( B2
= ( inf_inf_set_a @ K @ B ) )
=> ( ( inf_inf_set_a @ A @ B2 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_152_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_a,K: set_a,A: set_a,B: set_a] :
( ( A2
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_153_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X2 ) ) ) ).
% inf_commute
thf(fact_154_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_155_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_156_inf_Oassoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_157_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_158_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_159_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_160_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_161_additive__abelian__group_Osumset__is__empty__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B2 @ G )
= bot_bot_set_a ) ) ) ) ).
% additive_abelian_group.sumset_is_empty_iff
thf(fact_162_additive__abelian__group_Osumset__empty_H_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ B2 @ A2 )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_empty'(1)
thf(fact_163_additive__abelian__group_Osumset__empty_H_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_empty'(2)
thf(fact_164_additive__abelian__group_Ofinite__sumset_H,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,B2: set_nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ ( inf_inf_set_nat @ A2 @ G ) )
=> ( ( finite_finite_nat @ ( inf_inf_set_nat @ B2 @ G ) )
=> ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ B2 ) ) ) ) ) ).
% additive_abelian_group.finite_sumset'
thf(fact_165_additive__abelian__group_Ofinite__sumset_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ G ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B2 @ G ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) ) ) ) ) ).
% additive_abelian_group.finite_sumset'
thf(fact_166_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_167_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_168_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_169_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_170_mem__UnitsI,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) ) ) ) ).
% mem_UnitsI
thf(fact_171_mem__UnitsD,axiom,
! [U: a] :
( ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
& ( member_a @ U @ g ) ) ) ).
% mem_UnitsD
thf(fact_172_Int__subset__iff,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( ord_less_eq_set_a @ C2 @ A2 )
& ( ord_less_eq_set_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_173_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_174_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_175_sumset__subset__insert_I1_J,axiom,
! [A2: set_a,B2: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B2 ) ) ) ).
% sumset_subset_insert(1)
thf(fact_176_sumset__subset__insert_I2_J,axiom,
! [A2: set_a,B2: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B2 ) ) ).
% sumset_subset_insert(2)
thf(fact_177_group__of__Units,axiom,
group_group_a @ ( group_Units_a @ g @ addition @ zero ) @ addition @ zero ).
% group_of_Units
thf(fact_178_sumset__subset__Un_I1_J,axiom,
! [A2: set_a,B2: set_a,C2: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% sumset_subset_Un(1)
thf(fact_179_sumset__subset__Un_I2_J,axiom,
! [A2: set_a,B2: set_a,C2: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ C2 ) @ B2 ) ) ).
% sumset_subset_Un(2)
thf(fact_180_abelian__group__def,axiom,
( group_201663378560352916roup_a
= ( ^ [G2: set_a,Composition: a > a > a,Unit: a] :
( ( group_group_a @ G2 @ Composition @ Unit )
& ( group_4866109990395492029noid_a @ G2 @ Composition @ Unit ) ) ) ) ).
% abelian_group_def
thf(fact_181_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_182_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_183_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_184_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_185_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_186_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_187_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_188_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_189_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_190_sup_Oright__idem,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ B )
= ( sup_sup_set_a @ A @ B ) ) ).
% sup.right_idem
thf(fact_191_sup__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_192_sup_Oleft__idem,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% sup.left_idem
thf(fact_193_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_194_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_195_IntI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_196_Int__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_197_UnCI,axiom,
! [C: a,B2: set_a,A2: set_a] :
( ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ A2 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_198_Un__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
| ( member_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_199_sumset__subset__Un2,axiom,
! [A2: set_a,B2: set_a,B5: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B2 @ B5 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B5 ) ) ) ).
% sumset_subset_Un2
thf(fact_200_sumset__subset__Un1,axiom,
! [A2: set_a,A5: set_a,B2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ A5 ) @ B2 )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B2 ) ) ) ).
% sumset_subset_Un1
thf(fact_201_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_202_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_203_sup_Obounded__iff,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
= ( ( ord_less_eq_set_a @ B @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_204_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_205_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_206_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_207_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_208_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_209_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_210_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_211_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_212_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B ) )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_213_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_214_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_215_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_216_insert__subset,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( ( member_a @ X @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_217_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_218_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_219_Un__empty,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_220_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B2: set_a] :
( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_221_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B2: set_a] :
( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_222_insert__inter__insert,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_223_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_224_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_225_finite__Un,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) )
= ( ( finite_finite_nat @ F )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_226_finite__Un,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) )
= ( ( finite_finite_a @ F )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_227_Un__subset__iff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_a @ A2 @ C2 )
& ( ord_less_eq_set_a @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_228_Un__insert__left,axiom,
! [A: a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( insert_a @ A @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_229_Un__insert__right,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_230_Int__Un__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( sup_sup_set_a @ T @ ( inf_inf_set_a @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_231_Int__Un__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_232_Int__Un__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_233_Int__Un__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_234_Un__Int__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( inf_inf_set_a @ T @ ( sup_sup_set_a @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_235_Un__Int__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_236_Un__Int__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_237_Un__Int__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_238_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_239_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_240_disjoint__insert_I2_J,axiom,
! [A2: set_a,B: a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B @ B2 ) ) )
= ( ~ ( member_a @ B @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_241_disjoint__insert_I1_J,axiom,
! [B2: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B2 @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ B2 @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_242_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member_a @ A @ B2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_243_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B2 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_244_sumset__D_I2_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ zero @ bot_bot_set_a ) @ A2 )
= ( inf_inf_set_a @ A2 @ g ) ) ).
% sumset_D(2)
thf(fact_245_sumset__D_I1_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A2 @ g ) ) ).
% sumset_D(1)
thf(fact_246_insert__is__Un,axiom,
( insert_a
= ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_247_Un__singleton__iff,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2 = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_248_singleton__Un__iff,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2 = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_249_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_250_sup_Oleft__commute,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_251_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_a,K: set_a,B: set_a,A: set_a] :
( ( B2
= ( sup_sup_set_a @ K @ B ) )
=> ( ( sup_sup_set_a @ A @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_252_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_a,K: set_a,A: set_a,B: set_a] :
( ( A2
= ( sup_sup_set_a @ K @ A ) )
=> ( ( sup_sup_set_a @ A2 @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_253_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( sup_sup_set_a @ Y2 @ X2 ) ) ) ).
% sup_commute
thf(fact_254_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_255_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_256_sup_Oassoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sup.assoc
thf(fact_257_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( sup_sup_set_a @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_258_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_259_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_260_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_261_monoid_OUnits_Ocong,axiom,
group_Units_a = group_Units_a ).
% monoid.Units.cong
thf(fact_262_UnE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ~ ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_263_UnI1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_264_UnI2,axiom,
! [C: a,B2: set_a,A2: set_a] :
( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_265_bex__Un,axiom,
! [A2: set_a,B2: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: a] :
( ( member_a @ X2 @ B2 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_266_ball__Un,axiom,
! [A2: set_a,B2: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_267_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_268_Un__assoc,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_269_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a @ A @ ( insert_a @ A @ B2 ) ) ).
% insertI1
thf(fact_270_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a @ A @ B2 )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_271_Un__absorb,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_272_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] : ( sup_sup_set_a @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_273_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B7: set_a] :
( ( A2
= ( insert_a @ X @ B7 ) )
=> ( member_a @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_274_insert__ident,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_275_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_276_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_a] :
( ( A2
= ( insert_a @ B @ C3 ) )
& ~ ( member_a @ B @ C3 )
& ( B2
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_277_Un__left__absorb,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_278_insert__commute,axiom,
! [X: a,Y: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A2 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_279_Un__left__commute,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) )
= ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_280_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B7: set_a] :
( ( A2
= ( insert_a @ A @ B7 ) )
& ~ ( member_a @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_281_Un__empty__left,axiom,
! [B2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_282_Un__empty__right,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_283_Un__mono,axiom,
! [A2: set_a,C2: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_284_Un__least,axiom,
! [A2: set_a,C2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_285_Un__upper1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_286_Un__upper2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_287_Un__absorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_288_Un__absorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_289_subset__UnE,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A2 )
=> ! [B8: set_a] :
( ( ord_less_eq_set_a @ B8 @ B2 )
=> ( C2
!= ( sup_sup_set_a @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_290_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( sup_sup_set_a @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_291_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_292_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_293_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_294_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_295_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_296_insert__mono,axiom,
! [C2: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_297_subset__insert,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_298_subset__insertI,axiom,
! [B2: set_a,A: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_299_subset__insertI2,axiom,
! [A2: set_a,B2: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_300_Un__Int__crazy,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ B2 @ C2 ) ) @ ( inf_inf_set_a @ C2 @ A2 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ B2 @ C2 ) ) @ ( sup_sup_set_a @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_301_Int__Un__distrib,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_302_Un__Int__distrib,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_303_Int__Un__distrib2,axiom,
! [B2: set_a,C2: set_a,A2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A2 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B2 @ A2 ) @ ( inf_inf_set_a @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_304_Un__Int__distrib2,axiom,
! [B2: set_a,C2: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B2 @ C2 ) @ A2 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B2 @ A2 ) @ ( sup_sup_set_a @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_305_Int__insert__left,axiom,
! [A: a,C2: set_a,B2: set_a] :
( ( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) ) ) )
& ( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_306_Int__insert__right,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_307_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_308_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_309_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_310_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_311_le__supE,axiom,
! [A: set_a,B: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A @ X )
=> ~ ( ord_less_eq_set_a @ B @ X ) ) ) ).
% le_supE
thf(fact_312_le__supE,axiom,
! [A: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_313_le__supI,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_314_le__supI,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_315_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_316_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_317_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_318_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_319_le__supI1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI1
thf(fact_320_le__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_321_le__supI2,axiom,
! [X: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI2
thf(fact_322_le__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_323_sup_Omono,axiom,
! [C: set_a,A: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_324_sup_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_325_sup__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_326_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_327_sup__least,axiom,
! [Y: set_a,X: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_328_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_329_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X2 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_330_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( sup_sup_nat @ X2 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_331_sup_OorderE,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( A
= ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.orderE
thf(fact_332_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_333_sup_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% sup.orderI
thf(fact_334_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_335_sup__unique,axiom,
! [F2: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ( ( ord_less_eq_set_a @ Z2 @ X3 )
=> ( ord_less_eq_set_a @ ( F2 @ Y3 @ Z2 ) @ X3 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_336_sup__unique,axiom,
! [F2: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ( ord_less_eq_nat @ ( F2 @ Y3 @ Z2 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_337_sup_Oabsorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_338_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_339_sup_Oabsorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_340_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_341_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_342_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_343_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_344_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_345_sup_OboundedE,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B @ A )
=> ~ ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_346_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_347_sup_OboundedI,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_348_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_349_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_350_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_351_sup_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded1
thf(fact_352_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_353_sup_Ocobounded2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded2
thf(fact_354_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_355_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_356_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_357_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_358_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_359_sup_OcoboundedI1,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_360_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_361_sup_OcoboundedI2,axiom,
! [C: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_362_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_363_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_364_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ ( inf_inf_set_a @ X3 @ Z2 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_365_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y3 ) @ ( sup_sup_set_a @ X3 @ Z2 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_366_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_367_inf__sup__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_368_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_369_sup__inf__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_370_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_371_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_372_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_373_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_374_finite__UnI,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ F )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_375_finite__UnI,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ F )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_376_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_377_Un__infinite,axiom,
! [S: set_a,T: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_378_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_379_infinite__Un,axiom,
! [S: set_a,T: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T ) ) ) ).
% infinite_Un
thf(fact_380_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_381_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_382_subset__singleton__iff,axiom,
! [X4: set_a,A: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_383_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_384_Un__Int__assoc__eq,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_385_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_386_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_387_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_388_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_389_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A8: set_nat] :
( ? [A3: nat] :
( A
= ( insert_nat @ A3 @ A8 ) )
=> ~ ( finite_finite_nat @ A8 ) ) ) ) ).
% finite.cases
thf(fact_390_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A8: set_a] :
( ? [A3: a] :
( A
= ( insert_a @ A3 @ A8 ) )
=> ~ ( finite_finite_a @ A8 ) ) ) ) ).
% finite.cases
thf(fact_391_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A6: set_nat,B4: nat] :
( ( A4
= ( insert_nat @ B4 @ A6 ) )
& ( finite_finite_nat @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_392_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A6: set_a,B4: a] :
( ( A4
= ( insert_a @ B4 @ A6 ) )
& ( finite_finite_a @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_393_finite__induct,axiom,
! [F: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_394_finite__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_395_finite__ne__induct,axiom,
! [F: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( F != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_396_finite__ne__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( F != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_397_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A8: set_nat] :
( ~ ( finite_finite_nat @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_398_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A8: set_a] :
( ~ ( finite_finite_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_399_additive__abelian__group_Osumset__subset__Un1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A5: set_a,B2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( sup_sup_set_a @ A2 @ A5 ) @ B2 )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A5 @ B2 ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un1
thf(fact_400_additive__abelian__group_Osumset__subset__Un2,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a,B5: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( sup_sup_set_a @ B2 @ B5 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B5 ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un2
thf(fact_401_finite__subset__induct,axiom,
! [F: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A3 @ A2 )
=> ( ~ ( member_nat @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_402_finite__subset__induct,axiom,
! [F: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A2 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_403_finite__subset__induct_H,axiom,
! [F: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A3 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_404_finite__subset__induct_H,axiom,
! [F: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_405_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_406_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_407_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_408_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_409_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_410_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_411_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_412_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_413_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( member_a @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_414_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_415_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_416_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A6 )
=> ( member_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_417_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_418_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_419_subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_420_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_421_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_422_additive__abelian__group_Osumset__subset__Un_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a,C2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ) ).
% additive_abelian_group.sumset_subset_Un(1)
thf(fact_423_additive__abelian__group_Osumset__subset__Un_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a,C2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( sup_sup_set_a @ A2 @ C2 ) @ B2 ) ) ) ).
% additive_abelian_group.sumset_subset_Un(2)
thf(fact_424_additive__abelian__group_Osumset__subset__insert_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ B2 ) ) ) ) ).
% additive_abelian_group.sumset_subset_insert(1)
thf(fact_425_additive__abelian__group_Osumset__subset__insert_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A2 ) @ B2 ) ) ) ).
% additive_abelian_group.sumset_subset_insert(2)
thf(fact_426_IntE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_427_IntD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_428_IntD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ B2 ) ) ).
% IntD2
thf(fact_429_Int__assoc,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_430_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_431_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] : ( inf_inf_set_a @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_432_Int__left__absorb,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_433_Int__left__commute,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_434_monoid_Oinvertible_Ocong,axiom,
group_invertible_a = group_invertible_a ).
% monoid.invertible.cong
thf(fact_435_additive__abelian__group_Osumset__D_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ Zero @ bot_bot_set_a ) @ A2 )
= ( inf_inf_set_a @ A2 @ G ) ) ) ).
% additive_abelian_group.sumset_D(2)
thf(fact_436_additive__abelian__group_Osumset__D_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ Zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A2 @ G ) ) ) ).
% additive_abelian_group.sumset_D(1)
thf(fact_437_commutative__monoid_Ocommutative,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,X: a,Y: a] :
( ( group_4866109990395492029noid_a @ M @ Composition2 @ Unit2 )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( Composition2 @ X @ Y )
= ( Composition2 @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_438_Int__emptyI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_439_disjoint__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_440_Int__empty__left,axiom,
! [B2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B2 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_441_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_442_disjoint__iff__not__equal,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ B2 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_443_Int__mono,axiom,
! [A2: set_a,C2: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_444_Int__lower1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_445_Int__lower2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_446_Int__absorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_447_Int__absorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_448_Int__greatest,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_449_Int__Collect__mono,axiom,
! [A2: set_a,B2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_450_group_Oinvertible,axiom,
! [G: set_a,Composition2: a > a > a,Unit2: a,U: a] :
( ( group_group_a @ G @ Composition2 @ Unit2 )
=> ( ( member_a @ U @ G )
=> ( group_invertible_a @ G @ Composition2 @ Unit2 @ U ) ) ) ).
% group.invertible
thf(fact_451_abelian__group_Oaxioms_I1_J,axiom,
! [G: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_201663378560352916roup_a @ G @ Composition2 @ Unit2 )
=> ( group_group_a @ G @ Composition2 @ Unit2 ) ) ).
% abelian_group.axioms(1)
thf(fact_452_abelian__group_Oaxioms_I2_J,axiom,
! [G: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_201663378560352916roup_a @ G @ Composition2 @ Unit2 )
=> ( group_4866109990395492029noid_a @ G @ Composition2 @ Unit2 ) ) ).
% abelian_group.axioms(2)
thf(fact_453_abelian__group_Ointro,axiom,
! [G: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_group_a @ G @ Composition2 @ Unit2 )
=> ( ( group_4866109990395492029noid_a @ G @ Composition2 @ Unit2 )
=> ( group_201663378560352916roup_a @ G @ Composition2 @ Unit2 ) ) ) ).
% abelian_group.intro
thf(fact_454_sumset__insert1,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ A2 )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B2 )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ bot_bot_set_a ) @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ).
% sumset_insert1
thf(fact_455_sumset__insert2,axiom,
! [B2: set_a,A2: set_a,X: a] :
( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ B2 )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B2 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ).
% sumset_insert2
thf(fact_456_card__sumset__le,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ).
% card_sumset_le
thf(fact_457_sumsetdiff__sing,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% sumsetdiff_sing
thf(fact_458_inverse__composition__commute,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( group_invertible_a @ g @ addition @ zero @ Y )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ ( addition @ X @ Y ) )
= ( addition @ ( group_inverse_a @ g @ addition @ zero @ Y ) @ ( group_inverse_a @ g @ addition @ zero @ X ) ) ) ) ) ) ) ).
% inverse_composition_commute
thf(fact_459_invertible__left__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ ( addition @ U @ V2 ) )
= V2 ) ) ) ) ).
% invertible_left_inverse2
thf(fact_460_invertible__right__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( addition @ U @ ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ V2 ) )
= V2 ) ) ) ) ).
% invertible_right_inverse2
thf(fact_461_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F2: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F2 @ Y5 ) @ ( F2 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_462_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F2: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F2 @ Y5 ) @ ( F2 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_463_inverse__closed,axiom,
! [X: a] :
( ( member_a @ X @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ X ) @ g ) ) ).
% inverse_closed
thf(fact_464_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_465_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_466_Diff__idemp,axiom,
! [A2: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_467_inverse__equality,axiom,
! [U: a,V2: a] :
( ( ( addition @ U @ V2 )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ U )
= V2 ) ) ) ) ) ).
% inverse_equality
thf(fact_468_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_469_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_470_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_471_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_472_finite__Diff2,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_473_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_474_finite__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_475_insert__Diff1,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_476_Diff__insert0,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_477_Un__Diff__cancel2,axiom,
! [B2: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_a @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_478_Un__Diff__cancel,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B2 @ A2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_479_Diff__eq__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_480_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_481_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_482_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_483_Diff__disjoint,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B2 @ A2 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_484_inverse__unit,axiom,
( ( group_inverse_a @ g @ addition @ zero @ zero )
= zero ) ).
% inverse_unit
thf(fact_485_invertible__right__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( addition @ U @ ( group_inverse_a @ g @ addition @ zero @ U ) )
= zero ) ) ) ).
% invertible_right_inverse
thf(fact_486_invertible__left__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ U )
= zero ) ) ) ).
% invertible_left_inverse
thf(fact_487_invertible__inverse__invertible,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ ( group_inverse_a @ g @ addition @ zero @ U ) ) ) ) ).
% invertible_inverse_invertible
thf(fact_488_invertible__inverse__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ ( group_inverse_a @ g @ addition @ zero @ U ) )
= U ) ) ) ).
% invertible_inverse_inverse
thf(fact_489_invertible__inverse__closed,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ U ) @ g ) ) ) ).
% invertible_inverse_closed
thf(fact_490_card__le__sym__Diff,axiom,
! [A2: set_Product_unit,B2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A2 )
=> ( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A2 ) @ ( finite410649719033368117t_unit @ B2 ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ B2 ) ) @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_491_card__le__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_492_card__le__sym__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_493_monoid_Oinverse_Ocong,axiom,
group_inverse_a = group_inverse_a ).
% monoid.inverse.cong
thf(fact_494_DiffE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_495_DiffD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_496_DiffD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_497_card__Diff1__le,axiom,
! [A2: set_Product_unit,X: product_unit] : ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A2 ) ) ).
% card_Diff1_le
thf(fact_498_card__Diff1__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ).
% card_Diff1_le
thf(fact_499_card__insert__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( insert_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_500_card__insert__le,axiom,
! [A2: set_Product_unit,X: product_unit] : ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A2 ) @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_501_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_502_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_503_double__diff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_504_Diff__subset,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_505_Diff__mono,axiom,
! [A2: set_a,C2: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_506_insert__Diff__if,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_507_Diff__Int__distrib2,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C2 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C2 ) @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_508_Diff__Int__distrib,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C2 @ A2 ) @ ( inf_inf_set_a @ C2 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_509_Diff__Diff__Int,axiom,
! [A2: set_a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_510_Diff__Int2,axiom,
! [A2: set_a,C2: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C2 ) @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C2 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_511_Int__Diff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C2 ) ) ) ).
% Int_Diff
thf(fact_512_Un__Diff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ C2 ) @ ( minus_minus_set_a @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_513_additive__abelian__group_Oinverse__closed,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ X @ G )
=> ( member_a @ ( group_inverse_a @ G @ Addition @ Zero @ X ) @ G ) ) ) ).
% additive_abelian_group.inverse_closed
thf(fact_514_card__subset__eq,axiom,
! [B2: set_Product_unit,A2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ord_le3507040750410214029t_unit @ A2 @ B2 )
=> ( ( ( finite410649719033368117t_unit @ A2 )
= ( finite410649719033368117t_unit @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_515_card__subset__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_516_card__subset__eq,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_517_infinite__arbitrarily__large,axiom,
! [A2: set_Product_unit,N: nat] :
( ~ ( finite4290736615968046902t_unit @ A2 )
=> ? [B7: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B7 )
& ( ( finite410649719033368117t_unit @ B7 )
= N )
& ( ord_le3507040750410214029t_unit @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_518_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N )
& ( ord_less_eq_set_nat @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_519_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B7: set_a] :
( ( finite_finite_a @ B7 )
& ( ( finite_card_a @ B7 )
= N )
& ( ord_less_eq_set_a @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_520_card__mono,axiom,
! [B2: set_Product_unit,A2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ord_le3507040750410214029t_unit @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A2 ) @ ( finite410649719033368117t_unit @ B2 ) ) ) ) ).
% card_mono
thf(fact_521_card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_522_card__mono,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_523_card__seteq,axiom,
! [B2: set_Product_unit,A2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ord_le3507040750410214029t_unit @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B2 ) @ ( finite410649719033368117t_unit @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_524_card__seteq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_525_card__seteq,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_526_exists__subset__between,axiom,
! [A2: set_Product_unit,N: nat,C2: set_Product_unit] :
( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite410649719033368117t_unit @ C2 ) )
=> ( ( ord_le3507040750410214029t_unit @ A2 @ C2 )
=> ( ( finite4290736615968046902t_unit @ C2 )
=> ? [B7: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ B7 )
& ( ord_le3507040750410214029t_unit @ B7 @ C2 )
& ( ( finite410649719033368117t_unit @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_527_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C2 )
& ( ( finite_card_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_528_exists__subset__between,axiom,
! [A2: set_a,N: nat,C2: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C2 ) )
=> ( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( finite_finite_a @ C2 )
=> ? [B7: set_a] :
( ( ord_less_eq_set_a @ A2 @ B7 )
& ( ord_less_eq_set_a @ B7 @ C2 )
& ( ( finite_card_a @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_529_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_Product_unit] :
( ( ord_less_eq_nat @ N @ ( finite410649719033368117t_unit @ S ) )
=> ~ ! [T3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ T3 @ S )
=> ( ( ( finite410649719033368117t_unit @ T3 )
= N )
=> ~ ( finite4290736615968046902t_unit @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_530_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_531_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S )
=> ( ( ( finite_card_a @ T3 )
= N )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_532_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_Product_unit,C2: nat] :
( ! [G3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ G3 @ F )
=> ( ( finite4290736615968046902t_unit @ G3 )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ G3 ) @ C2 ) ) )
=> ( ( finite4290736615968046902t_unit @ F )
& ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ F ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_533_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_nat,C2: nat] :
( ! [G3: set_nat] :
( ( ord_less_eq_set_nat @ G3 @ F )
=> ( ( finite_finite_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_534_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_a,C2: nat] :
( ! [G3: set_a] :
( ( ord_less_eq_set_a @ G3 @ F )
=> ( ( finite_finite_a @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G3 ) @ C2 ) ) )
=> ( ( finite_finite_a @ F )
& ( ord_less_eq_nat @ ( finite_card_a @ F ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_535_diff__shunt__var,axiom,
! [X: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_536_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_537_Diff__insert2,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_538_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_539_Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_540_subset__Diff__insert,axiom,
! [A2: set_a,B2: set_a,X: a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ ( insert_a @ X @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C2 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_541_Int__Diff__disjoint,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ A2 @ B2 ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_542_Diff__triv,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_543_Diff__partition,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_544_Diff__subset__conv,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C2 )
= ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_545_Un__Diff__Int,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_546_Int__Diff__Un,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_547_Diff__Int,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( minus_minus_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_548_Diff__Un,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( minus_minus_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_549_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( member_nat @ A3 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_550_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: a,A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( member_a @ A3 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_551_infinite__coinduct,axiom,
! [X4: set_nat > $o,A2: set_nat] :
( ( X4 @ A2 )
=> ( ! [A8: set_nat] :
( ( X4 @ A8 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A8 )
& ( ( X4 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_552_infinite__coinduct,axiom,
! [X4: set_a > $o,A2: set_a] :
( ( X4 @ A2 )
=> ( ! [A8: set_a] :
( ( X4 @ A8 )
=> ? [X5: a] :
( ( member_a @ X5 @ A8 )
& ( ( X4 @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_553_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_554_infinite__remove,axiom,
! [S: set_a,A: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_555_subset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_556_Diff__single__insert,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_557_finite__remove__induct,axiom,
! [B2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_558_finite__remove__induct,axiom,
! [B2: set_a,P: set_a > $o] :
( ( finite_finite_a @ B2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_559_remove__induct,axiom,
! [P: set_nat > $o,B2: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_560_remove__induct,axiom,
! [P: set_a > $o,B2: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_561_additive__abelian__group_Osumsetdiff__sing,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B2: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% additive_abelian_group.sumsetdiff_sing
thf(fact_562_additive__abelian__group_Ocard__sumset__le,axiom,
! [G: set_Product_unit,Addition: product_unit > product_unit > product_unit,Zero: product_unit,A2: set_Product_unit,A: product_unit] :
( ( pluenn3635716580025208315t_unit @ G @ Addition @ Zero )
=> ( ( finite4290736615968046902t_unit @ A2 )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( pluenn1407455289632237236t_unit @ G @ Addition @ A2 @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A2 ) ) ) ) ).
% additive_abelian_group.card_sumset_le
thf(fact_563_additive__abelian__group_Ocard__sumset__le,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,A: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% additive_abelian_group.card_sumset_le
thf(fact_564_additive__abelian__group_Ocard__sumset__le,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ).
% additive_abelian_group.card_sumset_le
thf(fact_565_additive__abelian__group_Osumset__insert2,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,B2: set_a,A2: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ B2 )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ B2 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) ) ) ) ) ).
% additive_abelian_group.sumset_insert2
thf(fact_566_additive__abelian__group_Osumset__insert1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,X: a,B2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ A2 )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A2 ) @ B2 )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ bot_bot_set_a ) @ B2 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B2 ) ) ) ) ) ).
% additive_abelian_group.sumset_insert1
thf(fact_567_card__sumset__singleton__eq,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ A @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) ) ) )
& ( ~ ( member_a @ A @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= zero_zero_nat ) ) ) ) ).
% card_sumset_singleton_eq
thf(fact_568_subgroupI,axiom,
! [G: set_a] :
( ( ord_less_eq_set_a @ G @ g )
=> ( ( member_a @ zero @ G )
=> ( ! [G4: a,H: a] :
( ( member_a @ G4 @ G )
=> ( ( member_a @ H @ G )
=> ( member_a @ ( addition @ G4 @ H ) @ G ) ) )
=> ( ! [G4: a] :
( ( member_a @ G4 @ G )
=> ( group_invertible_a @ g @ addition @ zero @ G4 ) )
=> ( ! [G4: a] :
( ( member_a @ G4 @ G )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ G4 ) @ G ) )
=> ( group_subgroup_a @ G @ g @ addition @ zero ) ) ) ) ) ) ).
% subgroupI
thf(fact_569_monoid__axioms,axiom,
group_monoid_a @ g @ addition @ zero ).
% monoid_axioms
thf(fact_570_card__le__if__inj__on__rel,axiom,
! [B2: set_Product_unit,A2: set_a,R: a > product_unit > $o] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ? [B9: product_unit] :
( ( member_Product_unit @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B3: product_unit] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_Product_unit @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite410649719033368117t_unit @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_571_card__le__if__inj__on__rel,axiom,
! [B2: set_Product_unit,A2: set_Product_unit,R: product_unit > product_unit > $o] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ! [A3: product_unit] :
( ( member_Product_unit @ A3 @ A2 )
=> ? [B9: product_unit] :
( ( member_Product_unit @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: product_unit,A22: product_unit,B3: product_unit] :
( ( member_Product_unit @ A1 @ A2 )
=> ( ( member_Product_unit @ A22 @ A2 )
=> ( ( member_Product_unit @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A2 ) @ ( finite410649719033368117t_unit @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_572_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_a,R: a > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ? [B9: a] :
( ( member_a @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B3: a] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_573_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_Product_unit,R: product_unit > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A3: product_unit] :
( ( member_Product_unit @ A3 @ A2 )
=> ? [B9: a] :
( ( member_a @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: product_unit,A22: product_unit,B3: a] :
( ( member_Product_unit @ A1 @ A2 )
=> ( ( member_Product_unit @ A22 @ A2 )
=> ( ( member_a @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_574_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B3: nat] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_575_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_Product_unit,R: product_unit > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: product_unit] :
( ( member_Product_unit @ A3 @ A2 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: product_unit,A22: product_unit,B3: nat] :
( ( member_Product_unit @ A1 @ A2 )
=> ( ( member_Product_unit @ A22 @ A2 )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_576_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_577_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_578_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_579_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_580_card_Oempty,axiom,
( ( finite410649719033368117t_unit @ bot_bo3957492148770167129t_unit )
= zero_zero_nat ) ).
% card.empty
thf(fact_581_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_582_card_Oinfinite,axiom,
! [A2: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ A2 )
=> ( ( finite410649719033368117t_unit @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_583_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_584_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_585_card__0__eq,axiom,
! [A2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A2 )
=> ( ( ( finite410649719033368117t_unit @ A2 )
= zero_zero_nat )
= ( A2 = bot_bo3957492148770167129t_unit ) ) ) ).
% card_0_eq
thf(fact_586_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_587_card__0__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_588_subgroup__transitive,axiom,
! [K2: set_a,H2: set_a,Composition2: a > a > a,Unit2: a,G: set_a] :
( ( group_subgroup_a @ K2 @ H2 @ Composition2 @ Unit2 )
=> ( ( group_subgroup_a @ H2 @ G @ Composition2 @ Unit2 )
=> ( group_subgroup_a @ K2 @ G @ Composition2 @ Unit2 ) ) ) ).
% subgroup_transitive
thf(fact_589_Group__Theory_Omonoid__def,axiom,
( group_monoid_a
= ( ^ [M2: set_a,Composition: a > a > a,Unit: a] :
( ! [A4: a,B4: a] :
( ( member_a @ A4 @ M2 )
=> ( ( member_a @ B4 @ M2 )
=> ( member_a @ ( Composition @ A4 @ B4 ) @ M2 ) ) )
& ( member_a @ Unit @ M2 )
& ! [A4: a,B4: a,C4: a] :
( ( member_a @ A4 @ M2 )
=> ( ( member_a @ B4 @ M2 )
=> ( ( member_a @ C4 @ M2 )
=> ( ( Composition @ ( Composition @ A4 @ B4 ) @ C4 )
= ( Composition @ A4 @ ( Composition @ B4 @ C4 ) ) ) ) ) )
& ! [A4: a] :
( ( member_a @ A4 @ M2 )
=> ( ( Composition @ Unit @ A4 )
= A4 ) )
& ! [A4: a] :
( ( member_a @ A4 @ M2 )
=> ( ( Composition @ A4 @ Unit )
= A4 ) ) ) ) ) ).
% Group_Theory.monoid_def
thf(fact_590_monoid_Ocomposition__closed,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,A: a,B: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( member_a @ A @ M )
=> ( ( member_a @ B @ M )
=> ( member_a @ ( Composition2 @ A @ B ) @ M ) ) ) ) ).
% monoid.composition_closed
thf(fact_591_monoid_Oinverse__unique,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,U: a,V: a,V2: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( ( Composition2 @ U @ V )
= Unit2 )
=> ( ( ( Composition2 @ V2 @ U )
= Unit2 )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( member_a @ V @ M )
=> ( V2 = V ) ) ) ) ) ) ) ).
% monoid.inverse_unique
thf(fact_592_monoid_Ounit__closed,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( member_a @ Unit2 @ M ) ) ).
% monoid.unit_closed
thf(fact_593_monoid_Oassociative,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,A: a,B: a,C: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( member_a @ A @ M )
=> ( ( member_a @ B @ M )
=> ( ( member_a @ C @ M )
=> ( ( Composition2 @ ( Composition2 @ A @ B ) @ C )
= ( Composition2 @ A @ ( Composition2 @ B @ C ) ) ) ) ) ) ) ).
% monoid.associative
thf(fact_594_monoid_Oright__unit,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,A: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( member_a @ A @ M )
=> ( ( Composition2 @ A @ Unit2 )
= A ) ) ) ).
% monoid.right_unit
thf(fact_595_monoid_Oleft__unit,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,A: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( member_a @ A @ M )
=> ( ( Composition2 @ Unit2 @ A )
= A ) ) ) ).
% monoid.left_unit
thf(fact_596_Group__Theory_Omonoid_Ointro,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a] :
( ! [A3: a,B3: a] :
( ( member_a @ A3 @ M )
=> ( ( member_a @ B3 @ M )
=> ( member_a @ ( Composition2 @ A3 @ B3 ) @ M ) ) )
=> ( ( member_a @ Unit2 @ M )
=> ( ! [A3: a,B3: a,C5: a] :
( ( member_a @ A3 @ M )
=> ( ( member_a @ B3 @ M )
=> ( ( member_a @ C5 @ M )
=> ( ( Composition2 @ ( Composition2 @ A3 @ B3 ) @ C5 )
= ( Composition2 @ A3 @ ( Composition2 @ B3 @ C5 ) ) ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ M )
=> ( ( Composition2 @ Unit2 @ A3 )
= A3 ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ M )
=> ( ( Composition2 @ A3 @ Unit2 )
= A3 ) )
=> ( group_monoid_a @ M @ Composition2 @ Unit2 ) ) ) ) ) ) ).
% Group_Theory.monoid.intro
thf(fact_597_monoid_Oinverse__equality,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( ( Composition2 @ U @ V2 )
= Unit2 )
=> ( ( ( Composition2 @ V2 @ U )
= Unit2 )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( group_inverse_a @ M @ Composition2 @ Unit2 @ U )
= V2 ) ) ) ) ) ) ).
% monoid.inverse_equality
thf(fact_598_monoid_Oinverse__unit,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( group_inverse_a @ M @ Composition2 @ Unit2 @ Unit2 )
= Unit2 ) ) ).
% monoid.inverse_unit
thf(fact_599_monoid_OinvertibleE,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,U: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ U )
=> ( ! [V3: a] :
( ( ( ( Composition2 @ U @ V3 )
= Unit2 )
& ( ( Composition2 @ V3 @ U )
= Unit2 ) )
=> ~ ( member_a @ V3 @ M ) )
=> ~ ( member_a @ U @ M ) ) ) ) ).
% monoid.invertibleE
thf(fact_600_monoid_OinvertibleI,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( ( Composition2 @ U @ V2 )
= Unit2 )
=> ( ( ( Composition2 @ V2 @ U )
= Unit2 )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( group_invertible_a @ M @ Composition2 @ Unit2 @ U ) ) ) ) ) ) ).
% monoid.invertibleI
thf(fact_601_monoid_Oinvertible__def,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,U: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( member_a @ U @ M )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ U )
= ( ? [X2: a] :
( ( member_a @ X2 @ M )
& ( ( Composition2 @ U @ X2 )
= Unit2 )
& ( ( Composition2 @ X2 @ U )
= Unit2 ) ) ) ) ) ) ).
% monoid.invertible_def
thf(fact_602_monoid_Ounit__invertible,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( group_invertible_a @ M @ Composition2 @ Unit2 @ Unit2 ) ) ).
% monoid.unit_invertible
thf(fact_603_monoid_Ocomposition__invertible,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,X: a,Y: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ X )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ Y )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( group_invertible_a @ M @ Composition2 @ Unit2 @ ( Composition2 @ X @ Y ) ) ) ) ) ) ) ).
% monoid.composition_invertible
thf(fact_604_monoid_Oinvertible__left__cancel,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,X: a,Y: a,Z: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( member_a @ Z @ M )
=> ( ( ( Composition2 @ X @ Y )
= ( Composition2 @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_605_monoid_Oinvertible__right__cancel,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,X: a,Y: a,Z: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( member_a @ Z @ M )
=> ( ( ( Composition2 @ Y @ X )
= ( Composition2 @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_606_Group__Theory_Ogroup_Oaxioms_I1_J,axiom,
! [G: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_group_a @ G @ Composition2 @ Unit2 )
=> ( group_monoid_a @ G @ Composition2 @ Unit2 ) ) ).
% Group_Theory.group.axioms(1)
thf(fact_607_commutative__monoid_Oaxioms_I1_J,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_4866109990395492029noid_a @ M @ Composition2 @ Unit2 )
=> ( group_monoid_a @ M @ Composition2 @ Unit2 ) ) ).
% commutative_monoid.axioms(1)
thf(fact_608_subgroup_Osubgroup__inverse__equality,axiom,
! [G: set_a,M: set_a,Composition2: a > a > a,Unit2: a,U: a] :
( ( group_subgroup_a @ G @ M @ Composition2 @ Unit2 )
=> ( ( member_a @ U @ G )
=> ( ( group_inverse_a @ M @ Composition2 @ Unit2 @ U )
= ( group_inverse_a @ G @ Composition2 @ Unit2 @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_609_subgroup_Oaxioms_I2_J,axiom,
! [G: set_a,M: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_subgroup_a @ G @ M @ Composition2 @ Unit2 )
=> ( group_group_a @ G @ Composition2 @ Unit2 ) ) ).
% subgroup.axioms(2)
thf(fact_610_monoid_OsubgroupI,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,G: set_a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( ord_less_eq_set_a @ G @ M )
=> ( ( member_a @ Unit2 @ G )
=> ( ! [G4: a,H: a] :
( ( member_a @ G4 @ G )
=> ( ( member_a @ H @ G )
=> ( member_a @ ( Composition2 @ G4 @ H ) @ G ) ) )
=> ( ! [G4: a] :
( ( member_a @ G4 @ G )
=> ( group_invertible_a @ M @ Composition2 @ Unit2 @ G4 ) )
=> ( ! [G4: a] :
( ( member_a @ G4 @ G )
=> ( member_a @ ( group_inverse_a @ M @ Composition2 @ Unit2 @ G4 ) @ G ) )
=> ( group_subgroup_a @ G @ M @ Composition2 @ Unit2 ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_611_monoid_Oinvertible__left__inverse,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,U: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ U )
=> ( ( member_a @ U @ M )
=> ( ( Composition2 @ ( group_inverse_a @ M @ Composition2 @ Unit2 @ U ) @ U )
= Unit2 ) ) ) ) ).
% monoid.invertible_left_inverse
thf(fact_612_monoid_Oinvertible__left__inverse2,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ U )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( Composition2 @ ( group_inverse_a @ M @ Composition2 @ Unit2 @ U ) @ ( Composition2 @ U @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_left_inverse2
thf(fact_613_monoid_Oinvertible__right__inverse,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,U: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ U )
=> ( ( member_a @ U @ M )
=> ( ( Composition2 @ U @ ( group_inverse_a @ M @ Composition2 @ Unit2 @ U ) )
= Unit2 ) ) ) ) ).
% monoid.invertible_right_inverse
thf(fact_614_monoid_Oinvertible__inverse__closed,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,U: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ U )
=> ( ( member_a @ U @ M )
=> ( member_a @ ( group_inverse_a @ M @ Composition2 @ Unit2 @ U ) @ M ) ) ) ) ).
% monoid.invertible_inverse_closed
thf(fact_615_monoid_Oinvertible__right__inverse2,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ U )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( Composition2 @ U @ ( Composition2 @ ( group_inverse_a @ M @ Composition2 @ Unit2 @ U ) @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_right_inverse2
thf(fact_616_monoid_Oinvertible__inverse__inverse,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,U: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ U )
=> ( ( member_a @ U @ M )
=> ( ( group_inverse_a @ M @ Composition2 @ Unit2 @ ( group_inverse_a @ M @ Composition2 @ Unit2 @ U ) )
= U ) ) ) ) ).
% monoid.invertible_inverse_inverse
thf(fact_617_monoid_Oinverse__composition__commute,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,X: a,Y: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ X )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ Y )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( group_inverse_a @ M @ Composition2 @ Unit2 @ ( Composition2 @ X @ Y ) )
= ( Composition2 @ ( group_inverse_a @ M @ Composition2 @ Unit2 @ Y ) @ ( group_inverse_a @ M @ Composition2 @ Unit2 @ X ) ) ) ) ) ) ) ) ).
% monoid.inverse_composition_commute
thf(fact_618_monoid_Oinvertible__inverse__invertible,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,U: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ U )
=> ( ( member_a @ U @ M )
=> ( group_invertible_a @ M @ Composition2 @ Unit2 @ ( group_inverse_a @ M @ Composition2 @ Unit2 @ U ) ) ) ) ) ).
% monoid.invertible_inverse_invertible
thf(fact_619_monoid_Omem__UnitsI,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,U: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ U )
=> ( ( member_a @ U @ M )
=> ( member_a @ U @ ( group_Units_a @ M @ Composition2 @ Unit2 ) ) ) ) ) ).
% monoid.mem_UnitsI
thf(fact_620_monoid_Omem__UnitsD,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,U: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( member_a @ U @ ( group_Units_a @ M @ Composition2 @ Unit2 ) )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ U )
& ( member_a @ U @ M ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_621_monoid_Ogroup__of__Units,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( group_group_a @ ( group_Units_a @ M @ Composition2 @ Unit2 ) @ Composition2 @ Unit2 ) ) ).
% monoid.group_of_Units
thf(fact_622_subgroup_Osubgroup__inverse__iff,axiom,
! [G: set_a,M: set_a,Composition2: a > a > a,Unit2: a,X: a] :
( ( group_subgroup_a @ G @ M @ Composition2 @ Unit2 )
=> ( ( group_invertible_a @ M @ Composition2 @ Unit2 @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ ( group_inverse_a @ M @ Composition2 @ Unit2 @ X ) @ G )
= ( member_a @ X @ G ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_623_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_624_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_625_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
= ( ^ [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
& ( ord_less_eq_set_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_626_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_627_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_628_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_629_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_630_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_631_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_632_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_633_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_634_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_635_order__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_636_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_637_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_638_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_639_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_640_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_641_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_642_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_643_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_644_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_645_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_646_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_647_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_648_order__subst1,axiom,
! [A: set_a,F2: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_649_order__subst1,axiom,
! [A: set_a,F2: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_650_order__subst1,axiom,
! [A: nat,F2: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_651_order__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_652_order__subst2,axiom,
! [A: set_a,B: set_a,F2: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F2 @ B ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_653_order__subst2,axiom,
! [A: set_a,B: set_a,F2: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_654_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_655_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_656_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_657_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_658_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_659_ord__eq__le__subst,axiom,
! [A: set_a,F2: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_660_ord__eq__le__subst,axiom,
! [A: nat,F2: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_661_ord__eq__le__subst,axiom,
! [A: set_a,F2: nat > set_a,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_662_ord__eq__le__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_663_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F2: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_664_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F2: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_665_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_666_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_667_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_668_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_669_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_670_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_671_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N2: set_nat] :
? [M4: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_672_card__eq__0__iff,axiom,
! [A2: set_Product_unit] :
( ( ( finite410649719033368117t_unit @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bo3957492148770167129t_unit )
| ~ ( finite4290736615968046902t_unit @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_673_card__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_674_card__eq__0__iff,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_a )
| ~ ( finite_finite_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_675_card__Diff__subset,axiom,
! [B2: set_Product_unit,A2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ A2 )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite410649719033368117t_unit @ A2 ) @ ( finite410649719033368117t_unit @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_676_card__Diff__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_677_card__Diff__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_678_diff__card__le__card__Diff,axiom,
! [B2: set_Product_unit,A2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite410649719033368117t_unit @ A2 ) @ ( finite410649719033368117t_unit @ B2 ) ) @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_679_diff__card__le__card__Diff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_680_diff__card__le__card__Diff,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_681_card__Diff__subset__Int,axiom,
! [A2: set_Product_unit,B2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ ( inf_in4660618365625256667t_unit @ A2 @ B2 ) )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite410649719033368117t_unit @ A2 ) @ ( finite410649719033368117t_unit @ ( inf_in4660618365625256667t_unit @ A2 @ B2 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_682_card__Diff__subset__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_683_card__Diff__subset__Int,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_684_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_685_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_686_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_687_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_688_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_689_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_690_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
! [G: set_Product_unit,Addition: product_unit > product_unit > product_unit,Zero: product_unit,A2: set_Product_unit,A: product_unit] :
( ( pluenn3635716580025208315t_unit @ G @ Addition @ Zero )
=> ( ( finite4290736615968046902t_unit @ A2 )
=> ( ( ( member_Product_unit @ A @ G )
=> ( ( finite410649719033368117t_unit @ ( pluenn1407455289632237236t_unit @ G @ Addition @ A2 @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) )
= ( finite410649719033368117t_unit @ ( inf_in4660618365625256667t_unit @ A2 @ G ) ) ) )
& ( ~ ( member_Product_unit @ A @ G )
=> ( ( finite410649719033368117t_unit @ ( pluenn1407455289632237236t_unit @ G @ Addition @ A2 @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) )
= zero_zero_nat ) ) ) ) ) ).
% additive_abelian_group.card_sumset_singleton_eq
thf(fact_691_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,A: nat] :
( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
=> ( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ A @ G )
=> ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ ( inf_inf_set_nat @ A2 @ G ) ) ) )
& ( ~ ( member_nat @ A @ G )
=> ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= zero_zero_nat ) ) ) ) ) ).
% additive_abelian_group.card_sumset_singleton_eq
thf(fact_692_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ A @ G )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ G ) ) ) )
& ( ~ ( member_a @ A @ G )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= zero_zero_nat ) ) ) ) ) ).
% additive_abelian_group.card_sumset_singleton_eq
thf(fact_693_inverse__subgroupD,axiom,
! [H2: set_a] :
( ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ H2 ) @ g @ addition @ zero )
=> ( ( ord_less_eq_set_a @ H2 @ ( group_Units_a @ g @ addition @ zero ) )
=> ( group_subgroup_a @ H2 @ g @ addition @ zero ) ) ) ).
% inverse_subgroupD
thf(fact_694_inverse__subgroupI,axiom,
! [H2: set_a] :
( ( group_subgroup_a @ H2 @ g @ addition @ zero )
=> ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ H2 ) @ g @ addition @ zero ) ) ).
% inverse_subgroupI
thf(fact_695_diff__is__0__eq_H,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ( minus_minus_nat @ M5 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_696_image__eqI,axiom,
! [B: a,F2: a > a,X: a,A2: set_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ B @ ( image_a_a @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_697_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_698_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_699_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_700_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_701_image__empty,axiom,
! [F2: a > a] :
( ( image_a_a @ F2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_702_empty__is__image,axiom,
! [F2: a > a,A2: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_703_image__is__empty,axiom,
! [F2: a > a,A2: set_a] :
( ( ( image_a_a @ F2 @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_704_finite__imageI,axiom,
! [F: set_a,H3: a > a] :
( ( finite_finite_a @ F )
=> ( finite_finite_a @ ( image_a_a @ H3 @ F ) ) ) ).
% finite_imageI
thf(fact_705_finite__imageI,axiom,
! [F: set_a,H3: a > nat] :
( ( finite_finite_a @ F )
=> ( finite_finite_nat @ ( image_a_nat @ H3 @ F ) ) ) ).
% finite_imageI
thf(fact_706_finite__imageI,axiom,
! [F: set_nat,H3: nat > a] :
( ( finite_finite_nat @ F )
=> ( finite_finite_a @ ( image_nat_a @ H3 @ F ) ) ) ).
% finite_imageI
thf(fact_707_finite__imageI,axiom,
! [F: set_nat,H3: nat > nat] :
( ( finite_finite_nat @ F )
=> ( finite_finite_nat @ ( image_nat_nat @ H3 @ F ) ) ) ).
% finite_imageI
thf(fact_708_image__insert,axiom,
! [F2: a > a,A: a,B2: set_a] :
( ( image_a_a @ F2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ ( F2 @ A ) @ ( image_a_a @ F2 @ B2 ) ) ) ).
% image_insert
thf(fact_709_insert__image,axiom,
! [X: a,A2: set_a,F2: a > a] :
( ( member_a @ X @ A2 )
=> ( ( insert_a @ ( F2 @ X ) @ ( image_a_a @ F2 @ A2 ) )
= ( image_a_a @ F2 @ A2 ) ) ) ).
% insert_image
thf(fact_710_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_711_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_712_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_713_diff__self__eq__0,axiom,
! [M5: nat] :
( ( minus_minus_nat @ M5 @ M5 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_714_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_715_diff__is__0__eq,axiom,
! [M5: nat,N: nat] :
( ( ( minus_minus_nat @ M5 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M5 @ N ) ) ).
% diff_is_0_eq
thf(fact_716_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_717_imageI,axiom,
! [X: a,A2: set_a,F2: a > a] :
( ( member_a @ X @ A2 )
=> ( member_a @ ( F2 @ X ) @ ( image_a_a @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_718_image__iff,axiom,
! [Z: a,F2: a > a,A2: set_a] :
( ( member_a @ Z @ ( image_a_a @ F2 @ A2 ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( Z
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_719_bex__imageD,axiom,
! [F2: a > a,A2: set_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( image_a_a @ F2 @ A2 ) )
& ( P @ X5 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_720_image__cong,axiom,
! [M: set_a,N3: set_a,F2: a > a,G5: a > a] :
( ( M = N3 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N3 )
=> ( ( F2 @ X3 )
= ( G5 @ X3 ) ) )
=> ( ( image_a_a @ F2 @ M )
= ( image_a_a @ G5 @ N3 ) ) ) ) ).
% image_cong
thf(fact_721_ball__imageD,axiom,
! [F2: a > a,A2: set_a,P: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( image_a_a @ F2 @ A2 ) )
=> ( P @ X3 ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( P @ ( F2 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_722_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: a,F2: a > a] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_a @ B @ ( image_a_a @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_723_image__Un,axiom,
! [F2: a > a,A2: set_a,B2: set_a] :
( ( image_a_a @ F2 @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( sup_sup_set_a @ ( image_a_a @ F2 @ A2 ) @ ( image_a_a @ F2 @ B2 ) ) ) ).
% image_Un
thf(fact_724_image__mono,axiom,
! [A2: set_a,B2: set_a,F2: a > a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A2 ) @ ( image_a_a @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_725_image__subsetI,axiom,
! [A2: set_a,F2: a > a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ ( F2 @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_726_subset__imageE,axiom,
! [B2: set_a,F2: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F2 @ A2 ) )
=> ~ ! [C6: set_a] :
( ( ord_less_eq_set_a @ C6 @ A2 )
=> ( B2
!= ( image_a_a @ F2 @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_727_image__subset__iff,axiom,
! [F2: a > a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A2 ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F2 @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_728_subset__image__iff,axiom,
! [B2: set_a,F2: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F2 @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_729_all__subset__image,axiom,
! [F2: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F2 @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A2 )
=> ( P @ ( image_a_a @ F2 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_730_finite__surj,axiom,
! [A2: set_a,B2: set_nat,F2: a > nat] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F2 @ A2 ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_731_finite__surj,axiom,
! [A2: set_nat,B2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_732_finite__surj,axiom,
! [A2: set_a,B2: set_a,F2: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F2 @ A2 ) )
=> ( finite_finite_a @ B2 ) ) ) ).
% finite_surj
thf(fact_733_finite__surj,axiom,
! [A2: set_nat,B2: set_a,F2: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F2 @ A2 ) )
=> ( finite_finite_a @ B2 ) ) ) ).
% finite_surj
thf(fact_734_finite__subset__image,axiom,
! [B2: set_nat,F2: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F2 @ A2 ) )
=> ? [C6: set_nat] :
( ( ord_less_eq_set_nat @ C6 @ A2 )
& ( finite_finite_nat @ C6 )
& ( B2
= ( image_nat_nat @ F2 @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_735_finite__subset__image,axiom,
! [B2: set_nat,F2: a > nat,A2: set_a] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F2 @ A2 ) )
=> ? [C6: set_a] :
( ( ord_less_eq_set_a @ C6 @ A2 )
& ( finite_finite_a @ C6 )
& ( B2
= ( image_a_nat @ F2 @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_736_finite__subset__image,axiom,
! [B2: set_a,F2: nat > a,A2: set_nat] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F2 @ A2 ) )
=> ? [C6: set_nat] :
( ( ord_less_eq_set_nat @ C6 @ A2 )
& ( finite_finite_nat @ C6 )
& ( B2
= ( image_nat_a @ F2 @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_737_finite__subset__image,axiom,
! [B2: set_a,F2: a > a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F2 @ A2 ) )
=> ? [C6: set_a] :
( ( ord_less_eq_set_a @ C6 @ A2 )
& ( finite_finite_a @ C6 )
& ( B2
= ( image_a_a @ F2 @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_738_ex__finite__subset__image,axiom,
! [F2: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F2 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 )
& ( P @ ( image_nat_nat @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_739_ex__finite__subset__image,axiom,
! [F2: a > nat,A2: set_a,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_a_nat @ F2 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 )
& ( P @ ( image_a_nat @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_740_ex__finite__subset__image,axiom,
! [F2: nat > a,A2: set_nat,P: set_a > $o] :
( ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_nat_a @ F2 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 )
& ( P @ ( image_nat_a @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_741_ex__finite__subset__image,axiom,
! [F2: a > a,A2: set_a,P: set_a > $o] :
( ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F2 @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 )
& ( P @ ( image_a_a @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_742_all__finite__subset__image,axiom,
! [F2: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F2 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 ) )
=> ( P @ ( image_nat_nat @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_743_all__finite__subset__image,axiom,
! [F2: a > nat,A2: set_a,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_a_nat @ F2 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 ) )
=> ( P @ ( image_a_nat @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_744_all__finite__subset__image,axiom,
! [F2: nat > a,A2: set_nat,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_nat_a @ F2 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 ) )
=> ( P @ ( image_nat_a @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_745_all__finite__subset__image,axiom,
! [F2: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F2 @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A2 ) )
=> ( P @ ( image_a_a @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_746_image__Int__subset,axiom,
! [F2: a > a,A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F2 @ ( inf_inf_set_a @ A2 @ B2 ) ) @ ( inf_inf_set_a @ ( image_a_a @ F2 @ A2 ) @ ( image_a_a @ F2 @ B2 ) ) ) ).
% image_Int_subset
thf(fact_747_image__diff__subset,axiom,
! [F2: a > a,A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F2 @ A2 ) @ ( image_a_a @ F2 @ B2 ) ) @ ( image_a_a @ F2 @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_748_subgroup_Oimage__of__inverse,axiom,
! [G: set_a,M: set_a,Composition2: a > a > a,Unit2: a,X: a] :
( ( group_subgroup_a @ G @ M @ Composition2 @ Unit2 )
=> ( ( member_a @ X @ G )
=> ( member_a @ X @ ( image_a_a @ ( group_inverse_a @ M @ Composition2 @ Unit2 ) @ G ) ) ) ) ).
% subgroup.image_of_inverse
thf(fact_749_card__image__le,axiom,
! [A2: set_Product_unit,F2: product_unit > a] :
( ( finite4290736615968046902t_unit @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_Product_unit_a @ F2 @ A2 ) ) @ ( finite410649719033368117t_unit @ A2 ) ) ) ).
% card_image_le
thf(fact_750_card__image__le,axiom,
! [A2: set_Product_unit,F2: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ A2 )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( image_405062704495631173t_unit @ F2 @ A2 ) ) @ ( finite410649719033368117t_unit @ A2 ) ) ) ).
% card_image_le
thf(fact_751_card__image__le,axiom,
! [A2: set_a,F2: a > a] :
( ( finite_finite_a @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_a_a @ F2 @ A2 ) ) @ ( finite_card_a @ A2 ) ) ) ).
% card_image_le
thf(fact_752_card__image__le,axiom,
! [A2: set_a,F2: a > product_unit] :
( ( finite_finite_a @ A2 )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( image_a_Product_unit @ F2 @ A2 ) ) @ ( finite_card_a @ A2 ) ) ) ).
% card_image_le
thf(fact_753_card__image__le,axiom,
! [A2: set_nat,F2: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_nat_a @ F2 @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_754_card__image__le,axiom,
! [A2: set_nat,F2: nat > product_unit] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( image_8730104196221521654t_unit @ F2 @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_755_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_756_diff__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 ) ) ).
% diff_right_commute
thf(fact_757_minus__nat_Odiff__0,axiom,
! [M5: nat] :
( ( minus_minus_nat @ M5 @ zero_zero_nat )
= M5 ) ).
% minus_nat.diff_0
thf(fact_758_diffs0__imp__equal,axiom,
! [M5: nat,N: nat] :
( ( ( minus_minus_nat @ M5 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M5 )
= zero_zero_nat )
=> ( M5 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_759_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_760_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_761_nat__le__linear,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
| ( ord_less_eq_nat @ N @ M5 ) ) ).
% nat_le_linear
thf(fact_762_le__antisym,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ( ord_less_eq_nat @ N @ M5 )
=> ( M5 = N ) ) ) ).
% le_antisym
thf(fact_763_eq__imp__le,axiom,
! [M5: nat,N: nat] :
( ( M5 = N )
=> ( ord_less_eq_nat @ M5 @ N ) ) ).
% eq_imp_le
thf(fact_764_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_765_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_766_eq__diff__iff,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M5 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M5 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_767_le__diff__iff,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M5 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_768_Nat_Odiff__diff__eq,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M5 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M5 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_769_diff__le__mono,axiom,
! [M5: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_770_diff__le__self,axiom,
! [M5: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ N ) @ M5 ) ).
% diff_le_self
thf(fact_771_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_772_diff__le__mono2,axiom,
! [M5: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M5 ) ) ) ).
% diff_le_mono2
thf(fact_773_group_Oinverse__subgroupI,axiom,
! [G: set_a,Composition2: a > a > a,Unit2: a,H2: set_a] :
( ( group_group_a @ G @ Composition2 @ Unit2 )
=> ( ( group_subgroup_a @ H2 @ G @ Composition2 @ Unit2 )
=> ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ G @ Composition2 @ Unit2 ) @ H2 ) @ G @ Composition2 @ Unit2 ) ) ) ).
% group.inverse_subgroupI
thf(fact_774_surj__card__le,axiom,
! [A2: set_Product_unit,B2: set_Product_unit,F2: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ A2 )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( image_405062704495631173t_unit @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B2 ) @ ( finite410649719033368117t_unit @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_775_surj__card__le,axiom,
! [A2: set_a,B2: set_Product_unit,F2: a > product_unit] :
( ( finite_finite_a @ A2 )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( image_a_Product_unit @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B2 ) @ ( finite_card_a @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_776_surj__card__le,axiom,
! [A2: set_nat,B2: set_Product_unit,F2: nat > product_unit] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( image_8730104196221521654t_unit @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B2 ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_777_surj__card__le,axiom,
! [A2: set_Product_unit,B2: set_a,F2: product_unit > a] :
( ( finite4290736615968046902t_unit @ A2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_Product_unit_a @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite410649719033368117t_unit @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_778_surj__card__le,axiom,
! [A2: set_a,B2: set_a,F2: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_779_surj__card__le,axiom,
! [A2: set_nat,B2: set_a,F2: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_780_group_Oinverse__subgroupD,axiom,
! [G: set_a,Composition2: a > a > a,Unit2: a,H2: set_a] :
( ( group_group_a @ G @ Composition2 @ Unit2 )
=> ( ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ G @ Composition2 @ Unit2 ) @ H2 ) @ G @ Composition2 @ Unit2 )
=> ( ( ord_less_eq_set_a @ H2 @ ( group_Units_a @ G @ Composition2 @ Unit2 ) )
=> ( group_subgroup_a @ H2 @ G @ Composition2 @ Unit2 ) ) ) ) ).
% group.inverse_subgroupD
thf(fact_781_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_782_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_783_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_784_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_785_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_786_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_787_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_788_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_789_inverse__undefined,axiom,
! [U: a] :
( ~ ( member_a @ U @ g )
=> ( ( group_inverse_a @ g @ addition @ zero @ U )
= undefined_a ) ) ).
% inverse_undefined
thf(fact_790_monoid_Oinverse__undefined,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a,U: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ~ ( member_a @ U @ M )
=> ( ( group_inverse_a @ M @ Composition2 @ Unit2 @ U )
= undefined_a ) ) ) ).
% monoid.inverse_undefined
thf(fact_791_in__image__insert__iff,axiom,
! [B2: set_set_a,X: a,A2: set_a] :
( ! [C6: set_a] :
( ( member_set_a @ C6 @ B2 )
=> ~ ( member_a @ X @ C6 ) )
=> ( ( member_set_a @ A2 @ ( image_set_a_set_a @ ( insert_a @ X ) @ B2 ) )
= ( ( member_a @ X @ A2 )
& ( member_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_792_the__elem__image__unique,axiom,
! [A2: set_a,F2: a > a,X: a] :
( ( A2 != bot_bot_set_a )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ A2 )
=> ( ( F2 @ Y3 )
= ( F2 @ X ) ) )
=> ( ( the_elem_a @ ( image_a_a @ F2 @ A2 ) )
= ( F2 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_793_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
( A6
= ( insert_a @ ( the_elem_a @ A6 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_794_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_795_arg__min__least,axiom,
! [S: set_nat,Y: nat,F2: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ ( F2 @ ( lattic7446932960582359483at_nat @ F2 @ S ) ) @ ( F2 @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_796_arg__min__least,axiom,
! [S: set_a,Y: a,F2: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F2 @ ( lattic6340287419671400565_a_nat @ F2 @ S ) ) @ ( F2 @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_797_is__singletonI_H,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X3: a,Y3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( member_a @ Y3 @ A2 )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_798_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
? [X2: a] :
( A6
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_799_is__singletonE,axiom,
! [A2: set_a] :
( ( is_singleton_a @ A2 )
=> ~ ! [X3: a] :
( A2
!= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_800_insert__subsetI,axiom,
! [X: a,A2: set_a,X4: set_a] :
( ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X4 @ A2 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_801_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_802_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_803_commutative__monoid__def,axiom,
( group_4866109990395492029noid_a
= ( ^ [M2: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M2 @ Composition @ Unit )
& ( group_2081300317213596122ioms_a @ M2 @ Composition ) ) ) ) ).
% commutative_monoid_def
thf(fact_804_commutative__monoid_Ointro,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_monoid_a @ M @ Composition2 @ Unit2 )
=> ( ( group_2081300317213596122ioms_a @ M @ Composition2 )
=> ( group_4866109990395492029noid_a @ M @ Composition2 @ Unit2 ) ) ) ).
% commutative_monoid.intro
thf(fact_805_remove__def,axiom,
( remove_a
= ( ^ [X2: a,A6: set_a] : ( minus_minus_set_a @ A6 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_806_member__remove,axiom,
! [X: a,Y: a,A2: set_a] :
( ( member_a @ X @ ( remove_a @ Y @ A2 ) )
= ( ( member_a @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_807_commutative__monoid__axioms_Ointro,axiom,
! [M: set_a,Composition2: a > a > a] :
( ! [X3: a,Y3: a] :
( ( member_a @ X3 @ M )
=> ( ( member_a @ Y3 @ M )
=> ( ( Composition2 @ X3 @ Y3 )
= ( Composition2 @ Y3 @ X3 ) ) ) )
=> ( group_2081300317213596122ioms_a @ M @ Composition2 ) ) ).
% commutative_monoid_axioms.intro
thf(fact_808_commutative__monoid__axioms__def,axiom,
( group_2081300317213596122ioms_a
= ( ^ [M2: set_a,Composition: a > a > a] :
! [X2: a,Y2: a] :
( ( member_a @ X2 @ M2 )
=> ( ( member_a @ Y2 @ M2 )
=> ( ( Composition @ X2 @ Y2 )
= ( Composition @ Y2 @ X2 ) ) ) ) ) ) ).
% commutative_monoid_axioms_def
thf(fact_809_commutative__monoid_Oaxioms_I2_J,axiom,
! [M: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_4866109990395492029noid_a @ M @ Composition2 @ Unit2 )
=> ( group_2081300317213596122ioms_a @ M @ Composition2 ) ) ).
% commutative_monoid.axioms(2)
thf(fact_810_image__Fpow__mono,axiom,
! [F2: a > a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A2 ) @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ ( image_a_a @ F2 ) @ ( finite_Fpow_a @ A2 ) ) @ ( finite_Fpow_a @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_811_Group__Theory_Ogroup_Ointro,axiom,
! [G: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_monoid_a @ G @ Composition2 @ Unit2 )
=> ( ( group_group_axioms_a @ G @ Composition2 @ Unit2 )
=> ( group_group_a @ G @ Composition2 @ Unit2 ) ) ) ).
% Group_Theory.group.intro
thf(fact_812_Group__Theory_Ogroup__def,axiom,
( group_group_a
= ( ^ [G2: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ G2 @ Composition @ Unit )
& ( group_group_axioms_a @ G2 @ Composition @ Unit ) ) ) ) ).
% Group_Theory.group_def
thf(fact_813_empty__in__Fpow,axiom,
! [A2: set_a] : ( member_set_a @ bot_bot_set_a @ ( finite_Fpow_a @ A2 ) ) ).
% empty_in_Fpow
thf(fact_814_Fpow__mono,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A2 ) @ ( finite_Fpow_a @ B2 ) ) ) ).
% Fpow_mono
thf(fact_815_Group__Theory_Ogroup__axioms_Ointro,axiom,
! [G: set_a,Composition2: a > a > a,Unit2: a] :
( ! [U2: a] :
( ( member_a @ U2 @ G )
=> ( group_invertible_a @ G @ Composition2 @ Unit2 @ U2 ) )
=> ( group_group_axioms_a @ G @ Composition2 @ Unit2 ) ) ).
% Group_Theory.group_axioms.intro
thf(fact_816_Group__Theory_Ogroup__axioms__def,axiom,
( group_group_axioms_a
= ( ^ [G2: set_a,Composition: a > a > a,Unit: a] :
! [U3: a] :
( ( member_a @ U3 @ G2 )
=> ( group_invertible_a @ G2 @ Composition @ Unit @ U3 ) ) ) ) ).
% Group_Theory.group_axioms_def
thf(fact_817_Group__Theory_Ogroup_Oaxioms_I2_J,axiom,
! [G: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_group_a @ G @ Composition2 @ Unit2 )
=> ( group_group_axioms_a @ G @ Composition2 @ Unit2 ) ) ).
% Group_Theory.group.axioms(2)
thf(fact_818_Sup__fin_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A2 )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_819_Sup__fin_Oremove,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ A2 )
= X ) )
& ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ A2 )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_820_Sup__fin_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A2 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_821_Sup__fin_Oinsert__remove,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A2 ) )
= X ) )
& ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A2 ) )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_822_Inf__fin_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A2 )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_823_Inf__fin_Oremove,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ A2 )
= X ) )
& ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ A2 )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_824_inf__Sup__absorb,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_nat @ A @ ( lattic1093996805478795353in_nat @ A2 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_825_inf__Sup__absorb,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A @ ( lattic2918178356826803221_set_a @ A2 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_826_sup__Inf__absorb,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_827_sup__Inf__absorb,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ( ( sup_sup_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_828_Inf__fin_Oinsert,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A2 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_829_Inf__fin_Oinsert,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A2 ) )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_830_Sup__fin_Oinsert,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A2 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_831_Sup__fin_Oinsert,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A2 ) )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_832_Inf__fin__le__Sup__fin,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_833_Inf__fin__le__Sup__fin,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_834_Inf__fin_OcoboundedI,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_835_Inf__fin_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_836_Sup__fin_OcoboundedI,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ( ord_less_eq_set_a @ A @ ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_837_Sup__fin_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_eq_nat @ A @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_838_Inf__fin_Oin__idem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_839_Inf__fin_Oin__idem,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) )
= ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_840_Sup__fin_Oin__idem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) )
= ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_841_Sup__fin_Oin__idem,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ A2 ) )
= ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_842_Sup__fin_Obounded__iff,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ X )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_843_Sup__fin_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_844_Inf__fin_Obounded__iff,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_845_Inf__fin_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_846_Sup__fin_OboundedI,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ A2 )
=> ( ord_less_eq_set_a @ A3 @ X ) )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_847_Sup__fin_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ord_less_eq_nat @ A3 @ X ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_848_Sup__fin_OboundedE,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ X )
=> ! [A9: set_a] :
( ( member_set_a @ A9 @ A2 )
=> ( ord_less_eq_set_a @ A9 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_849_Sup__fin_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X )
=> ! [A9: nat] :
( ( member_nat @ A9 @ A2 )
=> ( ord_less_eq_nat @ A9 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_850_Inf__fin_OboundedI,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ A2 )
=> ( ord_less_eq_set_a @ X @ A3 ) )
=> ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_851_Inf__fin_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ord_less_eq_nat @ X @ A3 ) )
=> ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_852_Inf__fin_OboundedE,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) )
=> ! [A9: set_a] :
( ( member_set_a @ A9 @ A2 )
=> ( ord_less_eq_set_a @ X @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_853_Inf__fin_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
=> ! [A9: nat] :
( ( member_nat @ A9 @ A2 )
=> ( ord_less_eq_nat @ X @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_854_Sup__fin_Osubset__imp,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B2 )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ ( lattic2918178356826803221_set_a @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_855_Sup__fin_Osubset__imp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_856_Inf__fin_Osubset__imp,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B2 )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ B2 ) @ ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_857_Inf__fin_Osubset__imp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B2 ) @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_858_Inf__fin_Ohom__commute,axiom,
! [H3: nat > nat,N3: set_nat] :
( ! [X3: nat,Y3: nat] :
( ( H3 @ ( inf_inf_nat @ X3 @ Y3 ) )
= ( inf_inf_nat @ ( H3 @ X3 ) @ ( H3 @ Y3 ) ) )
=> ( ( finite_finite_nat @ N3 )
=> ( ( N3 != bot_bot_set_nat )
=> ( ( H3 @ ( lattic5238388535129920115in_nat @ N3 ) )
= ( lattic5238388535129920115in_nat @ ( image_nat_nat @ H3 @ N3 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_859_Inf__fin_Ohom__commute,axiom,
! [H3: set_a > set_a,N3: set_set_a] :
( ! [X3: set_a,Y3: set_a] :
( ( H3 @ ( inf_inf_set_a @ X3 @ Y3 ) )
= ( inf_inf_set_a @ ( H3 @ X3 ) @ ( H3 @ Y3 ) ) )
=> ( ( finite_finite_set_a @ N3 )
=> ( ( N3 != bot_bot_set_set_a )
=> ( ( H3 @ ( lattic8209813465164889211_set_a @ N3 ) )
= ( lattic8209813465164889211_set_a @ ( image_set_a_set_a @ H3 @ N3 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_860_Sup__fin_Ohom__commute,axiom,
! [H3: nat > nat,N3: set_nat] :
( ! [X3: nat,Y3: nat] :
( ( H3 @ ( sup_sup_nat @ X3 @ Y3 ) )
= ( sup_sup_nat @ ( H3 @ X3 ) @ ( H3 @ Y3 ) ) )
=> ( ( finite_finite_nat @ N3 )
=> ( ( N3 != bot_bot_set_nat )
=> ( ( H3 @ ( lattic1093996805478795353in_nat @ N3 ) )
= ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H3 @ N3 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_861_Sup__fin_Ohom__commute,axiom,
! [H3: set_a > set_a,N3: set_set_a] :
( ! [X3: set_a,Y3: set_a] :
( ( H3 @ ( sup_sup_set_a @ X3 @ Y3 ) )
= ( sup_sup_set_a @ ( H3 @ X3 ) @ ( H3 @ Y3 ) ) )
=> ( ( finite_finite_set_a @ N3 )
=> ( ( N3 != bot_bot_set_set_a )
=> ( ( H3 @ ( lattic2918178356826803221_set_a @ N3 ) )
= ( lattic2918178356826803221_set_a @ ( image_set_a_set_a @ H3 @ N3 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_862_Inf__fin_Osubset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B2 ) @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_863_Inf__fin_Osubset,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( B2 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ ( lattic8209813465164889211_set_a @ B2 ) @ ( lattic8209813465164889211_set_a @ A2 ) )
= ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_864_Sup__fin_Osubset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B2 ) @ ( lattic1093996805478795353in_nat @ A2 ) )
= ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_865_Sup__fin_Osubset,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( B2 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ ( lattic2918178356826803221_set_a @ B2 ) @ ( lattic2918178356826803221_set_a @ A2 ) )
= ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_866_Inf__fin_Oinsert__not__elem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A2 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_867_Inf__fin_Oinsert__not__elem,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ~ ( member_set_a @ X @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A2 ) )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ A2 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_868_Inf__fin_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat,Y3: nat] : ( member_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A2 ) ) ) ) ).
% Inf_fin.closed
thf(fact_869_Inf__fin_Oclosed,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [X3: set_a,Y3: set_a] : ( member_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ ( insert_set_a @ X3 @ ( insert_set_a @ Y3 @ bot_bot_set_set_a ) ) )
=> ( member_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ A2 ) ) ) ) ).
% Inf_fin.closed
thf(fact_870_Sup__fin_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat,Y3: nat] : ( member_nat @ ( sup_sup_nat @ X3 @ Y3 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_871_Sup__fin_Oclosed,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [X3: set_a,Y3: set_a] : ( member_set_a @ ( sup_sup_set_a @ X3 @ Y3 ) @ ( insert_set_a @ X3 @ ( insert_set_a @ Y3 @ bot_bot_set_set_a ) ) )
=> ( member_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_872_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A2 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_873_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ~ ( member_set_a @ X @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A2 ) )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_874_Inf__fin_Ounion,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ ( lattic5238388535129920115in_nat @ B2 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_875_Inf__fin_Ounion,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( B2 != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ ( lattic8209813465164889211_set_a @ A2 ) @ ( lattic8209813465164889211_set_a @ B2 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_876_Sup__fin_Ounion,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B2 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_877_Sup__fin_Ounion,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( B2 != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) )
= ( sup_sup_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ ( lattic2918178356826803221_set_a @ B2 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_878_Inf__fin_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A2 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_879_Inf__fin_Oinsert__remove,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A2 ) )
= X ) )
& ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X @ A2 ) )
= ( inf_inf_set_a @ X @ ( lattic8209813465164889211_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_880_card__Diff__singleton,axiom,
! [X: product_unit,A2: set_Product_unit] :
( ( member_Product_unit @ X @ A2 )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
= ( minus_minus_nat @ ( finite410649719033368117t_unit @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_881_card__Diff__singleton,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_882_card__Diff__singleton__if,axiom,
! [X: product_unit,A2: set_Product_unit] :
( ( ( member_Product_unit @ X @ A2 )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
= ( minus_minus_nat @ ( finite410649719033368117t_unit @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_Product_unit @ X @ A2 )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
= ( finite410649719033368117t_unit @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_883_card__Diff__singleton__if,axiom,
! [X: a,A2: set_a] :
( ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_884_card__Diff__insert,axiom,
! [A: product_unit,A2: set_Product_unit,B2: set_Product_unit] :
( ( member_Product_unit @ A @ A2 )
=> ( ~ ( member_Product_unit @ A @ B2 )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ ( insert_Product_unit @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_885_card__Diff__insert,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ A2 )
=> ( ~ ( member_a @ A @ B2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_886_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_887_is__singleton__altdef,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
( ( finite_card_a @ A6 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_888_is__singleton__altdef,axiom,
( is_sin2160648248035936513t_unit
= ( ^ [A6: set_Product_unit] :
( ( finite410649719033368117t_unit @ A6 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_889_card__1__singletonE,axiom,
! [A2: set_Product_unit] :
( ( ( finite410649719033368117t_unit @ A2 )
= one_one_nat )
=> ~ ! [X3: product_unit] :
( A2
!= ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ).
% card_1_singletonE
thf(fact_890_card__1__singletonE,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= one_one_nat )
=> ~ ! [X3: a] :
( A2
!= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% card_1_singletonE
thf(fact_891_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_892_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_893_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_894_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_895_card__insert__le__m1,axiom,
! [N: nat,Y: set_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( insert_a @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_896_card__insert__le__m1,axiom,
! [N: nat,Y: set_Product_unit,X: product_unit] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_897_card__Diff1__less__iff,axiom,
! [A2: set_Product_unit,X: product_unit] :
( ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A2 ) )
= ( ( finite4290736615968046902t_unit @ A2 )
& ( member_Product_unit @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_898_card__Diff1__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) )
= ( ( finite_finite_nat @ A2 )
& ( member_nat @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_899_card__Diff1__less__iff,axiom,
! [A2: set_a,X: a] :
( ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) )
= ( ( finite_finite_a @ A2 )
& ( member_a @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_900_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_901_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_902_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_903_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_904_zero__less__diff,axiom,
! [N: nat,M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M5 ) )
= ( ord_less_nat @ M5 @ N ) ) ).
% zero_less_diff
thf(fact_905_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_906_bot_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_907_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_908_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_909_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_910_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_911_diff__less__mono2,axiom,
! [M5: nat,N: nat,L: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( ( ord_less_nat @ M5 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M5 ) ) ) ) ).
% diff_less_mono2
thf(fact_912_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_913_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N2: set_nat] :
? [M4: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ord_less_nat @ X2 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_914_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_915_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_916_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_917_gr__implies__not__zero,axiom,
! [M5: nat,N: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_918_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_919_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_920_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_921_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_922_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_923_gr__implies__not0,axiom,
! [M5: nat,N: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_924_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N4 )
& ~ ( P @ M6 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_925_less__supI1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_set_a @ X @ A )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% less_supI1
thf(fact_926_less__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_nat @ X @ A )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_927_less__supI2,axiom,
! [X: set_a,B: set_a,A: set_a] :
( ( ord_less_set_a @ X @ B )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% less_supI2
thf(fact_928_less__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_nat @ X @ B )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_929_sup_Oabsorb3,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_930_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_931_sup_Oabsorb4,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_932_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_933_sup_Ostrict__boundedE,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
=> ~ ( ( ord_less_set_a @ B @ A )
=> ~ ( ord_less_set_a @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_934_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_935_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( A4
= ( sup_sup_set_a @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_936_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( A4
= ( sup_sup_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_937_sup_Ostrict__coboundedI1,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_set_a @ C @ A )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_938_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_939_sup_Ostrict__coboundedI2,axiom,
! [C: set_a,B: set_a,A: set_a] :
( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_940_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_941_inf_Ostrict__coboundedI2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_942_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_943_inf_Ostrict__coboundedI1,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_set_a @ A @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_944_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_945_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( A4
= ( inf_inf_set_a @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_946_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( A4
= ( inf_inf_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_947_inf_Ostrict__boundedE,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_948_inf_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_949_inf_Oabsorb4,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_950_inf_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_951_inf_Oabsorb3,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_952_inf_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_953_less__infI2,axiom,
! [B: set_a,X: set_a,A: set_a] :
( ( ord_less_set_a @ B @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_954_less__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_nat @ B @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_955_less__infI1,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_set_a @ A @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_956_less__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_nat @ A @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_957_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_958_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_959_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_960_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_961_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_962_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_963_order__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_964_order__less__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_965_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_966_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_967_ord__eq__less__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_968_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_969_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_970_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_971_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_972_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_973_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_974_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_975_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_976_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_977_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_978_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_979_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N5: nat] :
( ( P3 @ N5 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N5 )
=> ~ ( P3 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_980_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_981_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_982_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_983_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_984_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_985_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_986_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_987_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_988_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_989_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_990_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_991_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_992_le__neq__implies__less,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ( M5 != N )
=> ( ord_less_nat @ M5 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_993_less__or__eq__imp__le,axiom,
! [M5: nat,N: nat] :
( ( ( ord_less_nat @ M5 @ N )
| ( M5 = N ) )
=> ( ord_less_eq_nat @ M5 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_994_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N5: nat] :
( ( ord_less_nat @ M4 @ N5 )
| ( M4 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_995_less__imp__le__nat,axiom,
! [M5: nat,N: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( ord_less_eq_nat @ M5 @ N ) ) ).
% less_imp_le_nat
thf(fact_996_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N5: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
& ( M4 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_997_order__le__imp__less__or__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_998_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_999_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1000_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1001_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1002_order__less__le__subst1,axiom,
! [A: set_a,F2: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1003_order__less__le__subst1,axiom,
! [A: nat,F2: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1004_order__less__le__subst1,axiom,
! [A: set_a,F2: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1005_order__less__le__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1006_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F2: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F2 @ B ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1007_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F2: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1008_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1009_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1010_order__le__less__subst1,axiom,
! [A: set_a,F2: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1011_order__le__less__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1012_order__less__le__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1013_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1014_order__le__less__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1015_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1016_order__neq__le__trans,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1017_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1018_order__le__neq__trans,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1019_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1020_order__less__imp__le,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1021_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1022_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1023_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1024_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_1025_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_1026_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_1027_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_1028_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1029_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1030_order_Ostrict__implies__order,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1031_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1032_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1033_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1034_dual__order_Ostrict__trans2,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1035_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1036_dual__order_Ostrict__trans1,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1037_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1038_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1039_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1040_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_set_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1041_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1042_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1043_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1044_order_Ostrict__trans2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1045_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1046_order_Ostrict__trans1,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1047_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1048_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1049_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1050_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1051_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1052_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1053_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
& ~ ( ord_less_eq_set_a @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1054_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1055_antisym__conv2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ~ ( ord_less_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1056_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1057_antisym__conv1,axiom,
! [X: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1058_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1059_nless__le,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( ord_less_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1060_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1061_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_1062_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_1063_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_1064_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1065_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1066_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1067_infinite__growing,axiom,
! [X4: set_nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X4 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X4 ) ) ) ).
% infinite_growing
thf(fact_1068_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1069_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_1070_diff__less,axiom,
! [N: nat,M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ord_less_nat @ ( minus_minus_nat @ M5 @ N ) @ M5 ) ) ) ).
% diff_less
thf(fact_1071_less__diff__iff,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M5 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M5 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1072_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_1073_finite__linorder__max__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B3: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( ord_less_nat @ X5 @ B3 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B3 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1074_finite__linorder__min__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B3: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( ord_less_nat @ B3 @ X5 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B3 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1075_card__ge__0__finite,axiom,
! [A2: set_Product_unit] :
( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A2 ) )
=> ( finite4290736615968046902t_unit @ A2 ) ) ).
% card_ge_0_finite
thf(fact_1076_card__ge__0__finite,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ( finite_finite_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_1077_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_1078_card__less__sym__Diff,axiom,
! [A2: set_Product_unit,B2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A2 )
=> ( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ord_less_nat @ ( finite410649719033368117t_unit @ A2 ) @ ( finite410649719033368117t_unit @ B2 ) )
=> ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ B2 ) ) @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1079_card__less__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1080_card__less__sym__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1081_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X5: nat] :
( ( member_nat @ X5 @ S )
& ( ord_less_nat @ ( F2 @ X5 ) @ ( F2 @ ( lattic7446932960582359483at_nat @ F2 @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1082_arg__min__if__finite_I2_J,axiom,
! [S: set_a,F2: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ~ ? [X5: a] :
( ( member_a @ X5 @ S )
& ( ord_less_nat @ ( F2 @ X5 ) @ ( F2 @ ( lattic6340287419671400565_a_nat @ F2 @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1083_card__gt__0__iff,axiom,
! [A2: set_Product_unit] :
( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A2 ) )
= ( ( A2 != bot_bo3957492148770167129t_unit )
& ( finite4290736615968046902t_unit @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_1084_card__gt__0__iff,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
= ( ( A2 != bot_bot_set_nat )
& ( finite_finite_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_1085_card__gt__0__iff,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
= ( ( A2 != bot_bot_set_a )
& ( finite_finite_a @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_1086_card__Diff1__less,axiom,
! [A2: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A2 )
=> ( ( member_Product_unit @ X @ A2 )
=> ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1087_card__Diff1__less,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1088_card__Diff1__less,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1089_card__Diff2__less,axiom,
! [A2: set_Product_unit,X: product_unit,Y: product_unit] :
( ( finite4290736615968046902t_unit @ A2 )
=> ( ( member_Product_unit @ X @ A2 )
=> ( ( member_Product_unit @ Y @ A2 )
=> ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1090_card__Diff2__less,axiom,
! [A2: set_nat,X: nat,Y: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1091_card__Diff2__less,axiom,
! [A2: set_a,X: a,Y: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( insert_a @ Y @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1092_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M5: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K3 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K3 ) ) )
=> ( P @ M5 ) ) ) ).
% nat_descend_induct
thf(fact_1093_minf_I8_J,axiom,
! [T4: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ~ ( ord_less_eq_nat @ T4 @ X5 ) ) ).
% minf(8)
thf(fact_1094_psubsetI,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_1095_not__psubset__empty,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_1096_finite__psubset__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ! [B10: set_a] :
( ( ord_less_set_a @ B10 @ A8 )
=> ( P @ B10 ) )
=> ( P @ A8 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_1097_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [B10: set_nat] :
( ( ord_less_set_nat @ B10 @ A8 )
=> ( P @ B10 ) )
=> ( P @ A8 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_1098_psubset__imp__ex__mem,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ? [B3: a] : ( member_a @ B3 @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1099_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_set_a @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1100_subset__psubset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1101_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ~ ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1102_psubset__subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1103_psubset__imp__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_1104_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_1105_psubsetE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_1106_psubset__card__mono,axiom,
! [B2: set_Product_unit,A2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ord_le8056459307392131481t_unit @ A2 @ B2 )
=> ( ord_less_nat @ ( finite410649719033368117t_unit @ A2 ) @ ( finite410649719033368117t_unit @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_1107_psubset__card__mono,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_1108_psubset__card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_1109_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T3: set_nat] :
( ( ord_less_set_nat @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ ( minus_minus_set_nat @ S @ T3 ) )
& ( P @ ( insert_nat @ X5 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1110_finite__induct__select,axiom,
! [S: set_a,P: set_a > $o] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [T3: set_a] :
( ( ord_less_set_a @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X5: a] :
( ( member_a @ X5 @ ( minus_minus_set_a @ S @ T3 ) )
& ( P @ ( insert_a @ X5 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1111_psubset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( ( member_a @ X @ A2 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1112_card__psubset,axiom,
! [B2: set_Product_unit,A2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ord_le3507040750410214029t_unit @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite410649719033368117t_unit @ A2 ) @ ( finite410649719033368117t_unit @ B2 ) )
=> ( ord_le8056459307392131481t_unit @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_1113_card__psubset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_1114_card__psubset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_1115_pinf_I6_J,axiom,
! [T4: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T4 ) ) ).
% pinf(6)
thf(fact_1116_pinf_I8_J,axiom,
! [T4: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ord_less_eq_nat @ T4 @ X5 ) ) ).
% pinf(8)
thf(fact_1117_minf_I6_J,axiom,
! [T4: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ord_less_eq_nat @ X5 @ T4 ) ) ).
% minf(6)
thf(fact_1118_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C5: nat] :
( ( ord_less_eq_nat @ A @ C5 )
& ( ord_less_eq_nat @ C5 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C5 ) )
=> ( P @ X5 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1119_verit__comp__simplify1_I3_J,axiom,
! [B11: nat,A10: nat] :
( ( ~ ( ord_less_eq_nat @ B11 @ A10 ) )
= ( ord_less_nat @ A10 @ B11 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1120_psubsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1121_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1122_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1123_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1124_card__range__greater__zero,axiom,
! [F2: a > a] :
( ( finite_finite_a @ ( image_a_a @ F2 @ top_top_set_a ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( image_a_a @ F2 @ top_top_set_a ) ) ) ) ).
% card_range_greater_zero
thf(fact_1125_card__range__greater__zero,axiom,
! [F2: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ ( image_405062704495631173t_unit @ F2 @ top_to1996260823553986621t_unit ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ ( image_405062704495631173t_unit @ F2 @ top_to1996260823553986621t_unit ) ) ) ) ).
% card_range_greater_zero
thf(fact_1126_card__range__greater__zero,axiom,
! [F2: product_unit > a] :
( ( finite_finite_a @ ( image_Product_unit_a @ F2 @ top_to1996260823553986621t_unit ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( image_Product_unit_a @ F2 @ top_to1996260823553986621t_unit ) ) ) ) ).
% card_range_greater_zero
thf(fact_1127_card__range__greater__zero,axiom,
! [F2: product_unit > nat] :
( ( finite_finite_nat @ ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit ) ) ) ) ).
% card_range_greater_zero
thf(fact_1128_card__range__greater__zero,axiom,
! [F2: nat > product_unit] :
( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F2 @ top_top_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ ( image_8730104196221521654t_unit @ F2 @ top_top_set_nat ) ) ) ) ).
% card_range_greater_zero
thf(fact_1129_card__range__greater__zero,axiom,
! [F2: nat > a] :
( ( finite_finite_a @ ( image_nat_a @ F2 @ top_top_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( image_nat_a @ F2 @ top_top_set_nat ) ) ) ) ).
% card_range_greater_zero
thf(fact_1130_card__range__greater__zero,axiom,
! [F2: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) ) ) ) ).
% card_range_greater_zero
thf(fact_1131_card__Un__disjoint,axiom,
! [A2: set_Product_unit,B2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A2 )
=> ( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ( inf_in4660618365625256667t_unit @ A2 @ B2 )
= bot_bo3957492148770167129t_unit )
=> ( ( finite410649719033368117t_unit @ ( sup_su793286257634532545t_unit @ A2 @ B2 ) )
= ( plus_plus_nat @ ( finite410649719033368117t_unit @ A2 ) @ ( finite410649719033368117t_unit @ B2 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_1132_card__Un__disjoint,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( finite_card_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( plus_plus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_1133_card__Un__disjoint,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( finite_card_a @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( plus_plus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_1134_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_1135_UNIV__I,axiom,
! [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).
% UNIV_I
thf(fact_1136_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_1137_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_1138_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_1139_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_1140_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1141_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1142_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1143_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_1144_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_1145_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_1146_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_1147_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1148_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_1149_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_1150_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_1151_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_1152_finite__Plus__UNIV__iff,axiom,
( ( finite51705147264084924um_a_a @ top_to8848906000605539851um_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1153_finite__Plus__UNIV__iff,axiom,
( ( finite2069262655233506379t_unit @ top_to1755696212014396186t_unit )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1154_finite__Plus__UNIV__iff,axiom,
( ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1155_finite__Plus__UNIV__iff,axiom,
( ( finite1276461556078370925unit_a @ top_to5559247480540603964unit_a )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1156_finite__Plus__UNIV__iff,axiom,
( ( finite3146551501593861116t_unit @ top_to2771918933716375115t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1157_finite__Plus__UNIV__iff,axiom,
( ( finite4401952911629260215it_nat @ top_to2894617605782473790it_nat )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1158_finite__Plus__UNIV__iff,axiom,
( ( finite3740268481367103950_nat_a @ top_to54524901450547413_nat_a )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1159_finite__Plus__UNIV__iff,axiom,
( ( finite4327512606132785245t_unit @ top_to5465250082899874788t_unit )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1160_finite__Plus__UNIV__iff,axiom,
( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1161_inf__top__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ X )
= X ) ).
% inf_top_left
thf(fact_1162_inf__top__left,axiom,
! [X: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ X )
= X ) ).
% inf_top_left
thf(fact_1163_inf__top__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X )
= X ) ).
% inf_top_left
thf(fact_1164_inf__top__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ top_top_set_a )
= X ) ).
% inf_top_right
thf(fact_1165_inf__top__right,axiom,
! [X: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ X @ top_to1996260823553986621t_unit )
= X ) ).
% inf_top_right
thf(fact_1166_inf__top__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% inf_top_right
thf(fact_1167_inf__eq__top__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X @ Y )
= top_top_set_a )
= ( ( X = top_top_set_a )
& ( Y = top_top_set_a ) ) ) ).
% inf_eq_top_iff
thf(fact_1168_inf__eq__top__iff,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ X @ Y )
= top_to1996260823553986621t_unit )
= ( ( X = top_to1996260823553986621t_unit )
& ( Y = top_to1996260823553986621t_unit ) ) ) ).
% inf_eq_top_iff
thf(fact_1169_inf__eq__top__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= top_top_set_nat )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_1170_top__eq__inf__iff,axiom,
! [X: set_a,Y: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ X @ Y ) )
= ( ( X = top_top_set_a )
& ( Y = top_top_set_a ) ) ) ).
% top_eq_inf_iff
thf(fact_1171_top__eq__inf__iff,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( top_to1996260823553986621t_unit
= ( inf_in4660618365625256667t_unit @ X @ Y ) )
= ( ( X = top_to1996260823553986621t_unit )
& ( Y = top_to1996260823553986621t_unit ) ) ) ).
% top_eq_inf_iff
thf(fact_1172_top__eq__inf__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X @ Y ) )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_1173_inf__top_Oeq__neutr__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= top_top_set_a )
= ( ( A = top_top_set_a )
& ( B = top_top_set_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1174_inf__top_Oeq__neutr__iff,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A @ B )
= top_to1996260823553986621t_unit )
= ( ( A = top_to1996260823553986621t_unit )
& ( B = top_to1996260823553986621t_unit ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1175_inf__top_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1176_inf__top_Oleft__neutral,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_1177_inf__top_Oleft__neutral,axiom,
! [A: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_1178_inf__top_Oleft__neutral,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_1179_inf__top_Oneutr__eq__iff,axiom,
! [A: set_a,B: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ A @ B ) )
= ( ( A = top_top_set_a )
& ( B = top_top_set_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1180_inf__top_Oneutr__eq__iff,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( top_to1996260823553986621t_unit
= ( inf_in4660618365625256667t_unit @ A @ B ) )
= ( ( A = top_to1996260823553986621t_unit )
& ( B = top_to1996260823553986621t_unit ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1181_inf__top_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A @ B ) )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1182_inf__top_Oright__neutral,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ top_top_set_a )
= A ) ).
% inf_top.right_neutral
thf(fact_1183_inf__top_Oright__neutral,axiom,
! [A: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A @ top_to1996260823553986621t_unit )
= A ) ).
% inf_top.right_neutral
thf(fact_1184_inf__top_Oright__neutral,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% inf_top.right_neutral
thf(fact_1185_boolean__algebra_Odisj__one__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% boolean_algebra.disj_one_right
thf(fact_1186_boolean__algebra_Odisj__one__right,axiom,
! [X: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% boolean_algebra.disj_one_right
thf(fact_1187_boolean__algebra_Odisj__one__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_1188_boolean__algebra_Odisj__one__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X )
= top_top_set_a ) ).
% boolean_algebra.disj_one_left
thf(fact_1189_boolean__algebra_Odisj__one__left,axiom,
! [X: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ X )
= top_to1996260823553986621t_unit ) ).
% boolean_algebra.disj_one_left
thf(fact_1190_boolean__algebra_Odisj__one__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_1191_sup__top__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% sup_top_right
thf(fact_1192_sup__top__right,axiom,
! [X: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% sup_top_right
thf(fact_1193_sup__top__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_1194_sup__top__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X )
= top_top_set_a ) ).
% sup_top_left
thf(fact_1195_sup__top__left,axiom,
! [X: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ X )
= top_to1996260823553986621t_unit ) ).
% sup_top_left
thf(fact_1196_sup__top__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_1197_Int__UNIV,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= top_top_set_a )
= ( ( A2 = top_top_set_a )
& ( B2 = top_top_set_a ) ) ) ).
% Int_UNIV
thf(fact_1198_Int__UNIV,axiom,
! [A2: set_Product_unit,B2: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A2 @ B2 )
= top_to1996260823553986621t_unit )
= ( ( A2 = top_to1996260823553986621t_unit )
& ( B2 = top_to1996260823553986621t_unit ) ) ) ).
% Int_UNIV
thf(fact_1199_Int__UNIV,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= top_top_set_nat )
= ( ( A2 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_1200_Nat_Oadd__0__right,axiom,
! [M5: nat] :
( ( plus_plus_nat @ M5 @ zero_zero_nat )
= M5 ) ).
% Nat.add_0_right
thf(fact_1201_add__is__0,axiom,
! [M5: nat,N: nat] :
( ( ( plus_plus_nat @ M5 @ N )
= zero_zero_nat )
= ( ( M5 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1202_nat__add__left__cancel__le,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M5 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M5 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1203_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1204_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_1205_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_1206_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_1207_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_1208_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_1209_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_1210_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_1211_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_1212_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_1213_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_1214_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_1215_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_1216_Diff__UNIV,axiom,
! [A2: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A2 @ top_to1996260823553986621t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Diff_UNIV
thf(fact_1217_Diff__UNIV,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_1218_Diff__UNIV,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ top_top_set_a )
= bot_bot_set_a ) ).
% Diff_UNIV
thf(fact_1219_add__gr__0,axiom,
! [M5: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M5 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M5 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1220_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1221_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1222_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1223_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_1224_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_1225_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_1226_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C5: nat] :
( ( B
= ( plus_plus_nat @ A @ C5 ) )
=> ( C5 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1227_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_1228_top_Onot__eq__extremum,axiom,
! [A: set_Product_unit] :
( ( A != top_to1996260823553986621t_unit )
= ( ord_le8056459307392131481t_unit @ A @ top_to1996260823553986621t_unit ) ) ).
% top.not_eq_extremum
thf(fact_1229_top_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != top_top_set_nat )
= ( ord_less_set_nat @ A @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_1230_top_Oextremum__strict,axiom,
! [A: set_Product_unit] :
~ ( ord_le8056459307392131481t_unit @ top_to1996260823553986621t_unit @ A ) ).
% top.extremum_strict
thf(fact_1231_top_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A ) ).
% top.extremum_strict
thf(fact_1232_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_1233_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_1234_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1235_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1236_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_1237_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M5: nat,K: nat] :
( ! [M3: nat,N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ( ord_less_nat @ ( F2 @ M3 ) @ ( F2 @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F2 @ M5 ) @ K ) @ ( F2 @ ( plus_plus_nat @ M5 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1238_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1239_add__diff__inverse__nat,axiom,
! [M5: nat,N: nat] :
( ~ ( ord_less_nat @ M5 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M5 @ N ) )
= M5 ) ) ).
% add_diff_inverse_nat
thf(fact_1240_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1241_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_1242_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_1243_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_1244_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_1245_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_1246_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_1247_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1248_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1249_top__greatest,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ top_to1996260823553986621t_unit ) ).
% top_greatest
thf(fact_1250_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_1251_top__greatest,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% top_greatest
thf(fact_1252_top_Oextremum__unique,axiom,
! [A: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A )
= ( A = top_to1996260823553986621t_unit ) ) ).
% top.extremum_unique
thf(fact_1253_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_1254_top_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
= ( A = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_1255_add__leE,axiom,
! [M5: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M5 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M5 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1256_le__add1,axiom,
! [N: nat,M5: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M5 ) ) ).
% le_add1
thf(fact_1257_le__add2,axiom,
! [N: nat,M5: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M5 @ N ) ) ).
% le_add2
thf(fact_1258_add__leD1,axiom,
! [M5: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M5 @ K ) @ N )
=> ( ord_less_eq_nat @ M5 @ N ) ) ).
% add_leD1
thf(fact_1259_add__leD2,axiom,
! [M5: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M5 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1260_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1261_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1262_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1263_trans__le__add1,axiom,
! [I: nat,J: nat,M5: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M5 ) ) ) ).
% trans_le_add1
thf(fact_1264_trans__le__add2,axiom,
! [I: nat,J: nat,M5: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M5 @ J ) ) ) ).
% trans_le_add2
thf(fact_1265_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N5: nat] :
? [K4: nat] :
( N5
= ( plus_plus_nat @ M4 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1266_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1267_add__eq__self__zero,axiom,
! [M5: nat,N: nat] :
( ( ( plus_plus_nat @ M5 @ N )
= M5 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1268_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1269_Nat_Odiff__cancel,axiom,
! [K: nat,M5: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M5 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M5 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1270_diff__cancel2,axiom,
! [M5: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M5 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M5 @ N ) ) ).
% diff_cancel2
thf(fact_1271_diff__add__inverse,axiom,
! [N: nat,M5: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M5 ) @ N )
= M5 ) ).
% diff_add_inverse
% Conjectures (2)
thf(conj_0,hypothesis,
! [B9: a] :
( ( member_a @ B9 @ b )
=> ( ( member_a @ B9 @ g )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------