TPTP Problem File: SLH0122^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Pluennecke_Ruzsa_Inequality/0003_Pluennecke_Ruzsa_Inequality/prob_00527_021042__12332920_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1360 ( 508 unt; 92 typ; 0 def)
% Number of atoms : 3751 (1099 equ; 0 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 11128 ( 456 ~; 81 |; 188 &;8580 @)
% ( 0 <=>;1823 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 7 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 357 ( 357 >; 0 *; 0 +; 0 <<)
% Number of symbols : 84 ( 83 usr; 19 con; 0-5 aty)
% Number of variables : 3476 ( 226 ^;3182 !; 68 ?;3476 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:24:33.052
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (83)
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Real__Oreal,type,
finite_card_real: set_real > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Group__Theory_Oabelian__group_001tf__a,type,
group_201663378560352916roup_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001tf__a,type,
group_4866109990395492029noid_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ogroup_001tf__a,type,
group_group_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001tf__a,type,
group_Units_a: set_a > ( a > a > a ) > a > set_a ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001tf__a,type,
group_inverse_a: set_a > ( a > a > a ) > a > a > a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
inf_inf_int: int > int > int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Real__Oreal,type,
inf_inf_real: real > real > real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Real__Oreal_J,type,
inf_inf_set_real: set_real > set_real > set_real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_001tf__a,type,
pluenn1164192988769422572roup_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_ORuzsa__distance_001tf__a,type,
pluenn5761198478017115492ance_a: set_a > ( a > a > a ) > a > set_a > set_a > real ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Ominusset_001tf__a,type,
pluenn2534204936789923946sset_a: set_a > ( a > a > a ) > a > set_a > set_a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset_001tf__a,type,
pluenn3038260743871226533mset_a: set_a > ( a > a > a ) > set_a > set_a > set_a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset__iterated_001tf__a,type,
pluenn1960970773371692859ated_a: set_a > ( a > a > a ) > a > set_a > nat > set_a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumsetp_001tf__a,type,
pluenn895083305082786853setp_a: set_a > ( a > a > a ) > ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: set_a ).
thf(sy_v_A_H____,type,
a3: set_a ).
thf(sy_v_B,type,
b: set_a ).
thf(sy_v_G,type,
g: set_a ).
thf(sy_v_K,type,
k: real ).
thf(sy_v_K_H____,type,
k2: real ).
thf(sy_v_addition,type,
addition: a > a > a ).
thf(sy_v_thesis,type,
thesis: $o ).
thf(sy_v_zero,type,
zero: a ).
% Relevant facts (1267)
thf(fact_0_A_H_I2_J,axiom,
a3 != bot_bot_set_a ).
% A'(2)
thf(fact_1_assms_I3_J,axiom,
ord_less_eq_set_a @ a2 @ g ).
% assms(3)
thf(fact_2_A_H_I1_J,axiom,
ord_less_eq_set_a @ a3 @ a2 ).
% A'(1)
thf(fact_3_assms_I6_J,axiom,
ord_less_eq_set_a @ b @ g ).
% assms(6)
thf(fact_4_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( addition @ X @ Y )
= ( addition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_5_sumset__mono,axiom,
! [A: set_a,A2: set_a,B: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ A2 )
=> ( ( ord_less_eq_set_a @ B @ B2 )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ).
% sumset_mono
thf(fact_6_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_7_sumset_Ocases,axiom,
! [A3: a,A2: set_a,B2: set_a] :
( ( member_a @ A3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) )
=> ~ ! [A4: a,B3: a] :
( ( A3
= ( addition @ A4 @ B3 ) )
=> ( ( member_a @ A4 @ A2 )
=> ( ( member_a @ A4 @ g )
=> ( ( member_a @ B3 @ B2 )
=> ~ ( member_a @ B3 @ g ) ) ) ) ) ) ).
% sumset.cases
thf(fact_8_sumset_Osimps,axiom,
! [A3: a,A2: set_a,B2: set_a] :
( ( member_a @ A3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) )
= ( ? [A5: a,B4: a] :
( ( A3
= ( addition @ A5 @ B4 ) )
& ( member_a @ A5 @ A2 )
& ( member_a @ A5 @ g )
& ( member_a @ B4 @ B2 )
& ( member_a @ B4 @ g ) ) ) ) ).
% sumset.simps
thf(fact_9_sumset_OsumsetI,axiom,
! [A3: a,A2: set_a,B5: a,B2: set_a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ g )
=> ( ( member_a @ B5 @ B2 )
=> ( ( member_a @ B5 @ g )
=> ( member_a @ ( addition @ A3 @ B5 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ) ) ).
% sumset.sumsetI
thf(fact_10_sumset__assoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ C )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ C ) ) ) ).
% sumset_assoc
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_A_H__card,axiom,
! [C: set_a] :
( ( ord_less_eq_set_a @ C @ g )
=> ( ( finite_finite_a @ C )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ C ) ) ) ) @ ( times_times_real @ k2 @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ C ) ) ) ) ) ) ) ).
% A'_card
thf(fact_13_subsetI,axiom,
! [A2: set_real,B2: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( member_real @ X2 @ B2 ) )
=> ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% subsetI
thf(fact_14_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_15_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_16_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_17_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_18_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_19_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_20_assms_I4_J,axiom,
a2 != bot_bot_set_a ).
% assms(4)
thf(fact_21_assms_I2_J,axiom,
finite_finite_a @ a2 ).
% assms(2)
thf(fact_22_A_H_I3_J,axiom,
ord_less_real @ zero_zero_real @ k2 ).
% A'(3)
thf(fact_23_A_H_I4_J,axiom,
ord_less_eq_real @ k2 @ k ).
% A'(4)
thf(fact_24_False,axiom,
b != bot_bot_set_a ).
% False
thf(fact_25_assms_I5_J,axiom,
finite_finite_a @ b ).
% assms(5)
thf(fact_26_dual__order_Orefl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_27_dual__order_Orefl,axiom,
! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_28_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_29_dual__order_Orefl,axiom,
! [A3: int] : ( ord_less_eq_int @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_30_empty__iff,axiom,
! [C2: real] :
~ ( member_real @ C2 @ bot_bot_set_real ) ).
% empty_iff
thf(fact_31_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_32_all__not__in__conv,axiom,
! [A2: set_real] :
( ( ! [X3: real] :
~ ( member_real @ X3 @ A2 ) )
= ( A2 = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_33_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_34_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_35_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_36_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_37_K,axiom,
ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ b ) ) ) @ ( times_times_real @ k @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a2 ) ) ) ).
% K
thf(fact_38_card__le__sumset,axiom,
! [A2: set_a,A3: a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ g )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ g )
=> ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ) ) ) ) ).
% card_le_sumset
thf(fact_39_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_40_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_41_composition__closed,axiom,
! [A3: a,B5: a] :
( ( member_a @ A3 @ g )
=> ( ( member_a @ B5 @ g )
=> ( member_a @ ( addition @ A3 @ B5 ) @ g ) ) ) ).
% composition_closed
thf(fact_42_associative,axiom,
! [A3: a,B5: a,C2: a] :
( ( member_a @ A3 @ g )
=> ( ( member_a @ B5 @ g )
=> ( ( member_a @ C2 @ g )
=> ( ( addition @ ( addition @ A3 @ B5 ) @ C2 )
= ( addition @ A3 @ ( addition @ B5 @ C2 ) ) ) ) ) ) ).
% associative
thf(fact_43_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_44_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_45_Plu__2__2,axiom,
! [A0: set_a,B2: set_a,K0: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A0 @ B2 ) ) ) @ ( times_times_real @ K0 @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A0 ) ) ) )
=> ( ( finite_finite_a @ A0 )
=> ( ( ord_less_eq_set_a @ A0 @ g )
=> ( ( A0 != bot_bot_set_a )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ g )
=> ( ( B2 != bot_bot_set_a )
=> ~ ! [A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ A0 )
=> ( ( A6 != bot_bot_set_a )
=> ! [K: real] :
( ( ord_less_real @ zero_zero_real @ K )
=> ( ( ord_less_eq_real @ K @ K0 )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ g )
=> ( ( finite_finite_a @ C3 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A6 @ ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ C3 ) ) ) ) @ ( times_times_real @ K @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A6 @ C3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Plu_2_2
thf(fact_46_emptyE,axiom,
! [A3: real] :
~ ( member_real @ A3 @ bot_bot_set_real ) ).
% emptyE
thf(fact_47_emptyE,axiom,
! [A3: a] :
~ ( member_a @ A3 @ bot_bot_set_a ) ).
% emptyE
thf(fact_48_equals0D,axiom,
! [A2: set_real,A3: real] :
( ( A2 = bot_bot_set_real )
=> ~ ( member_real @ A3 @ A2 ) ) ).
% equals0D
thf(fact_49_equals0D,axiom,
! [A2: set_a,A3: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A3 @ A2 ) ) ).
% equals0D
thf(fact_50_equals0I,axiom,
! [A2: set_real] :
( ! [Y2: real] :
~ ( member_real @ Y2 @ A2 )
=> ( A2 = bot_bot_set_real ) ) ).
% equals0I
thf(fact_51_equals0I,axiom,
! [A2: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_52_ex__in__conv,axiom,
! [A2: set_real] :
( ( ? [X3: real] : ( member_real @ X3 @ A2 ) )
= ( A2 != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_53_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_54_additive__abelian__group_Osumset_Ocong,axiom,
pluenn3038260743871226533mset_a = pluenn3038260743871226533mset_a ).
% additive_abelian_group.sumset.cong
thf(fact_55_mem__Collect__eq,axiom,
! [A3: a,P: a > $o] :
( ( member_a @ A3 @ ( collect_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A3: real,P: real > $o] :
( ( member_real @ A3 @ ( collect_real @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_bot_Oextremum__uniqueI,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
=> ( A3 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_60_bot_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
=> ( A3 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_61_bot_Oextremum__unique,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_62_bot_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
= ( A3 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_63_bot_Oextremum,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% bot.extremum
thf(fact_64_bot_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A3 ) ).
% bot.extremum
thf(fact_65_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_66_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_67_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_68_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_69_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_70_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_71_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_72_ord__le__eq__subst,axiom,
! [A3: real,B5: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_73_ord__le__eq__subst,axiom,
! [A3: real,B5: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_74_ord__le__eq__subst,axiom,
! [A3: real,B5: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_75_ord__le__eq__subst,axiom,
! [A3: nat,B5: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_76_ord__le__eq__subst,axiom,
! [A3: nat,B5: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_77_ord__le__eq__subst,axiom,
! [A3: nat,B5: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_78_ord__le__eq__subst,axiom,
! [A3: int,B5: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_79_ord__le__eq__subst,axiom,
! [A3: int,B5: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_80_ord__le__eq__subst,axiom,
! [A3: int,B5: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_81_ord__le__eq__subst,axiom,
! [A3: set_a,B5: set_a,F: set_a > real,C2: real] :
( ( ord_less_eq_set_a @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_82_ord__eq__le__subst,axiom,
! [A3: real,F: real > real,B5: real,C2: real] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_eq_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_83_ord__eq__le__subst,axiom,
! [A3: nat,F: real > nat,B5: real,C2: real] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_eq_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_84_ord__eq__le__subst,axiom,
! [A3: int,F: real > int,B5: real,C2: real] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_eq_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_85_ord__eq__le__subst,axiom,
! [A3: real,F: nat > real,B5: nat,C2: nat] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_eq_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_86_ord__eq__le__subst,axiom,
! [A3: nat,F: nat > nat,B5: nat,C2: nat] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_eq_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_87_ord__eq__le__subst,axiom,
! [A3: int,F: nat > int,B5: nat,C2: nat] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_eq_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_88_ord__eq__le__subst,axiom,
! [A3: real,F: int > real,B5: int,C2: int] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_eq_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_89_ord__eq__le__subst,axiom,
! [A3: nat,F: int > nat,B5: int,C2: int] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_eq_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_90_ord__eq__le__subst,axiom,
! [A3: int,F: int > int,B5: int,C2: int] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_eq_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_91_ord__eq__le__subst,axiom,
! [A3: real,F: set_a > real,B5: set_a,C2: set_a] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_eq_set_a @ B5 @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_92_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_93_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_94_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_95_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_96_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_97_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_98_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_99_order__subst2,axiom,
! [A3: real,B5: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( ord_less_eq_real @ ( F @ B5 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_100_order__subst2,axiom,
! [A3: real,B5: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( ord_less_eq_nat @ ( F @ B5 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_101_order__subst2,axiom,
! [A3: real,B5: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( ord_less_eq_int @ ( F @ B5 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_102_order__subst2,axiom,
! [A3: nat,B5: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( ord_less_eq_real @ ( F @ B5 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_103_order__subst2,axiom,
! [A3: nat,B5: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( ord_less_eq_nat @ ( F @ B5 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_104_order__subst2,axiom,
! [A3: nat,B5: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( ord_less_eq_int @ ( F @ B5 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_105_order__subst2,axiom,
! [A3: int,B5: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ord_less_eq_real @ ( F @ B5 ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_106_order__subst2,axiom,
! [A3: int,B5: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ord_less_eq_nat @ ( F @ B5 ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_107_order__subst2,axiom,
! [A3: int,B5: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ord_less_eq_int @ ( F @ B5 ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_108_order__subst2,axiom,
! [A3: set_a,B5: set_a,F: set_a > real,C2: real] :
( ( ord_less_eq_set_a @ A3 @ B5 )
=> ( ( ord_less_eq_real @ ( F @ B5 ) @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_109_order__subst1,axiom,
! [A3: real,F: real > real,B5: real,C2: real] :
( ( ord_less_eq_real @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_110_order__subst1,axiom,
! [A3: real,F: nat > real,B5: nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_111_order__subst1,axiom,
! [A3: real,F: int > real,B5: int,C2: int] :
( ( ord_less_eq_real @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_112_order__subst1,axiom,
! [A3: nat,F: real > nat,B5: real,C2: real] :
( ( ord_less_eq_nat @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_113_order__subst1,axiom,
! [A3: nat,F: nat > nat,B5: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_114_order__subst1,axiom,
! [A3: nat,F: int > nat,B5: int,C2: int] :
( ( ord_less_eq_nat @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_115_order__subst1,axiom,
! [A3: int,F: real > int,B5: real,C2: real] :
( ( ord_less_eq_int @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_116_order__subst1,axiom,
! [A3: int,F: nat > int,B5: nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_117_order__subst1,axiom,
! [A3: int,F: int > int,B5: int,C2: int] :
( ( ord_less_eq_int @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_118_order__subst1,axiom,
! [A3: set_a,F: real > set_a,B5: real,C2: real] :
( ( ord_less_eq_set_a @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_119_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_120_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_121_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_122_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_123_antisym,axiom,
! [A3: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A3 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ A3 )
=> ( A3 = B5 ) ) ) ).
% antisym
thf(fact_124_antisym,axiom,
! [A3: real,B5: real] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( ord_less_eq_real @ B5 @ A3 )
=> ( A3 = B5 ) ) ) ).
% antisym
thf(fact_125_antisym,axiom,
! [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( ord_less_eq_nat @ B5 @ A3 )
=> ( A3 = B5 ) ) ) ).
% antisym
thf(fact_126_antisym,axiom,
! [A3: int,B5: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ord_less_eq_int @ B5 @ A3 )
=> ( A3 = B5 ) ) ) ).
% antisym
thf(fact_127_dual__order_Otrans,axiom,
! [B5: set_a,A3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B5 @ A3 )
=> ( ( ord_less_eq_set_a @ C2 @ B5 )
=> ( ord_less_eq_set_a @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_128_dual__order_Otrans,axiom,
! [B5: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B5 @ A3 )
=> ( ( ord_less_eq_real @ C2 @ B5 )
=> ( ord_less_eq_real @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_129_dual__order_Otrans,axiom,
! [B5: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B5 @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ B5 )
=> ( ord_less_eq_nat @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_130_dual__order_Otrans,axiom,
! [B5: int,A3: int,C2: int] :
( ( ord_less_eq_int @ B5 @ A3 )
=> ( ( ord_less_eq_int @ C2 @ B5 )
=> ( ord_less_eq_int @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_131_dual__order_Oantisym,axiom,
! [B5: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B5 @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ B5 )
=> ( A3 = B5 ) ) ) ).
% dual_order.antisym
thf(fact_132_dual__order_Oantisym,axiom,
! [B5: real,A3: real] :
( ( ord_less_eq_real @ B5 @ A3 )
=> ( ( ord_less_eq_real @ A3 @ B5 )
=> ( A3 = B5 ) ) ) ).
% dual_order.antisym
thf(fact_133_dual__order_Oantisym,axiom,
! [B5: nat,A3: nat] :
( ( ord_less_eq_nat @ B5 @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ B5 )
=> ( A3 = B5 ) ) ) ).
% dual_order.antisym
thf(fact_134_dual__order_Oantisym,axiom,
! [B5: int,A3: int] :
( ( ord_less_eq_int @ B5 @ A3 )
=> ( ( ord_less_eq_int @ A3 @ B5 )
=> ( A3 = B5 ) ) ) ).
% dual_order.antisym
thf(fact_135_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A5 )
& ( ord_less_eq_set_a @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_136_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_137_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_138_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_139_linorder__wlog,axiom,
! [P: real > real > $o,A3: real,B5: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A3 @ B5 ) ) ) ).
% linorder_wlog
thf(fact_140_linorder__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B5: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A3 @ B5 ) ) ) ).
% linorder_wlog
thf(fact_141_linorder__wlog,axiom,
! [P: int > int > $o,A3: int,B5: int] :
( ! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A3 @ B5 ) ) ) ).
% linorder_wlog
thf(fact_142_order__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_143_order__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_144_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_145_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_146_order_Otrans,axiom,
! [A3: set_a,B5: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_147_order_Otrans,axiom,
! [A3: real,B5: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( ord_less_eq_real @ B5 @ C2 )
=> ( ord_less_eq_real @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_148_order_Otrans,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( ord_less_eq_nat @ B5 @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_149_order_Otrans,axiom,
! [A3: int,B5: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ord_less_eq_int @ B5 @ C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_150_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_151_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_152_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_153_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_154_ord__le__eq__trans,axiom,
! [A3: set_a,B5: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B5 )
=> ( ( B5 = C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_155_ord__le__eq__trans,axiom,
! [A3: real,B5: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( B5 = C2 )
=> ( ord_less_eq_real @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_156_ord__le__eq__trans,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( B5 = C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_157_ord__le__eq__trans,axiom,
! [A3: int,B5: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( B5 = C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_158_ord__eq__le__trans,axiom,
! [A3: set_a,B5: set_a,C2: set_a] :
( ( A3 = B5 )
=> ( ( ord_less_eq_set_a @ B5 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_159_ord__eq__le__trans,axiom,
! [A3: real,B5: real,C2: real] :
( ( A3 = B5 )
=> ( ( ord_less_eq_real @ B5 @ C2 )
=> ( ord_less_eq_real @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_160_ord__eq__le__trans,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( A3 = B5 )
=> ( ( ord_less_eq_nat @ B5 @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_161_ord__eq__le__trans,axiom,
! [A3: int,B5: int,C2: int] :
( ( A3 = B5 )
=> ( ( ord_less_eq_int @ B5 @ C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_162_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_163_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_164_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_165_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_166_le__cases3,axiom,
! [X: real,Y: real,Z2: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_167_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_168_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_169_nle__le,axiom,
! [A3: real,B5: real] :
( ( ~ ( ord_less_eq_real @ A3 @ B5 ) )
= ( ( ord_less_eq_real @ B5 @ A3 )
& ( B5 != A3 ) ) ) ).
% nle_le
thf(fact_170_nle__le,axiom,
! [A3: nat,B5: nat] :
( ( ~ ( ord_less_eq_nat @ A3 @ B5 ) )
= ( ( ord_less_eq_nat @ B5 @ A3 )
& ( B5 != A3 ) ) ) ).
% nle_le
thf(fact_171_nle__le,axiom,
! [A3: int,B5: int] :
( ( ~ ( ord_less_eq_int @ A3 @ B5 ) )
= ( ( ord_less_eq_int @ B5 @ A3 )
& ( B5 != A3 ) ) ) ).
% nle_le
thf(fact_172_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_173_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A7: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_174_subset__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_175_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_176_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_177_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A7: set_real,B6: set_real] :
! [T: real] :
( ( member_real @ T @ A7 )
=> ( member_real @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_178_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B6: set_a] :
! [T: a] :
( ( member_a @ T @ A7 )
=> ( member_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_179_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_180_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_181_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A7: set_real,B6: set_real] :
! [X3: real] :
( ( member_real @ X3 @ A7 )
=> ( member_real @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_182_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B6: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A7 )
=> ( member_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_183_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_184_subsetD,axiom,
! [A2: set_real,B2: set_real,C2: real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_185_subsetD,axiom,
! [A2: set_a,B2: set_a,C2: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_186_in__mono,axiom,
! [A2: set_real,B2: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B2 ) ) ) ).
% in_mono
thf(fact_187_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_188__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062A_H_AK_H_O_A_092_060lbrakk_062A_H_A_092_060subseteq_062_AA_059_AA_H_A_092_060noteq_062_A_123_125_059_A0_A_060_AK_H_059_AK_H_A_092_060le_062_AK_059_A_092_060And_062C_O_A_092_060lbrakk_062C_A_092_060subseteq_062_AG_059_Afinite_AC_092_060rbrakk_062_A_092_060Longrightarrow_062_Areal_A_Icard_A_Isumset_AA_H_A_Isumset_AB_AC_J_J_J_A_092_060le_062_AK_H_A_K_Areal_A_Icard_A_Isumset_AA_H_AC_J_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [A8: set_a] :
( ( ord_less_eq_set_a @ A8 @ a2 )
=> ( ( A8 != bot_bot_set_a )
=> ! [K2: real] :
( ( ord_less_real @ zero_zero_real @ K2 )
=> ( ( ord_less_eq_real @ K2 @ k )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ g )
=> ( ( finite_finite_a @ C3 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A8 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ C3 ) ) ) ) @ ( times_times_real @ K2 @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A8 @ C3 ) ) ) ) ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>A' K'. \<lbrakk>A' \<subseteq> A; A' \<noteq> {}; 0 < K'; K' \<le> K; \<And>C. \<lbrakk>C \<subseteq> G; finite C\<rbrakk> \<Longrightarrow> real (card (sumset A' (sumset B C))) \<le> K' * real (card (sumset A' C))\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_189_sumsetp_OsumsetI,axiom,
! [A2: a > $o,A3: a,B2: a > $o,B5: a] :
( ( A2 @ A3 )
=> ( ( member_a @ A3 @ g )
=> ( ( B2 @ B5 )
=> ( ( member_a @ B5 @ g )
=> ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B2 @ ( addition @ A3 @ B5 ) ) ) ) ) ) ).
% sumsetp.sumsetI
thf(fact_190_sumsetp_Osimps,axiom,
! [A2: a > $o,B2: a > $o,A3: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B2 @ A3 )
= ( ? [A5: a,B4: a] :
( ( A3
= ( addition @ A5 @ B4 ) )
& ( A2 @ A5 )
& ( member_a @ A5 @ g )
& ( B2 @ B4 )
& ( member_a @ B4 @ g ) ) ) ) ).
% sumsetp.simps
thf(fact_191_sumsetp_Ocases,axiom,
! [A2: a > $o,B2: a > $o,A3: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B2 @ A3 )
=> ~ ! [A4: a,B3: a] :
( ( A3
= ( addition @ A4 @ B3 ) )
=> ( ( A2 @ A4 )
=> ( ( member_a @ A4 @ g )
=> ( ( B2 @ B3 )
=> ~ ( member_a @ B3 @ g ) ) ) ) ) ) ).
% sumsetp.cases
thf(fact_192_card__sumset__0__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ g )
=> ( ( ord_less_eq_set_a @ B2 @ g )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
| ( ( finite_card_a @ B2 )
= zero_zero_nat ) ) ) ) ) ).
% card_sumset_0_iff
thf(fact_193_card__sumset__le,axiom,
! [A2: set_a,A3: a] :
( ( finite_finite_a @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ).
% card_sumset_le
thf(fact_194_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_195_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_196_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_197_infinite__sumset__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) )
= ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
& ( ( inf_inf_set_a @ B2 @ g )
!= bot_bot_set_a ) )
| ( ( ( inf_inf_set_a @ A2 @ g )
!= bot_bot_set_a )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ B2 @ g ) ) ) ) ) ).
% infinite_sumset_iff
thf(fact_198_infinite__sumset__aux,axiom,
! [A2: set_a,B2: set_a] :
( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) )
= ( ( inf_inf_set_a @ B2 @ g )
!= bot_bot_set_a ) ) ) ).
% infinite_sumset_aux
thf(fact_199_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_200_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_201_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_202_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_203_insertCI,axiom,
! [A3: a,B2: set_a,B5: a] :
( ( ~ ( member_a @ A3 @ B2 )
=> ( A3 = B5 ) )
=> ( member_a @ A3 @ ( insert_a @ B5 @ B2 ) ) ) ).
% insertCI
thf(fact_204_insertCI,axiom,
! [A3: real,B2: set_real,B5: real] :
( ( ~ ( member_real @ A3 @ B2 )
=> ( A3 = B5 ) )
=> ( member_real @ A3 @ ( insert_real @ B5 @ B2 ) ) ) ).
% insertCI
thf(fact_205_insert__iff,axiom,
! [A3: a,B5: a,A2: set_a] :
( ( member_a @ A3 @ ( insert_a @ B5 @ A2 ) )
= ( ( A3 = B5 )
| ( member_a @ A3 @ A2 ) ) ) ).
% insert_iff
thf(fact_206_insert__iff,axiom,
! [A3: real,B5: real,A2: set_real] :
( ( member_real @ A3 @ ( insert_real @ B5 @ A2 ) )
= ( ( A3 = B5 )
| ( member_real @ A3 @ A2 ) ) ) ).
% insert_iff
thf(fact_207_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_208_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_209_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_210_IntI,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ A2 )
=> ( ( member_real @ C2 @ B2 )
=> ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_211_IntI,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ( member_a @ C2 @ B2 )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_212_Int__iff,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B2 ) )
= ( ( member_real @ C2 @ A2 )
& ( member_real @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_213_Int__iff,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( member_a @ C2 @ A2 )
& ( member_a @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_214_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_215_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_216_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_217_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_218_card__sumset__0__iff_H,axiom,
! [A2: set_a,B2: set_a] :
( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) )
= zero_zero_nat )
| ( ( finite_card_a @ ( inf_inf_set_a @ B2 @ g ) )
= zero_zero_nat ) ) ) ).
% card_sumset_0_iff'
thf(fact_219_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_220_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_221_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_222_singletonI,axiom,
! [A3: real] : ( member_real @ A3 @ ( insert_real @ A3 @ bot_bot_set_real ) ) ).
% singletonI
thf(fact_223_singletonI,axiom,
! [A3: a] : ( member_a @ A3 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_224_insert__subset,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ ( insert_real @ X @ A2 ) @ B2 )
= ( ( member_real @ X @ B2 )
& ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_225_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_226_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_227_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_228_Int__subset__iff,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ C @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_229_Int__insert__left__if0,axiom,
! [A3: real,C: set_real,B2: set_real] :
( ~ ( member_real @ A3 @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A3 @ B2 ) @ C )
= ( inf_inf_set_real @ B2 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_230_Int__insert__left__if0,axiom,
! [A3: a,C: set_a,B2: set_a] :
( ~ ( member_a @ A3 @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B2 ) @ C )
= ( inf_inf_set_a @ B2 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_231_Int__insert__left__if1,axiom,
! [A3: real,C: set_real,B2: set_real] :
( ( member_real @ A3 @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A3 @ B2 ) @ C )
= ( insert_real @ A3 @ ( inf_inf_set_real @ B2 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_232_Int__insert__left__if1,axiom,
! [A3: a,C: set_a,B2: set_a] :
( ( member_a @ A3 @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B2 ) @ C )
= ( insert_a @ A3 @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_233_insert__inter__insert,axiom,
! [A3: a,A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A3 @ A2 ) @ ( insert_a @ A3 @ B2 ) )
= ( insert_a @ A3 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_234_Int__insert__right__if0,axiom,
! [A3: real,A2: set_real,B2: set_real] :
( ~ ( member_real @ A3 @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A3 @ B2 ) )
= ( inf_inf_set_real @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_235_Int__insert__right__if0,axiom,
! [A3: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ A3 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A3 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_236_Int__insert__right__if1,axiom,
! [A3: real,A2: set_real,B2: set_real] :
( ( member_real @ A3 @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A3 @ B2 ) )
= ( insert_real @ A3 @ ( inf_inf_set_real @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_237_Int__insert__right__if1,axiom,
! [A3: a,A2: set_a,B2: set_a] :
( ( member_a @ A3 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A3 @ B2 ) )
= ( insert_a @ A3 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_238_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_239_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_240_mult__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K3 @ M )
= ( times_times_nat @ K3 @ N ) )
= ( ( M = N )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_241_mult__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ( times_times_nat @ M @ K3 )
= ( times_times_nat @ N @ K3 ) )
= ( ( M = N )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_242_card__sumset__singleton__eq,axiom,
! [A2: set_a,A3: a] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ A3 @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A3 @ bot_bot_set_a ) ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) ) ) )
& ( ~ ( member_a @ A3 @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A3 @ bot_bot_set_a ) ) )
= zero_zero_nat ) ) ) ) ).
% card_sumset_singleton_eq
thf(fact_243_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_244_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_245_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_246_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_247_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_248_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_249_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_250_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_251_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_252_singleton__insert__inj__eq,axiom,
! [B5: a,A3: a,A2: set_a] :
( ( ( insert_a @ B5 @ bot_bot_set_a )
= ( insert_a @ A3 @ A2 ) )
= ( ( A3 = B5 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B5 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_253_singleton__insert__inj__eq_H,axiom,
! [A3: a,A2: set_a,B5: a] :
( ( ( insert_a @ A3 @ A2 )
= ( insert_a @ B5 @ bot_bot_set_a ) )
= ( ( A3 = B5 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B5 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_254_insert__disjoint_I1_J,axiom,
! [A3: real,A2: set_real,B2: set_real] :
( ( ( inf_inf_set_real @ ( insert_real @ A3 @ A2 ) @ B2 )
= bot_bot_set_real )
= ( ~ ( member_real @ A3 @ B2 )
& ( ( inf_inf_set_real @ A2 @ B2 )
= bot_bot_set_real ) ) ) ).
% insert_disjoint(1)
thf(fact_255_insert__disjoint_I1_J,axiom,
! [A3: a,A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A3 @ A2 ) @ B2 )
= bot_bot_set_a )
= ( ~ ( member_a @ A3 @ B2 )
& ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_256_insert__disjoint_I2_J,axiom,
! [A3: real,A2: set_real,B2: set_real] :
( ( bot_bot_set_real
= ( inf_inf_set_real @ ( insert_real @ A3 @ A2 ) @ B2 ) )
= ( ~ ( member_real @ A3 @ B2 )
& ( bot_bot_set_real
= ( inf_inf_set_real @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_257_insert__disjoint_I2_J,axiom,
! [A3: a,A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A3 @ A2 ) @ B2 ) )
= ( ~ ( member_a @ A3 @ B2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_258_disjoint__insert_I1_J,axiom,
! [B2: set_real,A3: real,A2: set_real] :
( ( ( inf_inf_set_real @ B2 @ ( insert_real @ A3 @ A2 ) )
= bot_bot_set_real )
= ( ~ ( member_real @ A3 @ B2 )
& ( ( inf_inf_set_real @ B2 @ A2 )
= bot_bot_set_real ) ) ) ).
% disjoint_insert(1)
thf(fact_259_disjoint__insert_I1_J,axiom,
! [B2: set_a,A3: a,A2: set_a] :
( ( ( inf_inf_set_a @ B2 @ ( insert_a @ A3 @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A3 @ B2 )
& ( ( inf_inf_set_a @ B2 @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_260_disjoint__insert_I2_J,axiom,
! [A2: set_real,B5: real,B2: set_real] :
( ( bot_bot_set_real
= ( inf_inf_set_real @ A2 @ ( insert_real @ B5 @ B2 ) ) )
= ( ~ ( member_real @ B5 @ A2 )
& ( bot_bot_set_real
= ( inf_inf_set_real @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_261_disjoint__insert_I2_J,axiom,
! [A2: set_a,B5: a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B5 @ B2 ) ) )
= ( ~ ( member_a @ B5 @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_262_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_263_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_264_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_265_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_266_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_267_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_268_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_269_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_270_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_271_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_272_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_273_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_274_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_275_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_276_IntE,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B2 ) )
=> ~ ( ( member_real @ C2 @ A2 )
=> ~ ( member_real @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_277_IntE,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ~ ( member_a @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_278_IntD1,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B2 ) )
=> ( member_real @ C2 @ A2 ) ) ).
% IntD1
thf(fact_279_IntD1,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_280_IntD2,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B2 ) )
=> ( member_real @ C2 @ B2 ) ) ).
% IntD2
thf(fact_281_IntD2,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C2 @ B2 ) ) ).
% IntD2
thf(fact_282_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_283_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_284_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_285_insertE,axiom,
! [A3: a,B5: a,A2: set_a] :
( ( member_a @ A3 @ ( insert_a @ B5 @ A2 ) )
=> ( ( A3 != B5 )
=> ( member_a @ A3 @ A2 ) ) ) ).
% insertE
thf(fact_286_insertE,axiom,
! [A3: real,B5: real,A2: set_real] :
( ( member_real @ A3 @ ( insert_real @ B5 @ A2 ) )
=> ( ( A3 != B5 )
=> ( member_real @ A3 @ A2 ) ) ) ).
% insertE
thf(fact_287_le__trans,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K3 )
=> ( ord_less_eq_nat @ I @ K3 ) ) ) ).
% le_trans
thf(fact_288_insertI1,axiom,
! [A3: a,B2: set_a] : ( member_a @ A3 @ ( insert_a @ A3 @ B2 ) ) ).
% insertI1
thf(fact_289_insertI1,axiom,
! [A3: real,B2: set_real] : ( member_real @ A3 @ ( insert_real @ A3 @ B2 ) ) ).
% insertI1
thf(fact_290_insertI2,axiom,
! [A3: a,B2: set_a,B5: a] :
( ( member_a @ A3 @ B2 )
=> ( member_a @ A3 @ ( insert_a @ B5 @ B2 ) ) ) ).
% insertI2
thf(fact_291_insertI2,axiom,
! [A3: real,B2: set_real,B5: real] :
( ( member_real @ A3 @ B2 )
=> ( member_real @ A3 @ ( insert_real @ B5 @ B2 ) ) ) ).
% insertI2
thf(fact_292_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_293_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_294_Int__assoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% Int_assoc
thf(fact_295_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_296_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_297_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_298_Set_Oset__insert,axiom,
! [X: real,A2: set_real] :
( ( member_real @ X @ A2 )
=> ~ ! [B7: set_real] :
( ( A2
= ( insert_real @ X @ B7 ) )
=> ( member_real @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_299_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_300_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A7: set_a,B6: set_a] : ( inf_inf_set_a @ B6 @ A7 ) ) ) ).
% Int_commute
thf(fact_301_mult__le__mono,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K3 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_302_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_303_insert__ident,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ~ ( member_real @ X @ B2 )
=> ( ( ( insert_real @ X @ A2 )
= ( insert_real @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_304_mult__le__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ).
% mult_le_mono1
thf(fact_305_mult__le__mono2,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K3 @ I ) @ ( times_times_nat @ K3 @ J ) ) ) ).
% mult_le_mono2
thf(fact_306_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_307_insert__absorb,axiom,
! [A3: a,A2: set_a] :
( ( member_a @ A3 @ A2 )
=> ( ( insert_a @ A3 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_308_insert__absorb,axiom,
! [A3: real,A2: set_real] :
( ( member_real @ A3 @ A2 )
=> ( ( insert_real @ A3 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_309_insert__eq__iff,axiom,
! [A3: a,A2: set_a,B5: a,B2: set_a] :
( ~ ( member_a @ A3 @ A2 )
=> ( ~ ( member_a @ B5 @ B2 )
=> ( ( ( insert_a @ A3 @ A2 )
= ( insert_a @ B5 @ B2 ) )
= ( ( ( A3 = B5 )
=> ( A2 = B2 ) )
& ( ( A3 != B5 )
=> ? [C4: set_a] :
( ( A2
= ( insert_a @ B5 @ C4 ) )
& ~ ( member_a @ B5 @ C4 )
& ( B2
= ( insert_a @ A3 @ C4 ) )
& ~ ( member_a @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_310_insert__eq__iff,axiom,
! [A3: real,A2: set_real,B5: real,B2: set_real] :
( ~ ( member_real @ A3 @ A2 )
=> ( ~ ( member_real @ B5 @ B2 )
=> ( ( ( insert_real @ A3 @ A2 )
= ( insert_real @ B5 @ B2 ) )
= ( ( ( A3 = B5 )
=> ( A2 = B2 ) )
& ( ( A3 != B5 )
=> ? [C4: set_real] :
( ( A2
= ( insert_real @ B5 @ C4 ) )
& ~ ( member_real @ B5 @ C4 )
& ( B2
= ( insert_real @ A3 @ C4 ) )
& ~ ( member_real @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_311_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_312_Int__insert__left,axiom,
! [A3: real,C: set_real,B2: set_real] :
( ( ( member_real @ A3 @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A3 @ B2 ) @ C )
= ( insert_real @ A3 @ ( inf_inf_set_real @ B2 @ C ) ) ) )
& ( ~ ( member_real @ A3 @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A3 @ B2 ) @ C )
= ( inf_inf_set_real @ B2 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_313_Int__insert__left,axiom,
! [A3: a,C: set_a,B2: set_a] :
( ( ( member_a @ A3 @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B2 ) @ C )
= ( insert_a @ A3 @ ( inf_inf_set_a @ B2 @ C ) ) ) )
& ( ~ ( member_a @ A3 @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B2 ) @ C )
= ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_314_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_315_Int__insert__right,axiom,
! [A3: real,A2: set_real,B2: set_real] :
( ( ( member_real @ A3 @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A3 @ B2 ) )
= ( insert_real @ A3 @ ( inf_inf_set_real @ A2 @ B2 ) ) ) )
& ( ~ ( member_real @ A3 @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A3 @ B2 ) )
= ( inf_inf_set_real @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_316_Int__insert__right,axiom,
! [A3: a,A2: set_a,B2: set_a] :
( ( ( member_a @ A3 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A3 @ B2 ) )
= ( insert_a @ A3 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_a @ A3 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A3 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_317_Int__left__commute,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
= ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_left_commute
thf(fact_318_mk__disjoint__insert,axiom,
! [A3: a,A2: set_a] :
( ( member_a @ A3 @ A2 )
=> ? [B7: set_a] :
( ( A2
= ( insert_a @ A3 @ B7 ) )
& ~ ( member_a @ A3 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_319_mk__disjoint__insert,axiom,
! [A3: real,A2: set_real] :
( ( member_real @ A3 @ A2 )
=> ? [B7: set_real] :
( ( A2
= ( insert_real @ A3 @ B7 ) )
& ~ ( member_real @ A3 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_320_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K3: nat,B5: nat] :
( ( P @ K3 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B5 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_321_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_322_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_323_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_324_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_325_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_326_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_327_additive__abelian__group_Osumsetp_Ocong,axiom,
pluenn895083305082786853setp_a = pluenn895083305082786853setp_a ).
% additive_abelian_group.sumsetp.cong
thf(fact_328_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_329_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_330_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_331_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_332_disjoint__iff__not__equal,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ B2 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_333_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_334_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_335_disjoint__iff,axiom,
! [A2: set_real,B2: set_real] :
( ( ( inf_inf_set_real @ A2 @ B2 )
= bot_bot_set_real )
= ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ~ ( member_real @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_336_disjoint__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_337_Int__emptyI,axiom,
! [A2: set_real,B2: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ~ ( member_real @ X2 @ B2 ) )
=> ( ( inf_inf_set_real @ A2 @ B2 )
= bot_bot_set_real ) ) ).
% Int_emptyI
thf(fact_338_Int__emptyI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_339_singleton__inject,axiom,
! [A3: a,B5: a] :
( ( ( insert_a @ A3 @ bot_bot_set_a )
= ( insert_a @ B5 @ bot_bot_set_a ) )
=> ( A3 = B5 ) ) ).
% singleton_inject
thf(fact_340_insert__not__empty,axiom,
! [A3: a,A2: set_a] :
( ( insert_a @ A3 @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_341_doubleton__eq__iff,axiom,
! [A3: a,B5: a,C2: a,D: a] :
( ( ( insert_a @ A3 @ ( insert_a @ B5 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A3 = C2 )
& ( B5 = D ) )
| ( ( A3 = D )
& ( B5 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_342_singleton__iff,axiom,
! [B5: real,A3: real] :
( ( member_real @ B5 @ ( insert_real @ A3 @ bot_bot_set_real ) )
= ( B5 = A3 ) ) ).
% singleton_iff
thf(fact_343_singleton__iff,axiom,
! [B5: a,A3: a] :
( ( member_a @ B5 @ ( insert_a @ A3 @ bot_bot_set_a ) )
= ( B5 = A3 ) ) ).
% singleton_iff
thf(fact_344_singletonD,axiom,
! [B5: real,A3: real] :
( ( member_real @ B5 @ ( insert_real @ A3 @ bot_bot_set_real ) )
=> ( B5 = A3 ) ) ).
% singletonD
thf(fact_345_singletonD,axiom,
! [B5: a,A3: a] :
( ( member_a @ B5 @ ( insert_a @ A3 @ bot_bot_set_a ) )
=> ( B5 = A3 ) ) ).
% singletonD
thf(fact_346_Int__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_347_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_348_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_349_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_350_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_351_Int__greatest,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_352_Int__Collect__mono,axiom,
! [A2: set_real,B2: set_real,P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_real @ ( inf_inf_set_real @ A2 @ ( collect_real @ P ) ) @ ( inf_inf_set_real @ B2 @ ( collect_real @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_353_Int__Collect__mono,axiom,
! [A2: set_a,B2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( 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_354_insert__mono,axiom,
! [C: set_a,D2: set_a,A3: a] :
( ( ord_less_eq_set_a @ C @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A3 @ C ) @ ( insert_a @ A3 @ D2 ) ) ) ).
% insert_mono
thf(fact_355_subset__insert,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) )
= ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_356_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_357_subset__insertI,axiom,
! [B2: set_a,A3: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A3 @ B2 ) ) ).
% subset_insertI
thf(fact_358_subset__insertI2,axiom,
! [A2: set_a,B2: set_a,B5: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B5 @ B2 ) ) ) ).
% subset_insertI2
thf(fact_359_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_360_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_361_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_362_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_363_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_364_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_365_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_366_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_367_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_368_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_369_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_370_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_371_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_372_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_373_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_374_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_375_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_376_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_377_order__less__subst2,axiom,
! [A3: real,B5: real,F: real > real,C2: real] :
( ( ord_less_real @ A3 @ B5 )
=> ( ( ord_less_real @ ( F @ B5 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_378_order__less__subst2,axiom,
! [A3: real,B5: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A3 @ B5 )
=> ( ( ord_less_nat @ ( F @ B5 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_379_order__less__subst2,axiom,
! [A3: real,B5: real,F: real > int,C2: int] :
( ( ord_less_real @ A3 @ B5 )
=> ( ( ord_less_int @ ( F @ B5 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_380_order__less__subst2,axiom,
! [A3: nat,B5: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ord_less_real @ ( F @ B5 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_381_order__less__subst2,axiom,
! [A3: nat,B5: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ord_less_nat @ ( F @ B5 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_382_order__less__subst2,axiom,
! [A3: nat,B5: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ord_less_int @ ( F @ B5 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_383_order__less__subst2,axiom,
! [A3: int,B5: int,F: int > real,C2: real] :
( ( ord_less_int @ A3 @ B5 )
=> ( ( ord_less_real @ ( F @ B5 ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_384_order__less__subst2,axiom,
! [A3: int,B5: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A3 @ B5 )
=> ( ( ord_less_nat @ ( F @ B5 ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_385_order__less__subst2,axiom,
! [A3: int,B5: int,F: int > int,C2: int] :
( ( ord_less_int @ A3 @ B5 )
=> ( ( ord_less_int @ ( F @ B5 ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_386_order__less__subst1,axiom,
! [A3: real,F: real > real,B5: real,C2: real] :
( ( ord_less_real @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_387_order__less__subst1,axiom,
! [A3: real,F: nat > real,B5: nat,C2: nat] :
( ( ord_less_real @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_388_order__less__subst1,axiom,
! [A3: real,F: int > real,B5: int,C2: int] :
( ( ord_less_real @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_389_order__less__subst1,axiom,
! [A3: nat,F: real > nat,B5: real,C2: real] :
( ( ord_less_nat @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_390_order__less__subst1,axiom,
! [A3: nat,F: nat > nat,B5: nat,C2: nat] :
( ( ord_less_nat @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_391_order__less__subst1,axiom,
! [A3: nat,F: int > nat,B5: int,C2: int] :
( ( ord_less_nat @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_392_order__less__subst1,axiom,
! [A3: int,F: real > int,B5: real,C2: real] :
( ( ord_less_int @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_393_order__less__subst1,axiom,
! [A3: int,F: nat > int,B5: nat,C2: nat] :
( ( ord_less_int @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_394_order__less__subst1,axiom,
! [A3: int,F: int > int,B5: int,C2: int] :
( ( ord_less_int @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_395_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_396_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_397_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_398_ord__less__eq__subst,axiom,
! [A3: real,B5: real,F: real > real,C2: real] :
( ( ord_less_real @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_399_ord__less__eq__subst,axiom,
! [A3: real,B5: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_400_ord__less__eq__subst,axiom,
! [A3: real,B5: real,F: real > int,C2: int] :
( ( ord_less_real @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_401_ord__less__eq__subst,axiom,
! [A3: nat,B5: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_402_ord__less__eq__subst,axiom,
! [A3: nat,B5: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_403_ord__less__eq__subst,axiom,
! [A3: nat,B5: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_404_ord__less__eq__subst,axiom,
! [A3: int,B5: int,F: int > real,C2: real] :
( ( ord_less_int @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_405_ord__less__eq__subst,axiom,
! [A3: int,B5: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_406_ord__less__eq__subst,axiom,
! [A3: int,B5: int,F: int > int,C2: int] :
( ( ord_less_int @ A3 @ B5 )
=> ( ( ( F @ B5 )
= C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_407_ord__eq__less__subst,axiom,
! [A3: real,F: real > real,B5: real,C2: real] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_408_ord__eq__less__subst,axiom,
! [A3: nat,F: real > nat,B5: real,C2: real] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_409_ord__eq__less__subst,axiom,
! [A3: int,F: real > int,B5: real,C2: real] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_410_ord__eq__less__subst,axiom,
! [A3: real,F: nat > real,B5: nat,C2: nat] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_411_ord__eq__less__subst,axiom,
! [A3: nat,F: nat > nat,B5: nat,C2: nat] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_412_ord__eq__less__subst,axiom,
! [A3: int,F: nat > int,B5: nat,C2: nat] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_413_ord__eq__less__subst,axiom,
! [A3: real,F: int > real,B5: int,C2: int] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_414_ord__eq__less__subst,axiom,
! [A3: nat,F: int > nat,B5: int,C2: int] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_415_ord__eq__less__subst,axiom,
! [A3: int,F: int > int,B5: int,C2: int] :
( ( A3
= ( F @ B5 ) )
=> ( ( ord_less_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_416_order__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_417_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_418_order__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_419_order__less__asym_H,axiom,
! [A3: real,B5: real] :
( ( ord_less_real @ A3 @ B5 )
=> ~ ( ord_less_real @ B5 @ A3 ) ) ).
% order_less_asym'
thf(fact_420_order__less__asym_H,axiom,
! [A3: nat,B5: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ~ ( ord_less_nat @ B5 @ A3 ) ) ).
% order_less_asym'
thf(fact_421_order__less__asym_H,axiom,
! [A3: int,B5: int] :
( ( ord_less_int @ A3 @ B5 )
=> ~ ( ord_less_int @ B5 @ A3 ) ) ).
% order_less_asym'
thf(fact_422_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_423_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_424_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_425_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_426_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_427_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_428_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_429_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_430_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_431_dual__order_Ostrict__implies__not__eq,axiom,
! [B5: real,A3: real] :
( ( ord_less_real @ B5 @ A3 )
=> ( A3 != B5 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_432_dual__order_Ostrict__implies__not__eq,axiom,
! [B5: nat,A3: nat] :
( ( ord_less_nat @ B5 @ A3 )
=> ( A3 != B5 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_433_dual__order_Ostrict__implies__not__eq,axiom,
! [B5: int,A3: int] :
( ( ord_less_int @ B5 @ A3 )
=> ( A3 != B5 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_434_order_Ostrict__implies__not__eq,axiom,
! [A3: real,B5: real] :
( ( ord_less_real @ A3 @ B5 )
=> ( A3 != B5 ) ) ).
% order.strict_implies_not_eq
thf(fact_435_order_Ostrict__implies__not__eq,axiom,
! [A3: nat,B5: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( A3 != B5 ) ) ).
% order.strict_implies_not_eq
thf(fact_436_order_Ostrict__implies__not__eq,axiom,
! [A3: int,B5: int] :
( ( ord_less_int @ A3 @ B5 )
=> ( A3 != B5 ) ) ).
% order.strict_implies_not_eq
thf(fact_437_dual__order_Ostrict__trans,axiom,
! [B5: real,A3: real,C2: real] :
( ( ord_less_real @ B5 @ A3 )
=> ( ( ord_less_real @ C2 @ B5 )
=> ( ord_less_real @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_438_dual__order_Ostrict__trans,axiom,
! [B5: nat,A3: nat,C2: nat] :
( ( ord_less_nat @ B5 @ A3 )
=> ( ( ord_less_nat @ C2 @ B5 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_439_dual__order_Ostrict__trans,axiom,
! [B5: int,A3: int,C2: int] :
( ( ord_less_int @ B5 @ A3 )
=> ( ( ord_less_int @ C2 @ B5 )
=> ( ord_less_int @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_440_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_441_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_442_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_443_order_Ostrict__trans,axiom,
! [A3: real,B5: real,C2: real] :
( ( ord_less_real @ A3 @ B5 )
=> ( ( ord_less_real @ B5 @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_444_order_Ostrict__trans,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ord_less_nat @ B5 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_445_order_Ostrict__trans,axiom,
! [A3: int,B5: int,C2: int] :
( ( ord_less_int @ A3 @ B5 )
=> ( ( ord_less_int @ B5 @ C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_446_linorder__less__wlog,axiom,
! [P: real > real > $o,A3: real,B5: real] :
( ! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A3 @ B5 ) ) ) ) ).
% linorder_less_wlog
thf(fact_447_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B5: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A3 @ B5 ) ) ) ) ).
% linorder_less_wlog
thf(fact_448_linorder__less__wlog,axiom,
! [P: int > int > $o,A3: int,B5: int] :
( ! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A3 @ B5 ) ) ) ) ).
% linorder_less_wlog
thf(fact_449_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_450_dual__order_Oirrefl,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_451_dual__order_Oirrefl,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_452_dual__order_Oirrefl,axiom,
! [A3: int] :
~ ( ord_less_int @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_453_dual__order_Oasym,axiom,
! [B5: real,A3: real] :
( ( ord_less_real @ B5 @ A3 )
=> ~ ( ord_less_real @ A3 @ B5 ) ) ).
% dual_order.asym
thf(fact_454_dual__order_Oasym,axiom,
! [B5: nat,A3: nat] :
( ( ord_less_nat @ B5 @ A3 )
=> ~ ( ord_less_nat @ A3 @ B5 ) ) ).
% dual_order.asym
thf(fact_455_dual__order_Oasym,axiom,
! [B5: int,A3: int] :
( ( ord_less_int @ B5 @ A3 )
=> ~ ( ord_less_int @ A3 @ B5 ) ) ).
% dual_order.asym
thf(fact_456_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_457_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_458_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_459_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_460_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_461_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_462_less__induct,axiom,
! [P: nat > $o,A3: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ).
% less_induct
thf(fact_463_ord__less__eq__trans,axiom,
! [A3: real,B5: real,C2: real] :
( ( ord_less_real @ A3 @ B5 )
=> ( ( B5 = C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_464_ord__less__eq__trans,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( B5 = C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_465_ord__less__eq__trans,axiom,
! [A3: int,B5: int,C2: int] :
( ( ord_less_int @ A3 @ B5 )
=> ( ( B5 = C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_466_ord__eq__less__trans,axiom,
! [A3: real,B5: real,C2: real] :
( ( A3 = B5 )
=> ( ( ord_less_real @ B5 @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_467_ord__eq__less__trans,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( A3 = B5 )
=> ( ( ord_less_nat @ B5 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_468_ord__eq__less__trans,axiom,
! [A3: int,B5: int,C2: int] :
( ( A3 = B5 )
=> ( ( ord_less_int @ B5 @ C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_469_order_Oasym,axiom,
! [A3: real,B5: real] :
( ( ord_less_real @ A3 @ B5 )
=> ~ ( ord_less_real @ B5 @ A3 ) ) ).
% order.asym
thf(fact_470_order_Oasym,axiom,
! [A3: nat,B5: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ~ ( ord_less_nat @ B5 @ A3 ) ) ).
% order.asym
thf(fact_471_order_Oasym,axiom,
! [A3: int,B5: int] :
( ( ord_less_int @ A3 @ B5 )
=> ~ ( ord_less_int @ B5 @ A3 ) ) ).
% order.asym
thf(fact_472_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_473_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_474_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_475_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y ) ) ) ).
% dense
thf(fact_476_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_477_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_478_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_479_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_480_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_481_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_482_subset__singleton__iff,axiom,
! [X5: set_a,A3: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A3 @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_483_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_484_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_485_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_486_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_487_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_488_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_489_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_490_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_491_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_492_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_493_nless__le,axiom,
! [A3: set_a,B5: set_a] :
( ( ~ ( ord_less_set_a @ A3 @ B5 ) )
= ( ~ ( ord_less_eq_set_a @ A3 @ B5 )
| ( A3 = B5 ) ) ) ).
% nless_le
thf(fact_494_nless__le,axiom,
! [A3: real,B5: real] :
( ( ~ ( ord_less_real @ A3 @ B5 ) )
= ( ~ ( ord_less_eq_real @ A3 @ B5 )
| ( A3 = B5 ) ) ) ).
% nless_le
thf(fact_495_nless__le,axiom,
! [A3: nat,B5: nat] :
( ( ~ ( ord_less_nat @ A3 @ B5 ) )
= ( ~ ( ord_less_eq_nat @ A3 @ B5 )
| ( A3 = B5 ) ) ) ).
% nless_le
thf(fact_496_nless__le,axiom,
! [A3: int,B5: int] :
( ( ~ ( ord_less_int @ A3 @ B5 ) )
= ( ~ ( ord_less_eq_int @ A3 @ B5 )
| ( A3 = B5 ) ) ) ).
% nless_le
thf(fact_497_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_498_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_499_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_500_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_501_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_502_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_503_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_504_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_505_dense__ge,axiom,
! [Z2: real,Y: real] :
( ! [X2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ord_less_eq_real @ Y @ X2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_506_dense__le,axiom,
! [Y: real,Z2: real] :
( ! [X2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_le
thf(fact_507_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_508_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_509_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_510_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_511_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_512_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_513_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_514_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_set_a @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_515_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_real @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_516_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_517_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_int @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_518_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_519_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_520_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_521_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_522_order_Ostrict__trans1,axiom,
! [A3: set_a,B5: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B5 )
=> ( ( ord_less_set_a @ B5 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_523_order_Ostrict__trans1,axiom,
! [A3: real,B5: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( ord_less_real @ B5 @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_524_order_Ostrict__trans1,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( ord_less_nat @ B5 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_525_order_Ostrict__trans1,axiom,
! [A3: int,B5: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ord_less_int @ B5 @ C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_526_order_Ostrict__trans2,axiom,
! [A3: set_a,B5: set_a,C2: set_a] :
( ( ord_less_set_a @ A3 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_527_order_Ostrict__trans2,axiom,
! [A3: real,B5: real,C2: real] :
( ( ord_less_real @ A3 @ B5 )
=> ( ( ord_less_eq_real @ B5 @ C2 )
=> ( ord_less_real @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_528_order_Ostrict__trans2,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ord_less_eq_nat @ B5 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_529_order_Ostrict__trans2,axiom,
! [A3: int,B5: int,C2: int] :
( ( ord_less_int @ A3 @ B5 )
=> ( ( ord_less_eq_int @ B5 @ C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_530_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_531_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ~ ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_532_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_533_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_534_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_535_dense__le__bounded,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_536_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A5: set_a] :
( ( ord_less_set_a @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_537_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_real @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_538_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_539_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_int @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_540_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A5: set_a] :
( ( ord_less_eq_set_a @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_541_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_542_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_543_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_544_dual__order_Ostrict__trans1,axiom,
! [B5: set_a,A3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B5 @ A3 )
=> ( ( ord_less_set_a @ C2 @ B5 )
=> ( ord_less_set_a @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_545_dual__order_Ostrict__trans1,axiom,
! [B5: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B5 @ A3 )
=> ( ( ord_less_real @ C2 @ B5 )
=> ( ord_less_real @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_546_dual__order_Ostrict__trans1,axiom,
! [B5: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B5 @ A3 )
=> ( ( ord_less_nat @ C2 @ B5 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_547_dual__order_Ostrict__trans1,axiom,
! [B5: int,A3: int,C2: int] :
( ( ord_less_eq_int @ B5 @ A3 )
=> ( ( ord_less_int @ C2 @ B5 )
=> ( ord_less_int @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_548_dual__order_Ostrict__trans2,axiom,
! [B5: set_a,A3: set_a,C2: set_a] :
( ( ord_less_set_a @ B5 @ A3 )
=> ( ( ord_less_eq_set_a @ C2 @ B5 )
=> ( ord_less_set_a @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_549_dual__order_Ostrict__trans2,axiom,
! [B5: real,A3: real,C2: real] :
( ( ord_less_real @ B5 @ A3 )
=> ( ( ord_less_eq_real @ C2 @ B5 )
=> ( ord_less_real @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_550_dual__order_Ostrict__trans2,axiom,
! [B5: nat,A3: nat,C2: nat] :
( ( ord_less_nat @ B5 @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ B5 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_551_dual__order_Ostrict__trans2,axiom,
! [B5: int,A3: int,C2: int] :
( ( ord_less_int @ B5 @ A3 )
=> ( ( ord_less_eq_int @ C2 @ B5 )
=> ( ord_less_int @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_552_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A5: set_a] :
( ( ord_less_eq_set_a @ B4 @ A5 )
& ~ ( ord_less_eq_set_a @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_553_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ~ ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_554_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_555_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_556_order_Ostrict__implies__order,axiom,
! [A3: set_a,B5: set_a] :
( ( ord_less_set_a @ A3 @ B5 )
=> ( ord_less_eq_set_a @ A3 @ B5 ) ) ).
% order.strict_implies_order
thf(fact_557_order_Ostrict__implies__order,axiom,
! [A3: real,B5: real] :
( ( ord_less_real @ A3 @ B5 )
=> ( ord_less_eq_real @ A3 @ B5 ) ) ).
% order.strict_implies_order
thf(fact_558_order_Ostrict__implies__order,axiom,
! [A3: nat,B5: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ord_less_eq_nat @ A3 @ B5 ) ) ).
% order.strict_implies_order
thf(fact_559_order_Ostrict__implies__order,axiom,
! [A3: int,B5: int] :
( ( ord_less_int @ A3 @ B5 )
=> ( ord_less_eq_int @ A3 @ B5 ) ) ).
% order.strict_implies_order
thf(fact_560_dual__order_Ostrict__implies__order,axiom,
! [B5: set_a,A3: set_a] :
( ( ord_less_set_a @ B5 @ A3 )
=> ( ord_less_eq_set_a @ B5 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_561_dual__order_Ostrict__implies__order,axiom,
! [B5: real,A3: real] :
( ( ord_less_real @ B5 @ A3 )
=> ( ord_less_eq_real @ B5 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_562_dual__order_Ostrict__implies__order,axiom,
! [B5: nat,A3: nat] :
( ( ord_less_nat @ B5 @ A3 )
=> ( ord_less_eq_nat @ B5 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_563_dual__order_Ostrict__implies__order,axiom,
! [B5: int,A3: int] :
( ( ord_less_int @ B5 @ A3 )
=> ( ord_less_eq_int @ B5 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_564_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_set_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_565_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_566_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_567_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_568_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_569_order__less__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_570_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_571_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_572_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_573_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_574_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_575_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_576_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_577_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_578_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_579_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_580_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_581_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_582_order__le__neq__trans,axiom,
! [A3: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A3 @ B5 )
=> ( ( A3 != B5 )
=> ( ord_less_set_a @ A3 @ B5 ) ) ) ).
% order_le_neq_trans
thf(fact_583_order__le__neq__trans,axiom,
! [A3: real,B5: real] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( A3 != B5 )
=> ( ord_less_real @ A3 @ B5 ) ) ) ).
% order_le_neq_trans
thf(fact_584_order__le__neq__trans,axiom,
! [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( A3 != B5 )
=> ( ord_less_nat @ A3 @ B5 ) ) ) ).
% order_le_neq_trans
thf(fact_585_order__le__neq__trans,axiom,
! [A3: int,B5: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( A3 != B5 )
=> ( ord_less_int @ A3 @ B5 ) ) ) ).
% order_le_neq_trans
thf(fact_586_order__neq__le__trans,axiom,
! [A3: set_a,B5: set_a] :
( ( A3 != B5 )
=> ( ( ord_less_eq_set_a @ A3 @ B5 )
=> ( ord_less_set_a @ A3 @ B5 ) ) ) ).
% order_neq_le_trans
thf(fact_587_order__neq__le__trans,axiom,
! [A3: real,B5: real] :
( ( A3 != B5 )
=> ( ( ord_less_eq_real @ A3 @ B5 )
=> ( ord_less_real @ A3 @ B5 ) ) ) ).
% order_neq_le_trans
thf(fact_588_order__neq__le__trans,axiom,
! [A3: nat,B5: nat] :
( ( A3 != B5 )
=> ( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ord_less_nat @ A3 @ B5 ) ) ) ).
% order_neq_le_trans
thf(fact_589_order__neq__le__trans,axiom,
! [A3: int,B5: int] :
( ( A3 != B5 )
=> ( ( ord_less_eq_int @ A3 @ B5 )
=> ( ord_less_int @ A3 @ B5 ) ) ) ).
% order_neq_le_trans
thf(fact_590_order__le__less__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_591_order__le__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_592_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_593_order__le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_594_order__less__le__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_595_order__less__le__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_596_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_597_order__less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_598_order__le__less__subst1,axiom,
! [A3: real,F: real > real,B5: real,C2: real] :
( ( ord_less_eq_real @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_599_order__le__less__subst1,axiom,
! [A3: real,F: nat > real,B5: nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_600_order__le__less__subst1,axiom,
! [A3: real,F: int > real,B5: int,C2: int] :
( ( ord_less_eq_real @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_601_order__le__less__subst1,axiom,
! [A3: nat,F: real > nat,B5: real,C2: real] :
( ( ord_less_eq_nat @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_602_order__le__less__subst1,axiom,
! [A3: nat,F: nat > nat,B5: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_603_order__le__less__subst1,axiom,
! [A3: nat,F: int > nat,B5: int,C2: int] :
( ( ord_less_eq_nat @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_604_order__le__less__subst1,axiom,
! [A3: int,F: real > int,B5: real,C2: real] :
( ( ord_less_eq_int @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_605_order__le__less__subst1,axiom,
! [A3: int,F: nat > int,B5: nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_606_order__le__less__subst1,axiom,
! [A3: int,F: int > int,B5: int,C2: int] :
( ( ord_less_eq_int @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_607_order__le__less__subst1,axiom,
! [A3: set_a,F: real > set_a,B5: real,C2: real] :
( ( ord_less_eq_set_a @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_608_order__le__less__subst2,axiom,
! [A3: real,B5: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( ord_less_real @ ( F @ B5 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_609_order__le__less__subst2,axiom,
! [A3: real,B5: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( ord_less_nat @ ( F @ B5 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_610_order__le__less__subst2,axiom,
! [A3: real,B5: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( ord_less_int @ ( F @ B5 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_611_order__le__less__subst2,axiom,
! [A3: nat,B5: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( ord_less_real @ ( F @ B5 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_612_order__le__less__subst2,axiom,
! [A3: nat,B5: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( ord_less_nat @ ( F @ B5 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_613_order__le__less__subst2,axiom,
! [A3: nat,B5: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( ord_less_int @ ( F @ B5 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_614_order__le__less__subst2,axiom,
! [A3: int,B5: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ord_less_real @ ( F @ B5 ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_615_order__le__less__subst2,axiom,
! [A3: int,B5: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ord_less_nat @ ( F @ B5 ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_616_order__le__less__subst2,axiom,
! [A3: int,B5: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ord_less_int @ ( F @ B5 ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_617_order__le__less__subst2,axiom,
! [A3: set_a,B5: set_a,F: set_a > real,C2: real] :
( ( ord_less_eq_set_a @ A3 @ B5 )
=> ( ( ord_less_real @ ( F @ B5 ) @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_618_order__less__le__subst1,axiom,
! [A3: real,F: real > real,B5: real,C2: real] :
( ( ord_less_real @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_619_order__less__le__subst1,axiom,
! [A3: nat,F: real > nat,B5: real,C2: real] :
( ( ord_less_nat @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_620_order__less__le__subst1,axiom,
! [A3: int,F: real > int,B5: real,C2: real] :
( ( ord_less_int @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_real @ B5 @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_621_order__less__le__subst1,axiom,
! [A3: real,F: nat > real,B5: nat,C2: nat] :
( ( ord_less_real @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_622_order__less__le__subst1,axiom,
! [A3: nat,F: nat > nat,B5: nat,C2: nat] :
( ( ord_less_nat @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_623_order__less__le__subst1,axiom,
! [A3: int,F: nat > int,B5: nat,C2: nat] :
( ( ord_less_int @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_nat @ B5 @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_624_order__less__le__subst1,axiom,
! [A3: real,F: int > real,B5: int,C2: int] :
( ( ord_less_real @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_625_order__less__le__subst1,axiom,
! [A3: nat,F: int > nat,B5: int,C2: int] :
( ( ord_less_nat @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_626_order__less__le__subst1,axiom,
! [A3: int,F: int > int,B5: int,C2: int] :
( ( ord_less_int @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_int @ B5 @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_627_order__less__le__subst1,axiom,
! [A3: real,F: set_a > real,B5: set_a,C2: set_a] :
( ( ord_less_real @ A3 @ ( F @ B5 ) )
=> ( ( ord_less_eq_set_a @ B5 @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_628_order__less__le__subst2,axiom,
! [A3: real,B5: real,F: real > real,C2: real] :
( ( ord_less_real @ A3 @ B5 )
=> ( ( ord_less_eq_real @ ( F @ B5 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_629_order__less__le__subst2,axiom,
! [A3: nat,B5: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ord_less_eq_real @ ( F @ B5 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_630_order__less__le__subst2,axiom,
! [A3: int,B5: int,F: int > real,C2: real] :
( ( ord_less_int @ A3 @ B5 )
=> ( ( ord_less_eq_real @ ( F @ B5 ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_631_order__less__le__subst2,axiom,
! [A3: real,B5: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A3 @ B5 )
=> ( ( ord_less_eq_nat @ ( F @ B5 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_632_order__less__le__subst2,axiom,
! [A3: nat,B5: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ord_less_eq_nat @ ( F @ B5 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_633_order__less__le__subst2,axiom,
! [A3: int,B5: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A3 @ B5 )
=> ( ( ord_less_eq_nat @ ( F @ B5 ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_634_order__less__le__subst2,axiom,
! [A3: real,B5: real,F: real > int,C2: int] :
( ( ord_less_real @ A3 @ B5 )
=> ( ( ord_less_eq_int @ ( F @ B5 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_635_order__less__le__subst2,axiom,
! [A3: nat,B5: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ord_less_eq_int @ ( F @ B5 ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_636_order__less__le__subst2,axiom,
! [A3: int,B5: int,F: int > int,C2: int] :
( ( ord_less_int @ A3 @ B5 )
=> ( ( ord_less_eq_int @ ( F @ B5 ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_637_order__less__le__subst2,axiom,
! [A3: real,B5: real,F: real > set_a,C2: set_a] :
( ( ord_less_real @ A3 @ B5 )
=> ( ( ord_less_eq_set_a @ ( F @ B5 ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_638_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_639_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_640_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_641_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_642_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_643_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_644_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_645_bot_Onot__eq__extremum,axiom,
! [A3: set_a] :
( ( A3 != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A3 ) ) ).
% bot.not_eq_extremum
thf(fact_646_bot_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A3 ) ) ).
% bot.not_eq_extremum
thf(fact_647_bot_Oextremum__strict,axiom,
! [A3: set_a] :
~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_648_bot_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_649_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_650_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_651_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_652_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_653_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_654_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_655_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_656_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_657_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_658_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_659_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_660_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_661_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_662_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_663_finite__Int,axiom,
! [F2: set_a,G: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_664_finite__insert,axiom,
! [A3: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A3 @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_665_finite__insert,axiom,
! [A3: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A3 @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_666_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_667_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_668_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_669_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_670_inf_Oidem,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_671_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_672_inf_Oleft__idem,axiom,
! [A3: set_a,B5: set_a] :
( ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ A3 @ B5 ) )
= ( inf_inf_set_a @ A3 @ B5 ) ) ).
% inf.left_idem
thf(fact_673_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_674_inf_Oright__idem,axiom,
! [A3: set_a,B5: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B5 ) @ B5 )
= ( inf_inf_set_a @ A3 @ B5 ) ) ).
% inf.right_idem
thf(fact_675_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_676_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_677_Diff__iff,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B2 ) )
= ( ( member_real @ C2 @ A2 )
& ~ ( member_real @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_678_Diff__iff,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C2 @ A2 )
& ~ ( member_a @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_679_DiffI,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ A2 )
=> ( ~ ( member_real @ C2 @ B2 )
=> ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_680_DiffI,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ~ ( member_a @ C2 @ B2 )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_681_inf_Obounded__iff,axiom,
! [A3: set_a,B5: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B5 @ C2 ) )
= ( ( ord_less_eq_set_a @ A3 @ B5 )
& ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_682_inf_Obounded__iff,axiom,
! [A3: real,B5: real,C2: real] :
( ( ord_less_eq_real @ A3 @ ( inf_inf_real @ B5 @ C2 ) )
= ( ( ord_less_eq_real @ A3 @ B5 )
& ( ord_less_eq_real @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_683_inf_Obounded__iff,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B5 @ C2 ) )
= ( ( ord_less_eq_nat @ A3 @ B5 )
& ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_684_inf_Obounded__iff,axiom,
! [A3: int,B5: int,C2: int] :
( ( ord_less_eq_int @ A3 @ ( inf_inf_int @ B5 @ C2 ) )
= ( ( ord_less_eq_int @ A3 @ B5 )
& ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_685_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_686_le__inf__iff,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z2 ) )
= ( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_687_le__inf__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_688_le__inf__iff,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z2 ) )
= ( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_689_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_690_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_691_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_692_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_693_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_694_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_695_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_696_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_697_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_698_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_699_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_700_insert__Diff1,axiom,
! [X: real,B2: set_real,A2: set_real] :
( ( member_real @ X @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
= ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_701_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_702_Diff__insert0,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ( minus_minus_set_real @ A2 @ ( insert_real @ X @ B2 ) )
= ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_703_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_704_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_705_insert__Diff__single,axiom,
! [A3: a,A2: set_a] :
( ( insert_a @ A3 @ ( minus_minus_set_a @ A2 @ ( insert_a @ A3 @ bot_bot_set_a ) ) )
= ( insert_a @ A3 @ A2 ) ) ).
% insert_Diff_single
thf(fact_706_finite__Diff__insert,axiom,
! [A2: set_nat,A3: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A3 @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_707_finite__Diff__insert,axiom,
! [A2: set_a,A3: a,B2: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A3 @ B2 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_708_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_709_mult__less__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K3 ) @ ( times_times_nat @ N @ K3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_710_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_711_mult__le__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K3 ) @ ( times_times_nat @ N @ K3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_712_psubset__imp__ex__mem,axiom,
! [A2: set_real,B2: set_real] :
( ( ord_less_set_real @ A2 @ B2 )
=> ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_713_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_714_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_715_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_716_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_717_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_718_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_719_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_720_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_721_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_722_DiffD2,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ~ ( member_real @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_723_DiffD2,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_724_DiffD1,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ( member_real @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_725_DiffD1,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_726_DiffE,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ~ ( ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_727_DiffE,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_728_Diff__infinite__finite,axiom,
! [T3: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T3 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_729_Diff__infinite__finite,axiom,
! [T3: set_a,S2: set_a] :
( ( finite_finite_a @ T3 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_730_finite__psubset__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ! [B8: set_a] :
( ( ord_less_set_a @ B8 @ A6 )
=> ( P @ B8 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_731_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B8: set_nat] :
( ( ord_less_set_nat @ B8 @ A6 )
=> ( P @ B8 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_732_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_733_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_734_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_735_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_736_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_737_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% of_nat_diff
thf(fact_738_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_739_finite__induct__select,axiom,
! [S2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T4: set_nat] :
( ( ord_less_set_nat @ T4 @ S2 )
=> ( ( P @ T4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ ( minus_minus_set_nat @ S2 @ T4 ) )
& ( P @ ( insert_nat @ X6 @ T4 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_740_finite__induct__select,axiom,
! [S2: set_a,P: set_a > $o] :
( ( finite_finite_a @ S2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [T4: set_a] :
( ( ord_less_set_a @ T4 @ S2 )
=> ( ( P @ T4 )
=> ? [X6: a] :
( ( member_a @ X6 @ ( minus_minus_set_a @ S2 @ T4 ) )
& ( P @ ( insert_a @ X6 @ T4 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_741_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_742_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_743_double__diff,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_744_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_745_Diff__mono,axiom,
! [A2: set_a,C: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_746_insert__Diff__if,axiom,
! [X: real,B2: set_real,A2: set_real] :
( ( ( member_real @ X @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
= ( minus_minus_set_real @ A2 @ B2 ) ) )
& ( ~ ( member_real @ X @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
= ( insert_real @ X @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_747_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_748_not__psubset__empty,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_749_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B6: set_a] :
( ( ord_less_set_a @ A7 @ B6 )
| ( A7 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_750_subset__psubset__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_751_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A7: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
& ~ ( ord_less_eq_set_a @ B6 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_752_psubset__subset__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_753_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_754_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A7: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
& ( A7 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_755_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_756_Diff__Int__distrib2,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_757_Diff__Int__distrib,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C @ A2 ) @ ( inf_inf_set_a @ C @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_758_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_759_Diff__Int2,axiom,
! [A2: set_a,C: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B2 @ C ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ B2 ) ) ).
% Diff_Int2
thf(fact_760_Int__Diff,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C )
= ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C ) ) ) ).
% Int_Diff
thf(fact_761_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_762_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_763_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_764_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_765_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_766_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_767_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_768_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_769_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_770_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_771_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_772_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_773_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_774_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_775_psubset__insert__iff,axiom,
! [A2: set_real,X: real,B2: set_real] :
( ( ord_less_set_real @ A2 @ ( insert_real @ X @ B2 ) )
= ( ( ( member_real @ X @ B2 )
=> ( ord_less_set_real @ A2 @ B2 ) )
& ( ~ ( member_real @ X @ B2 )
=> ( ( ( member_real @ X @ A2 )
=> ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) )
& ( ~ ( member_real @ X @ A2 )
=> ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_776_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_777_card__Diff1__less,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_778_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_779_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_780_card__Diff2__less,axiom,
! [A2: set_real,X: real,Y: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ X @ A2 )
=> ( ( member_real @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ ( insert_real @ Y @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_781_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_782_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_783_card__Diff1__less__iff,axiom,
! [A2: set_real,X: real] :
( ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) )
= ( ( finite_finite_real @ A2 )
& ( member_real @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_784_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_785_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_786_infinite__remove,axiom,
! [S2: set_nat,A3: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_787_infinite__remove,axiom,
! [S2: set_a,A3: a] :
( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_788_infinite__coinduct,axiom,
! [X5: set_nat > $o,A2: set_nat] :
( ( X5 @ A2 )
=> ( ! [A6: set_nat] :
( ( X5 @ A6 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A6 )
& ( ( X5 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_789_infinite__coinduct,axiom,
! [X5: set_a > $o,A2: set_a] :
( ( X5 @ A2 )
=> ( ! [A6: set_a] :
( ( X5 @ A6 )
=> ? [X6: a] :
( ( member_a @ X6 @ A6 )
& ( ( X5 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X6 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A6 @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_790_finite__empty__induct,axiom,
! [A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( P @ A2 )
=> ( ! [A4: real,A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ( member_real @ A4 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_real @ A6 @ ( insert_real @ A4 @ bot_bot_set_real ) ) ) ) ) )
=> ( P @ bot_bot_set_real ) ) ) ) ).
% finite_empty_induct
thf(fact_791_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A4: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat @ A4 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_792_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A4: a,A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( member_a @ A4 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_793_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_794_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_795_Diff__insert,axiom,
! [A2: set_a,A3: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A3 @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_796_insert__Diff,axiom,
! [A3: real,A2: set_real] :
( ( member_real @ A3 @ A2 )
=> ( ( insert_real @ A3 @ ( minus_minus_set_real @ A2 @ ( insert_real @ A3 @ bot_bot_set_real ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_797_insert__Diff,axiom,
! [A3: a,A2: set_a] :
( ( member_a @ A3 @ A2 )
=> ( ( insert_a @ A3 @ ( minus_minus_set_a @ A2 @ ( insert_a @ A3 @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_798_Diff__insert2,axiom,
! [A2: set_a,A3: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A3 @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A3 @ bot_bot_set_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_799_Diff__insert__absorb,axiom,
! [X: real,A2: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_800_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_801_subset__Diff__insert,axiom,
! [A2: set_real,B2: set_real,X: real,C: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ ( insert_real @ X @ C ) ) )
= ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ C ) )
& ~ ( member_real @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_802_subset__Diff__insert,axiom,
! [A2: set_a,B2: set_a,X: a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ ( insert_a @ X @ C ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_803_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_804_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_805_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_806_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_807_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K4 )
=> ~ ( P @ I3 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_808_mult__less__mono2,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_nat @ ( times_times_nat @ K3 @ I ) @ ( times_times_nat @ K3 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_809_mult__less__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ) ).
% mult_less_mono1
thf(fact_810_remove__induct,axiom,
! [P: set_real > $o,B2: set_real] :
( ( P @ bot_bot_set_real )
=> ( ( ~ ( finite_finite_real @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ( A6 != bot_bot_set_real )
=> ( ( ord_less_eq_set_real @ A6 @ B2 )
=> ( ! [X6: real] :
( ( member_real @ X6 @ A6 )
=> ( P @ ( minus_minus_set_real @ A6 @ ( insert_real @ X6 @ bot_bot_set_real ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_811_remove__induct,axiom,
! [P: set_nat > $o,B2: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B2 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_812_remove__induct,axiom,
! [P: set_a > $o,B2: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B2 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_813_finite__remove__induct,axiom,
! [B2: set_real,P: set_real > $o] :
( ( finite_finite_real @ B2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ( A6 != bot_bot_set_real )
=> ( ( ord_less_eq_set_real @ A6 @ B2 )
=> ( ! [X6: real] :
( ( member_real @ X6 @ A6 )
=> ( P @ ( minus_minus_set_real @ A6 @ ( insert_real @ X6 @ bot_bot_set_real ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_814_finite__remove__induct,axiom,
! [B2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B2 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_815_finite__remove__induct,axiom,
! [B2: set_a,P: set_a > $o] :
( ( finite_finite_a @ B2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B2 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_816_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_817_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_818_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_819_subset__insert__iff,axiom,
! [A2: set_real,X: real,B2: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) )
= ( ( ( member_real @ X @ A2 )
=> ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) )
& ( ~ ( member_real @ X @ A2 )
=> ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_820_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_821_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_822_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_823_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_824_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_825_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_826_inf_Oassoc,axiom,
! [A3: set_a,B5: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B5 ) @ C2 )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B5 @ C2 ) ) ) ).
% inf.assoc
thf(fact_827_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_828_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A5 ) ) ) ).
% inf.commute
thf(fact_829_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X3 ) ) ) ).
% inf_commute
thf(fact_830_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_a,K3: set_a,A3: set_a,B5: set_a] :
( ( A2
= ( inf_inf_set_a @ K3 @ A3 ) )
=> ( ( inf_inf_set_a @ A2 @ B5 )
= ( inf_inf_set_a @ K3 @ ( inf_inf_set_a @ A3 @ B5 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_831_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_a,K3: set_a,B5: set_a,A3: set_a] :
( ( B2
= ( inf_inf_set_a @ K3 @ B5 ) )
=> ( ( inf_inf_set_a @ A3 @ B2 )
= ( inf_inf_set_a @ K3 @ ( inf_inf_set_a @ A3 @ B5 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_832_inf_Oleft__commute,axiom,
! [B5: set_a,A3: set_a,C2: set_a] :
( ( inf_inf_set_a @ B5 @ ( inf_inf_set_a @ A3 @ C2 ) )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B5 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_833_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_834_finite__has__minimal2,axiom,
! [A2: set_set_a,A3: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A3 @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ X2 @ A3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_835_finite__has__minimal2,axiom,
! [A2: set_real,A3: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A3 @ A2 )
=> ? [X2: real] :
( ( member_real @ X2 @ A2 )
& ( ord_less_eq_real @ X2 @ A3 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_836_finite__has__minimal2,axiom,
! [A2: set_nat,A3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A3 @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_837_finite__has__minimal2,axiom,
! [A2: set_int,A3: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A3 @ A2 )
=> ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( ord_less_eq_int @ X2 @ A3 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_838_finite__has__maximal2,axiom,
! [A2: set_set_a,A3: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A3 @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ A3 @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_839_finite__has__maximal2,axiom,
! [A2: set_real,A3: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A3 @ A2 )
=> ? [X2: real] :
( ( member_real @ X2 @ A2 )
& ( ord_less_eq_real @ A3 @ X2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_840_finite__has__maximal2,axiom,
! [A2: set_nat,A3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A3 @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ A3 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_841_finite__has__maximal2,axiom,
! [A2: set_int,A3: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A3 @ A2 )
=> ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( ord_less_eq_int @ A3 @ X2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_842_inf_OcoboundedI2,axiom,
! [B5: set_a,C2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B5 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B5 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_843_inf_OcoboundedI2,axiom,
! [B5: real,C2: real,A3: real] :
( ( ord_less_eq_real @ B5 @ C2 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A3 @ B5 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_844_inf_OcoboundedI2,axiom,
! [B5: nat,C2: nat,A3: nat] :
( ( ord_less_eq_nat @ B5 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B5 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_845_inf_OcoboundedI2,axiom,
! [B5: int,C2: int,A3: int] :
( ( ord_less_eq_int @ B5 @ C2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A3 @ B5 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_846_inf_OcoboundedI1,axiom,
! [A3: set_a,C2: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B5 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_847_inf_OcoboundedI1,axiom,
! [A3: real,C2: real,B5: real] :
( ( ord_less_eq_real @ A3 @ C2 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A3 @ B5 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_848_inf_OcoboundedI1,axiom,
! [A3: nat,C2: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B5 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_849_inf_OcoboundedI1,axiom,
! [A3: int,C2: int,B5: int] :
( ( ord_less_eq_int @ A3 @ C2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A3 @ B5 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_850_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A5: set_a] :
( ( inf_inf_set_a @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_851_inf_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A5: real] :
( ( inf_inf_real @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_852_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_853_inf_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( ( inf_inf_int @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_854_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
( ( inf_inf_set_a @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_855_inf_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] :
( ( inf_inf_real @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_856_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( inf_inf_nat @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_857_inf_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( ( inf_inf_int @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_858_inf_Ocobounded2,axiom,
! [A3: set_a,B5: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B5 ) @ B5 ) ).
% inf.cobounded2
thf(fact_859_inf_Ocobounded2,axiom,
! [A3: real,B5: real] : ( ord_less_eq_real @ ( inf_inf_real @ A3 @ B5 ) @ B5 ) ).
% inf.cobounded2
thf(fact_860_inf_Ocobounded2,axiom,
! [A3: nat,B5: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B5 ) @ B5 ) ).
% inf.cobounded2
thf(fact_861_inf_Ocobounded2,axiom,
! [A3: int,B5: int] : ( ord_less_eq_int @ ( inf_inf_int @ A3 @ B5 ) @ B5 ) ).
% inf.cobounded2
thf(fact_862_inf_Ocobounded1,axiom,
! [A3: set_a,B5: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B5 ) @ A3 ) ).
% inf.cobounded1
thf(fact_863_inf_Ocobounded1,axiom,
! [A3: real,B5: real] : ( ord_less_eq_real @ ( inf_inf_real @ A3 @ B5 ) @ A3 ) ).
% inf.cobounded1
thf(fact_864_inf_Ocobounded1,axiom,
! [A3: nat,B5: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B5 ) @ A3 ) ).
% inf.cobounded1
thf(fact_865_inf_Ocobounded1,axiom,
! [A3: int,B5: int] : ( ord_less_eq_int @ ( inf_inf_int @ A3 @ B5 ) @ A3 ) ).
% inf.cobounded1
thf(fact_866_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
( A5
= ( inf_inf_set_a @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_867_inf_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] :
( A5
= ( inf_inf_real @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_868_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( A5
= ( inf_inf_nat @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_869_inf_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( A5
= ( inf_inf_int @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_870_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_871_inf__greatest,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Z2 )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_872_inf__greatest,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_873_inf__greatest,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Z2 )
=> ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_874_inf_OboundedI,axiom,
! [A3: set_a,B5: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B5 )
=> ( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B5 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_875_inf_OboundedI,axiom,
! [A3: real,B5: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( ord_less_eq_real @ A3 @ C2 )
=> ( ord_less_eq_real @ A3 @ ( inf_inf_real @ B5 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_876_inf_OboundedI,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B5 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_877_inf_OboundedI,axiom,
! [A3: int,B5: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ord_less_eq_int @ A3 @ C2 )
=> ( ord_less_eq_int @ A3 @ ( inf_inf_int @ B5 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_878_inf_OboundedE,axiom,
! [A3: set_a,B5: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B5 @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B5 )
=> ~ ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_879_inf_OboundedE,axiom,
! [A3: real,B5: real,C2: real] :
( ( ord_less_eq_real @ A3 @ ( inf_inf_real @ B5 @ C2 ) )
=> ~ ( ( ord_less_eq_real @ A3 @ B5 )
=> ~ ( ord_less_eq_real @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_880_inf_OboundedE,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B5 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A3 @ B5 )
=> ~ ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_881_inf_OboundedE,axiom,
! [A3: int,B5: int,C2: int] :
( ( ord_less_eq_int @ A3 @ ( inf_inf_int @ B5 @ C2 ) )
=> ~ ( ( ord_less_eq_int @ A3 @ B5 )
=> ~ ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_882_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_883_inf__absorb2,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( inf_inf_real @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_884_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_885_inf__absorb2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( inf_inf_int @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_886_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_887_inf__absorb1,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( inf_inf_real @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_888_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_889_inf__absorb1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( inf_inf_int @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_890_inf_Oabsorb2,axiom,
! [B5: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B5 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B5 )
= B5 ) ) ).
% inf.absorb2
thf(fact_891_inf_Oabsorb2,axiom,
! [B5: real,A3: real] :
( ( ord_less_eq_real @ B5 @ A3 )
=> ( ( inf_inf_real @ A3 @ B5 )
= B5 ) ) ).
% inf.absorb2
thf(fact_892_inf_Oabsorb2,axiom,
! [B5: nat,A3: nat] :
( ( ord_less_eq_nat @ B5 @ A3 )
=> ( ( inf_inf_nat @ A3 @ B5 )
= B5 ) ) ).
% inf.absorb2
thf(fact_893_inf_Oabsorb2,axiom,
! [B5: int,A3: int] :
( ( ord_less_eq_int @ B5 @ A3 )
=> ( ( inf_inf_int @ A3 @ B5 )
= B5 ) ) ).
% inf.absorb2
thf(fact_894_inf_Oabsorb1,axiom,
! [A3: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A3 @ B5 )
=> ( ( inf_inf_set_a @ A3 @ B5 )
= A3 ) ) ).
% inf.absorb1
thf(fact_895_inf_Oabsorb1,axiom,
! [A3: real,B5: real] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( inf_inf_real @ A3 @ B5 )
= A3 ) ) ).
% inf.absorb1
thf(fact_896_inf_Oabsorb1,axiom,
! [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( inf_inf_nat @ A3 @ B5 )
= A3 ) ) ).
% inf.absorb1
thf(fact_897_inf_Oabsorb1,axiom,
! [A3: int,B5: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( inf_inf_int @ A3 @ B5 )
= A3 ) ) ).
% inf.absorb1
thf(fact_898_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_899_le__iff__inf,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y4: real] :
( ( inf_inf_real @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_900_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( inf_inf_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_901_le__iff__inf,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y4: int] :
( ( inf_inf_int @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_902_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ X2 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_903_inf__unique,axiom,
! [F: real > real > real,X: real,Y: real] :
( ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ X2 @ Z3 )
=> ( ord_less_eq_real @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_real @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_904_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_905_inf__unique,axiom,
! [F: int > int > int,X: int,Y: int] :
( ! [X2: int,Y2: int] : ( ord_less_eq_int @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: int,Y2: int] : ( ord_less_eq_int @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ X2 @ Z3 )
=> ( ord_less_eq_int @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_int @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_906_inf_OorderI,axiom,
! [A3: set_a,B5: set_a] :
( ( A3
= ( inf_inf_set_a @ A3 @ B5 ) )
=> ( ord_less_eq_set_a @ A3 @ B5 ) ) ).
% inf.orderI
thf(fact_907_inf_OorderI,axiom,
! [A3: real,B5: real] :
( ( A3
= ( inf_inf_real @ A3 @ B5 ) )
=> ( ord_less_eq_real @ A3 @ B5 ) ) ).
% inf.orderI
thf(fact_908_inf_OorderI,axiom,
! [A3: nat,B5: nat] :
( ( A3
= ( inf_inf_nat @ A3 @ B5 ) )
=> ( ord_less_eq_nat @ A3 @ B5 ) ) ).
% inf.orderI
thf(fact_909_inf_OorderI,axiom,
! [A3: int,B5: int] :
( ( A3
= ( inf_inf_int @ A3 @ B5 ) )
=> ( ord_less_eq_int @ A3 @ B5 ) ) ).
% inf.orderI
thf(fact_910_inf_OorderE,axiom,
! [A3: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A3 @ B5 )
=> ( A3
= ( inf_inf_set_a @ A3 @ B5 ) ) ) ).
% inf.orderE
thf(fact_911_inf_OorderE,axiom,
! [A3: real,B5: real] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( A3
= ( inf_inf_real @ A3 @ B5 ) ) ) ).
% inf.orderE
thf(fact_912_inf_OorderE,axiom,
! [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( A3
= ( inf_inf_nat @ A3 @ B5 ) ) ) ).
% inf.orderE
thf(fact_913_inf_OorderE,axiom,
! [A3: int,B5: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( A3
= ( inf_inf_int @ A3 @ B5 ) ) ) ).
% inf.orderE
thf(fact_914_le__infI2,axiom,
! [B5: set_a,X: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B5 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B5 ) @ X ) ) ).
% le_infI2
thf(fact_915_le__infI2,axiom,
! [B5: real,X: real,A3: real] :
( ( ord_less_eq_real @ B5 @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A3 @ B5 ) @ X ) ) ).
% le_infI2
thf(fact_916_le__infI2,axiom,
! [B5: nat,X: nat,A3: nat] :
( ( ord_less_eq_nat @ B5 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B5 ) @ X ) ) ).
% le_infI2
thf(fact_917_le__infI2,axiom,
! [B5: int,X: int,A3: int] :
( ( ord_less_eq_int @ B5 @ X )
=> ( ord_less_eq_int @ ( inf_inf_int @ A3 @ B5 ) @ X ) ) ).
% le_infI2
thf(fact_918_le__infI1,axiom,
! [A3: set_a,X: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A3 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B5 ) @ X ) ) ).
% le_infI1
thf(fact_919_le__infI1,axiom,
! [A3: real,X: real,B5: real] :
( ( ord_less_eq_real @ A3 @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A3 @ B5 ) @ X ) ) ).
% le_infI1
thf(fact_920_le__infI1,axiom,
! [A3: nat,X: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B5 ) @ X ) ) ).
% le_infI1
thf(fact_921_le__infI1,axiom,
! [A3: int,X: int,B5: int] :
( ( ord_less_eq_int @ A3 @ X )
=> ( ord_less_eq_int @ ( inf_inf_int @ A3 @ B5 ) @ X ) ) ).
% le_infI1
thf(fact_922_inf__mono,axiom,
! [A3: set_a,C2: set_a,B5: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B5 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B5 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_923_inf__mono,axiom,
! [A3: real,C2: real,B5: real,D: real] :
( ( ord_less_eq_real @ A3 @ C2 )
=> ( ( ord_less_eq_real @ B5 @ D )
=> ( ord_less_eq_real @ ( inf_inf_real @ A3 @ B5 ) @ ( inf_inf_real @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_924_inf__mono,axiom,
! [A3: nat,C2: nat,B5: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ( ord_less_eq_nat @ B5 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B5 ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_925_inf__mono,axiom,
! [A3: int,C2: int,B5: int,D: int] :
( ( ord_less_eq_int @ A3 @ C2 )
=> ( ( ord_less_eq_int @ B5 @ D )
=> ( ord_less_eq_int @ ( inf_inf_int @ A3 @ B5 ) @ ( inf_inf_int @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_926_le__infI,axiom,
! [X: set_a,A3: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ X @ B5 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A3 @ B5 ) ) ) ) ).
% le_infI
thf(fact_927_le__infI,axiom,
! [X: real,A3: real,B5: real] :
( ( ord_less_eq_real @ X @ A3 )
=> ( ( ord_less_eq_real @ X @ B5 )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ A3 @ B5 ) ) ) ) ).
% le_infI
thf(fact_928_le__infI,axiom,
! [X: nat,A3: nat,B5: nat] :
( ( ord_less_eq_nat @ X @ A3 )
=> ( ( ord_less_eq_nat @ X @ B5 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A3 @ B5 ) ) ) ) ).
% le_infI
thf(fact_929_le__infI,axiom,
! [X: int,A3: int,B5: int] :
( ( ord_less_eq_int @ X @ A3 )
=> ( ( ord_less_eq_int @ X @ B5 )
=> ( ord_less_eq_int @ X @ ( inf_inf_int @ A3 @ B5 ) ) ) ) ).
% le_infI
thf(fact_930_le__infE,axiom,
! [X: set_a,A3: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A3 @ B5 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A3 )
=> ~ ( ord_less_eq_set_a @ X @ B5 ) ) ) ).
% le_infE
thf(fact_931_le__infE,axiom,
! [X: real,A3: real,B5: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ A3 @ B5 ) )
=> ~ ( ( ord_less_eq_real @ X @ A3 )
=> ~ ( ord_less_eq_real @ X @ B5 ) ) ) ).
% le_infE
thf(fact_932_le__infE,axiom,
! [X: nat,A3: nat,B5: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A3 @ B5 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A3 )
=> ~ ( ord_less_eq_nat @ X @ B5 ) ) ) ).
% le_infE
thf(fact_933_le__infE,axiom,
! [X: int,A3: int,B5: int] :
( ( ord_less_eq_int @ X @ ( inf_inf_int @ A3 @ B5 ) )
=> ~ ( ( ord_less_eq_int @ X @ A3 )
=> ~ ( ord_less_eq_int @ X @ B5 ) ) ) ).
% le_infE
thf(fact_934_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_935_inf__le2,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_936_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_937_inf__le2,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_938_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_939_inf__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_940_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_941_inf__le1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_942_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_943_inf__sup__ord_I1_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_944_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_945_inf__sup__ord_I1_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_946_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_947_inf__sup__ord_I2_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_948_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_949_inf__sup__ord_I2_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_950_less__infI1,axiom,
! [A3: set_a,X: set_a,B5: set_a] :
( ( ord_less_set_a @ A3 @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A3 @ B5 ) @ X ) ) ).
% less_infI1
thf(fact_951_less__infI1,axiom,
! [A3: real,X: real,B5: real] :
( ( ord_less_real @ A3 @ X )
=> ( ord_less_real @ ( inf_inf_real @ A3 @ B5 ) @ X ) ) ).
% less_infI1
thf(fact_952_less__infI1,axiom,
! [A3: nat,X: nat,B5: nat] :
( ( ord_less_nat @ A3 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A3 @ B5 ) @ X ) ) ).
% less_infI1
thf(fact_953_less__infI1,axiom,
! [A3: int,X: int,B5: int] :
( ( ord_less_int @ A3 @ X )
=> ( ord_less_int @ ( inf_inf_int @ A3 @ B5 ) @ X ) ) ).
% less_infI1
thf(fact_954_less__infI2,axiom,
! [B5: set_a,X: set_a,A3: set_a] :
( ( ord_less_set_a @ B5 @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A3 @ B5 ) @ X ) ) ).
% less_infI2
thf(fact_955_less__infI2,axiom,
! [B5: real,X: real,A3: real] :
( ( ord_less_real @ B5 @ X )
=> ( ord_less_real @ ( inf_inf_real @ A3 @ B5 ) @ X ) ) ).
% less_infI2
thf(fact_956_less__infI2,axiom,
! [B5: nat,X: nat,A3: nat] :
( ( ord_less_nat @ B5 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A3 @ B5 ) @ X ) ) ).
% less_infI2
thf(fact_957_less__infI2,axiom,
! [B5: int,X: int,A3: int] :
( ( ord_less_int @ B5 @ X )
=> ( ord_less_int @ ( inf_inf_int @ A3 @ B5 ) @ X ) ) ).
% less_infI2
thf(fact_958_inf_Oabsorb3,axiom,
! [A3: set_a,B5: set_a] :
( ( ord_less_set_a @ A3 @ B5 )
=> ( ( inf_inf_set_a @ A3 @ B5 )
= A3 ) ) ).
% inf.absorb3
thf(fact_959_inf_Oabsorb3,axiom,
! [A3: real,B5: real] :
( ( ord_less_real @ A3 @ B5 )
=> ( ( inf_inf_real @ A3 @ B5 )
= A3 ) ) ).
% inf.absorb3
thf(fact_960_inf_Oabsorb3,axiom,
! [A3: nat,B5: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( inf_inf_nat @ A3 @ B5 )
= A3 ) ) ).
% inf.absorb3
thf(fact_961_inf_Oabsorb3,axiom,
! [A3: int,B5: int] :
( ( ord_less_int @ A3 @ B5 )
=> ( ( inf_inf_int @ A3 @ B5 )
= A3 ) ) ).
% inf.absorb3
thf(fact_962_inf_Oabsorb4,axiom,
! [B5: set_a,A3: set_a] :
( ( ord_less_set_a @ B5 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B5 )
= B5 ) ) ).
% inf.absorb4
thf(fact_963_inf_Oabsorb4,axiom,
! [B5: real,A3: real] :
( ( ord_less_real @ B5 @ A3 )
=> ( ( inf_inf_real @ A3 @ B5 )
= B5 ) ) ).
% inf.absorb4
thf(fact_964_inf_Oabsorb4,axiom,
! [B5: nat,A3: nat] :
( ( ord_less_nat @ B5 @ A3 )
=> ( ( inf_inf_nat @ A3 @ B5 )
= B5 ) ) ).
% inf.absorb4
thf(fact_965_inf_Oabsorb4,axiom,
! [B5: int,A3: int] :
( ( ord_less_int @ B5 @ A3 )
=> ( ( inf_inf_int @ A3 @ B5 )
= B5 ) ) ).
% inf.absorb4
thf(fact_966_inf_Ostrict__boundedE,axiom,
! [A3: set_a,B5: set_a,C2: set_a] :
( ( ord_less_set_a @ A3 @ ( inf_inf_set_a @ B5 @ C2 ) )
=> ~ ( ( ord_less_set_a @ A3 @ B5 )
=> ~ ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_967_inf_Ostrict__boundedE,axiom,
! [A3: real,B5: real,C2: real] :
( ( ord_less_real @ A3 @ ( inf_inf_real @ B5 @ C2 ) )
=> ~ ( ( ord_less_real @ A3 @ B5 )
=> ~ ( ord_less_real @ A3 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_968_inf_Ostrict__boundedE,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( ord_less_nat @ A3 @ ( inf_inf_nat @ B5 @ C2 ) )
=> ~ ( ( ord_less_nat @ A3 @ B5 )
=> ~ ( ord_less_nat @ A3 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_969_inf_Ostrict__boundedE,axiom,
! [A3: int,B5: int,C2: int] :
( ( ord_less_int @ A3 @ ( inf_inf_int @ B5 @ C2 ) )
=> ~ ( ( ord_less_int @ A3 @ B5 )
=> ~ ( ord_less_int @ A3 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_970_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B4: set_a] :
( ( A5
= ( inf_inf_set_a @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_971_inf_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( A5
= ( inf_inf_real @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_972_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( A5
= ( inf_inf_nat @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_973_inf_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( A5
= ( inf_inf_int @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_974_inf_Ostrict__coboundedI1,axiom,
! [A3: set_a,C2: set_a,B5: set_a] :
( ( ord_less_set_a @ A3 @ C2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A3 @ B5 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_975_inf_Ostrict__coboundedI1,axiom,
! [A3: real,C2: real,B5: real] :
( ( ord_less_real @ A3 @ C2 )
=> ( ord_less_real @ ( inf_inf_real @ A3 @ B5 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_976_inf_Ostrict__coboundedI1,axiom,
! [A3: nat,C2: nat,B5: nat] :
( ( ord_less_nat @ A3 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A3 @ B5 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_977_inf_Ostrict__coboundedI1,axiom,
! [A3: int,C2: int,B5: int] :
( ( ord_less_int @ A3 @ C2 )
=> ( ord_less_int @ ( inf_inf_int @ A3 @ B5 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_978_inf_Ostrict__coboundedI2,axiom,
! [B5: set_a,C2: set_a,A3: set_a] :
( ( ord_less_set_a @ B5 @ C2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A3 @ B5 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_979_inf_Ostrict__coboundedI2,axiom,
! [B5: real,C2: real,A3: real] :
( ( ord_less_real @ B5 @ C2 )
=> ( ord_less_real @ ( inf_inf_real @ A3 @ B5 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_980_inf_Ostrict__coboundedI2,axiom,
! [B5: nat,C2: nat,A3: nat] :
( ( ord_less_nat @ B5 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A3 @ B5 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_981_inf_Ostrict__coboundedI2,axiom,
! [B5: int,C2: int,A3: int] :
( ( ord_less_int @ B5 @ C2 )
=> ( ord_less_int @ ( inf_inf_int @ A3 @ B5 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_982_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_983_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_984_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_985_infinite__imp__nonempty,axiom,
! [S2: set_a] :
( ~ ( finite_finite_a @ S2 )
=> ( S2 != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_986_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_987_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_988_infinite__super,axiom,
! [S2: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_989_infinite__super,axiom,
! [S2: set_a,T3: set_a] :
( ( ord_less_eq_set_a @ S2 @ T3 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ T3 ) ) ) ).
% infinite_super
thf(fact_990_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_991_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_992_finite_OinsertI,axiom,
! [A2: set_a,A3: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A3 @ A2 ) ) ) ).
% finite.insertI
thf(fact_993_finite_OinsertI,axiom,
! [A2: set_nat,A3: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A3 @ A2 ) ) ) ).
% finite.insertI
thf(fact_994_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_995_finite__has__minimal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X2: real] :
( ( member_real @ X2 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_996_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_997_finite__has__minimal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_998_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_999_finite__has__maximal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X2: real] :
( ( member_real @ X2 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1000_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1001_finite__has__maximal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1002_finite_Ocases,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A4: nat] :
( A3
= ( insert_nat @ A4 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_1003_finite_Ocases,axiom,
! [A3: set_a] :
( ( finite_finite_a @ A3 )
=> ( ( A3 != bot_bot_set_a )
=> ~ ! [A6: set_a] :
( ? [A4: a] :
( A3
= ( insert_a @ A4 @ A6 ) )
=> ~ ( finite_finite_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_1004_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A5: set_nat] :
( ( A5 = bot_bot_set_nat )
| ? [A7: set_nat,B4: nat] :
( ( A5
= ( insert_nat @ B4 @ A7 ) )
& ( finite_finite_nat @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_1005_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A5: set_a] :
( ( A5 = bot_bot_set_a )
| ? [A7: set_a,B4: a] :
( ( A5
= ( insert_a @ B4 @ A7 ) )
& ( finite_finite_a @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_1006_finite__induct,axiom,
! [F2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X2: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ~ ( member_real @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1007_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1008_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1009_finite__ne__induct,axiom,
! [F2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( F2 != bot_bot_set_real )
=> ( ! [X2: real] : ( P @ ( insert_real @ X2 @ bot_bot_set_real ) )
=> ( ! [X2: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( F3 != bot_bot_set_real )
=> ( ~ ( member_real @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1010_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1011_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1012_infinite__finite__induct,axiom,
! [P: set_real > $o,A2: set_real] :
( ! [A6: set_real] :
( ~ ( finite_finite_real @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X2: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ~ ( member_real @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1013_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1014_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A6: set_a] :
( ~ ( finite_finite_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1015_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_1016_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_1017_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_1018_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_1019_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_1020_finite__subset__induct,axiom,
! [F2: set_real,A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( ord_less_eq_set_real @ F2 @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A4: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( member_real @ A4 @ A2 )
=> ( ~ ( member_real @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1021_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A4 @ A2 )
=> ( ~ ( member_nat @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1022_finite__subset__induct,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A4 @ A2 )
=> ( ~ ( member_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1023_finite__subset__induct_H,axiom,
! [F2: set_real,A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( ord_less_eq_set_real @ F2 @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A4: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( member_real @ A4 @ A2 )
=> ( ( ord_less_eq_set_real @ F3 @ A2 )
=> ( ~ ( member_real @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1024_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A4 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1025_finite__subset__induct_H,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A4 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1026_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_1027_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_1028_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_1029_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_1030_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_1031_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_1032_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C )
& ( ( finite_card_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1033_exists__subset__between,axiom,
! [A2: set_a,N: nat,C: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C ) )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( finite_finite_a @ C )
=> ? [B7: set_a] :
( ( ord_less_eq_set_a @ A2 @ B7 )
& ( ord_less_eq_set_a @ B7 @ C )
& ( ( finite_card_a @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1034_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S2 ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S2 )
=> ( ( ( finite_card_nat @ T4 )
= N )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1035_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S2 ) )
=> ~ ! [T4: set_a] :
( ( ord_less_eq_set_a @ T4 @ S2 )
=> ( ( ( finite_card_a @ T4 )
= N )
=> ~ ( finite_finite_a @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1036_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1037_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C: nat] :
( ! [G2: set_a] :
( ( ord_less_eq_set_a @ G2 @ F2 )
=> ( ( finite_finite_a @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G2 ) @ C ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1038_nat__mult__le__cancel__disj,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1039_nat__mult__less__cancel__disj,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1040_diff__gt__0__iff__gt,axiom,
! [A3: real,B5: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B5 ) )
= ( ord_less_real @ B5 @ A3 ) ) ).
% diff_gt_0_iff_gt
thf(fact_1041_diff__gt__0__iff__gt,axiom,
! [A3: int,B5: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B5 ) )
= ( ord_less_int @ B5 @ A3 ) ) ).
% diff_gt_0_iff_gt
thf(fact_1042_diff__ge__0__iff__ge,axiom,
! [A3: real,B5: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B5 ) )
= ( ord_less_eq_real @ B5 @ A3 ) ) ).
% diff_ge_0_iff_ge
thf(fact_1043_diff__ge__0__iff__ge,axiom,
! [A3: int,B5: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B5 ) )
= ( ord_less_eq_int @ B5 @ A3 ) ) ).
% diff_ge_0_iff_ge
thf(fact_1044_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C2 ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1045_diff__self,axiom,
! [A3: int] :
( ( minus_minus_int @ A3 @ A3 )
= zero_zero_int ) ).
% diff_self
thf(fact_1046_diff__self,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ A3 )
= zero_zero_real ) ).
% diff_self
thf(fact_1047_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1048_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1049_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ A3 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1050_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: int] :
( ( minus_minus_int @ A3 @ A3 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1051_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ A3 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1052_diff__zero,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% diff_zero
thf(fact_1053_diff__zero,axiom,
! [A3: int] :
( ( minus_minus_int @ A3 @ zero_zero_int )
= A3 ) ).
% diff_zero
thf(fact_1054_diff__zero,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ zero_zero_real )
= A3 ) ).
% diff_zero
thf(fact_1055_zero__diff,axiom,
! [A3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1056_diff__0__right,axiom,
! [A3: int] :
( ( minus_minus_int @ A3 @ zero_zero_int )
= A3 ) ).
% diff_0_right
thf(fact_1057_diff__0__right,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ zero_zero_real )
= A3 ) ).
% diff_0_right
thf(fact_1058_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1059_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1060_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_1061_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1062_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1063_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1064_psubsetD,axiom,
! [A2: set_a,B2: set_a,C2: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_1065_psubsetD,axiom,
! [A2: set_real,B2: set_real,C2: real] :
( ( ord_less_set_real @ A2 @ B2 )
=> ( ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_1066_diff__commute,axiom,
! [I: nat,J: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K3 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K3 ) @ J ) ) ).
% diff_commute
thf(fact_1067_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1068_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1069_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1070_less__imp__diff__less,axiom,
! [J: nat,K3: nat,N: nat] :
( ( ord_less_nat @ J @ K3 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K3 ) ) ).
% less_imp_diff_less
thf(fact_1071_eq__diff__iff,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ( minus_minus_nat @ M @ K3 )
= ( minus_minus_nat @ N @ K3 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1072_le__diff__iff,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1073_Nat_Odiff__diff__eq,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1074_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1075_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1076_le__diff__iff_H,axiom,
! [A3: nat,C2: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ( ord_less_eq_nat @ B5 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A3 ) @ ( minus_minus_nat @ C2 @ B5 ) )
= ( ord_less_eq_nat @ B5 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_1077_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1078_diff__mult__distrib,axiom,
! [M: nat,N: nat,K3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K3 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K3 ) @ ( times_times_nat @ N @ K3 ) ) ) ).
% diff_mult_distrib
thf(fact_1079_diff__mult__distrib2,axiom,
! [K3: nat,M: nat,N: nat] :
( ( times_times_nat @ K3 @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1080_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1081_diff__less__mono,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B5 )
=> ( ( ord_less_eq_nat @ C2 @ A3 )
=> ( ord_less_nat @ ( minus_minus_nat @ A3 @ C2 ) @ ( minus_minus_nat @ B5 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1082_less__diff__iff,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1083_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_1084_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_1085_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_1086_mult_Oleft__commute,axiom,
! [B5: real,A3: real,C2: real] :
( ( times_times_real @ B5 @ ( times_times_real @ A3 @ C2 ) )
= ( times_times_real @ A3 @ ( times_times_real @ B5 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_1087_mult_Oleft__commute,axiom,
! [B5: nat,A3: nat,C2: nat] :
( ( times_times_nat @ B5 @ ( times_times_nat @ A3 @ C2 ) )
= ( times_times_nat @ A3 @ ( times_times_nat @ B5 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_1088_mult_Oleft__commute,axiom,
! [B5: int,A3: int,C2: int] :
( ( times_times_int @ B5 @ ( times_times_int @ A3 @ C2 ) )
= ( times_times_int @ A3 @ ( times_times_int @ B5 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_1089_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A5: real,B4: real] : ( times_times_real @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_1090_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A5: nat,B4: nat] : ( times_times_nat @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_1091_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A5: int,B4: int] : ( times_times_int @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_1092_mult_Oassoc,axiom,
! [A3: real,B5: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B5 ) @ C2 )
= ( times_times_real @ A3 @ ( times_times_real @ B5 @ C2 ) ) ) ).
% mult.assoc
thf(fact_1093_mult_Oassoc,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B5 ) @ C2 )
= ( times_times_nat @ A3 @ ( times_times_nat @ B5 @ C2 ) ) ) ).
% mult.assoc
thf(fact_1094_mult_Oassoc,axiom,
! [A3: int,B5: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B5 ) @ C2 )
= ( times_times_int @ A3 @ ( times_times_int @ B5 @ C2 ) ) ) ).
% mult.assoc
thf(fact_1095_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: real,B5: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B5 ) @ C2 )
= ( times_times_real @ A3 @ ( times_times_real @ B5 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1096_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B5 ) @ C2 )
= ( times_times_nat @ A3 @ ( times_times_nat @ B5 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1097_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: int,B5: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B5 ) @ C2 )
= ( times_times_int @ A3 @ ( times_times_int @ B5 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1098_diff__right__commute,axiom,
! [A3: nat,C2: nat,B5: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ C2 ) @ B5 )
= ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B5 ) @ C2 ) ) ).
% diff_right_commute
thf(fact_1099_diff__right__commute,axiom,
! [A3: int,C2: int,B5: int] :
( ( minus_minus_int @ ( minus_minus_int @ A3 @ C2 ) @ B5 )
= ( minus_minus_int @ ( minus_minus_int @ A3 @ B5 ) @ C2 ) ) ).
% diff_right_commute
thf(fact_1100_diff__right__commute,axiom,
! [A3: real,C2: real,B5: real] :
( ( minus_minus_real @ ( minus_minus_real @ A3 @ C2 ) @ B5 )
= ( minus_minus_real @ ( minus_minus_real @ A3 @ B5 ) @ C2 ) ) ).
% diff_right_commute
thf(fact_1101_diff__eq__diff__eq,axiom,
! [A3: int,B5: int,C2: int,D: int] :
( ( ( minus_minus_int @ A3 @ B5 )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( A3 = B5 )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_1102_diff__eq__diff__eq,axiom,
! [A3: real,B5: real,C2: real,D: real] :
( ( ( minus_minus_real @ A3 @ B5 )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( A3 = B5 )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_1103_complete__real,axiom,
! [S2: set_real] :
( ? [X6: real] : ( member_real @ X6 @ S2 )
=> ( ? [Z4: real] :
! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ? [Y2: real] :
( ! [X6: real] :
( ( member_real @ X6 @ S2 )
=> ( ord_less_eq_real @ X6 @ Y2 ) )
& ! [Z4: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ( ord_less_eq_real @ Y2 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_1104_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_1105_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_1106_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_1107_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_1108_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_1109_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_1110_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_1111_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1112_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_1113_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1114_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1115_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_1116_diff__mono,axiom,
! [A3: real,B5: real,D: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( ord_less_eq_real @ D @ C2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A3 @ C2 ) @ ( minus_minus_real @ B5 @ D ) ) ) ) ).
% diff_mono
thf(fact_1117_diff__mono,axiom,
! [A3: int,B5: int,D: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ord_less_eq_int @ D @ C2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A3 @ C2 ) @ ( minus_minus_int @ B5 @ D ) ) ) ) ).
% diff_mono
thf(fact_1118_diff__left__mono,axiom,
! [B5: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B5 @ A3 )
=> ( ord_less_eq_real @ ( minus_minus_real @ C2 @ A3 ) @ ( minus_minus_real @ C2 @ B5 ) ) ) ).
% diff_left_mono
thf(fact_1119_diff__left__mono,axiom,
! [B5: int,A3: int,C2: int] :
( ( ord_less_eq_int @ B5 @ A3 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A3 ) @ ( minus_minus_int @ C2 @ B5 ) ) ) ).
% diff_left_mono
thf(fact_1120_diff__right__mono,axiom,
! [A3: real,B5: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A3 @ C2 ) @ ( minus_minus_real @ B5 @ C2 ) ) ) ).
% diff_right_mono
thf(fact_1121_diff__right__mono,axiom,
! [A3: int,B5: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A3 @ C2 ) @ ( minus_minus_int @ B5 @ C2 ) ) ) ).
% diff_right_mono
thf(fact_1122_diff__eq__diff__less__eq,axiom,
! [A3: real,B5: real,C2: real,D: real] :
( ( ( minus_minus_real @ A3 @ B5 )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( ord_less_eq_real @ A3 @ B5 )
= ( ord_less_eq_real @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1123_diff__eq__diff__less__eq,axiom,
! [A3: int,B5: int,C2: int,D: int] :
( ( ( minus_minus_int @ A3 @ B5 )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_eq_int @ A3 @ B5 )
= ( ord_less_eq_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1124_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A5: int,B4: int] :
( ( minus_minus_int @ A5 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1125_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A5: real,B4: real] :
( ( minus_minus_real @ A5 @ B4 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1126_diff__strict__right__mono,axiom,
! [A3: real,B5: real,C2: real] :
( ( ord_less_real @ A3 @ B5 )
=> ( ord_less_real @ ( minus_minus_real @ A3 @ C2 ) @ ( minus_minus_real @ B5 @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_1127_diff__strict__right__mono,axiom,
! [A3: int,B5: int,C2: int] :
( ( ord_less_int @ A3 @ B5 )
=> ( ord_less_int @ ( minus_minus_int @ A3 @ C2 ) @ ( minus_minus_int @ B5 @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_1128_diff__strict__left__mono,axiom,
! [B5: real,A3: real,C2: real] :
( ( ord_less_real @ B5 @ A3 )
=> ( ord_less_real @ ( minus_minus_real @ C2 @ A3 ) @ ( minus_minus_real @ C2 @ B5 ) ) ) ).
% diff_strict_left_mono
thf(fact_1129_diff__strict__left__mono,axiom,
! [B5: int,A3: int,C2: int] :
( ( ord_less_int @ B5 @ A3 )
=> ( ord_less_int @ ( minus_minus_int @ C2 @ A3 ) @ ( minus_minus_int @ C2 @ B5 ) ) ) ).
% diff_strict_left_mono
thf(fact_1130_diff__eq__diff__less,axiom,
! [A3: real,B5: real,C2: real,D: real] :
( ( ( minus_minus_real @ A3 @ B5 )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( ord_less_real @ A3 @ B5 )
= ( ord_less_real @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1131_diff__eq__diff__less,axiom,
! [A3: int,B5: int,C2: int,D: int] :
( ( ( minus_minus_int @ A3 @ B5 )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_int @ A3 @ B5 )
= ( ord_less_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1132_diff__strict__mono,axiom,
! [A3: real,B5: real,D: real,C2: real] :
( ( ord_less_real @ A3 @ B5 )
=> ( ( ord_less_real @ D @ C2 )
=> ( ord_less_real @ ( minus_minus_real @ A3 @ C2 ) @ ( minus_minus_real @ B5 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1133_diff__strict__mono,axiom,
! [A3: int,B5: int,D: int,C2: int] :
( ( ord_less_int @ A3 @ B5 )
=> ( ( ord_less_int @ D @ C2 )
=> ( ord_less_int @ ( minus_minus_int @ A3 @ C2 ) @ ( minus_minus_int @ B5 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1134_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_1135_nat__mult__eq__cancel__disj,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K3 @ M )
= ( times_times_nat @ K3 @ N ) )
= ( ( K3 = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1136_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_1137_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1138_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_1139_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_1140_nat__mult__eq__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ( times_times_nat @ K3 @ M )
= ( times_times_nat @ K3 @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1141_nat__mult__less__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1142_nat__mult__le__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1143_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y5: real] :
? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1144_mult__cancel__right,axiom,
! [A3: real,C2: real,B5: real] :
( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B5 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B5 ) ) ) ).
% mult_cancel_right
thf(fact_1145_mult__cancel__right,axiom,
! [A3: nat,C2: nat,B5: nat] :
( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B5 @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B5 ) ) ) ).
% mult_cancel_right
thf(fact_1146_mult__cancel__right,axiom,
! [A3: int,C2: int,B5: int] :
( ( ( times_times_int @ A3 @ C2 )
= ( times_times_int @ B5 @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A3 = B5 ) ) ) ).
% mult_cancel_right
thf(fact_1147_mult__cancel__left,axiom,
! [C2: real,A3: real,B5: real] :
( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B5 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B5 ) ) ) ).
% mult_cancel_left
thf(fact_1148_mult__cancel__left,axiom,
! [C2: nat,A3: nat,B5: nat] :
( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B5 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B5 ) ) ) ).
% mult_cancel_left
thf(fact_1149_mult__cancel__left,axiom,
! [C2: int,A3: int,B5: int] :
( ( ( times_times_int @ C2 @ A3 )
= ( times_times_int @ C2 @ B5 ) )
= ( ( C2 = zero_zero_int )
| ( A3 = B5 ) ) ) ).
% mult_cancel_left
thf(fact_1150_mult__eq__0__iff,axiom,
! [A3: real,B5: real] :
( ( ( times_times_real @ A3 @ B5 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B5 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_1151_mult__eq__0__iff,axiom,
! [A3: nat,B5: nat] :
( ( ( times_times_nat @ A3 @ B5 )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B5 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1152_mult__eq__0__iff,axiom,
! [A3: int,B5: int] :
( ( ( times_times_int @ A3 @ B5 )
= zero_zero_int )
= ( ( A3 = zero_zero_int )
| ( B5 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_1153_mult__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_1154_mult__zero__right,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1155_mult__zero__right,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1156_mult__zero__left,axiom,
! [A3: real] :
( ( times_times_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_1157_mult__zero__left,axiom,
! [A3: nat] :
( ( times_times_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1158_mult__zero__left,axiom,
! [A3: int] :
( ( times_times_int @ zero_zero_int @ A3 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1159_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1160_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1161_mult__not__zero,axiom,
! [A3: real,B5: real] :
( ( ( times_times_real @ A3 @ B5 )
!= zero_zero_real )
=> ( ( A3 != zero_zero_real )
& ( B5 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_1162_mult__not__zero,axiom,
! [A3: nat,B5: nat] :
( ( ( times_times_nat @ A3 @ B5 )
!= zero_zero_nat )
=> ( ( A3 != zero_zero_nat )
& ( B5 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_1163_mult__not__zero,axiom,
! [A3: int,B5: int] :
( ( ( times_times_int @ A3 @ B5 )
!= zero_zero_int )
=> ( ( A3 != zero_zero_int )
& ( B5 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_1164_divisors__zero,axiom,
! [A3: real,B5: real] :
( ( ( times_times_real @ A3 @ B5 )
= zero_zero_real )
=> ( ( A3 = zero_zero_real )
| ( B5 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_1165_divisors__zero,axiom,
! [A3: nat,B5: nat] :
( ( ( times_times_nat @ A3 @ B5 )
= zero_zero_nat )
=> ( ( A3 = zero_zero_nat )
| ( B5 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_1166_divisors__zero,axiom,
! [A3: int,B5: int] :
( ( ( times_times_int @ A3 @ B5 )
= zero_zero_int )
=> ( ( A3 = zero_zero_int )
| ( B5 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_1167_no__zero__divisors,axiom,
! [A3: real,B5: real] :
( ( A3 != zero_zero_real )
=> ( ( B5 != zero_zero_real )
=> ( ( times_times_real @ A3 @ B5 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_1168_no__zero__divisors,axiom,
! [A3: nat,B5: nat] :
( ( A3 != zero_zero_nat )
=> ( ( B5 != zero_zero_nat )
=> ( ( times_times_nat @ A3 @ B5 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_1169_no__zero__divisors,axiom,
! [A3: int,B5: int] :
( ( A3 != zero_zero_int )
=> ( ( B5 != zero_zero_int )
=> ( ( times_times_int @ A3 @ B5 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_1170_mult__left__cancel,axiom,
! [C2: real,A3: real,B5: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B5 ) )
= ( A3 = B5 ) ) ) ).
% mult_left_cancel
thf(fact_1171_mult__left__cancel,axiom,
! [C2: nat,A3: nat,B5: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B5 ) )
= ( A3 = B5 ) ) ) ).
% mult_left_cancel
thf(fact_1172_mult__left__cancel,axiom,
! [C2: int,A3: int,B5: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ C2 @ A3 )
= ( times_times_int @ C2 @ B5 ) )
= ( A3 = B5 ) ) ) ).
% mult_left_cancel
thf(fact_1173_mult__right__cancel,axiom,
! [C2: real,A3: real,B5: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B5 @ C2 ) )
= ( A3 = B5 ) ) ) ).
% mult_right_cancel
thf(fact_1174_mult__right__cancel,axiom,
! [C2: nat,A3: nat,B5: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B5 @ C2 ) )
= ( A3 = B5 ) ) ) ).
% mult_right_cancel
thf(fact_1175_mult__right__cancel,axiom,
! [C2: int,A3: int,B5: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ A3 @ C2 )
= ( times_times_int @ B5 @ C2 ) )
= ( A3 = B5 ) ) ) ).
% mult_right_cancel
thf(fact_1176_left__diff__distrib,axiom,
! [A3: real,B5: real,C2: real] :
( ( times_times_real @ ( minus_minus_real @ A3 @ B5 ) @ C2 )
= ( minus_minus_real @ ( times_times_real @ A3 @ C2 ) @ ( times_times_real @ B5 @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_1177_left__diff__distrib,axiom,
! [A3: int,B5: int,C2: int] :
( ( times_times_int @ ( minus_minus_int @ A3 @ B5 ) @ C2 )
= ( minus_minus_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B5 @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_1178_right__diff__distrib,axiom,
! [A3: real,B5: real,C2: real] :
( ( times_times_real @ A3 @ ( minus_minus_real @ B5 @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A3 @ B5 ) @ ( times_times_real @ A3 @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_1179_right__diff__distrib,axiom,
! [A3: int,B5: int,C2: int] :
( ( times_times_int @ A3 @ ( minus_minus_int @ B5 @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ A3 @ B5 ) @ ( times_times_int @ A3 @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_1180_left__diff__distrib_H,axiom,
! [B5: real,C2: real,A3: real] :
( ( times_times_real @ ( minus_minus_real @ B5 @ C2 ) @ A3 )
= ( minus_minus_real @ ( times_times_real @ B5 @ A3 ) @ ( times_times_real @ C2 @ A3 ) ) ) ).
% left_diff_distrib'
thf(fact_1181_left__diff__distrib_H,axiom,
! [B5: nat,C2: nat,A3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B5 @ C2 ) @ A3 )
= ( minus_minus_nat @ ( times_times_nat @ B5 @ A3 ) @ ( times_times_nat @ C2 @ A3 ) ) ) ).
% left_diff_distrib'
thf(fact_1182_left__diff__distrib_H,axiom,
! [B5: int,C2: int,A3: int] :
( ( times_times_int @ ( minus_minus_int @ B5 @ C2 ) @ A3 )
= ( minus_minus_int @ ( times_times_int @ B5 @ A3 ) @ ( times_times_int @ C2 @ A3 ) ) ) ).
% left_diff_distrib'
thf(fact_1183_right__diff__distrib_H,axiom,
! [A3: real,B5: real,C2: real] :
( ( times_times_real @ A3 @ ( minus_minus_real @ B5 @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A3 @ B5 ) @ ( times_times_real @ A3 @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_1184_right__diff__distrib_H,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( times_times_nat @ A3 @ ( minus_minus_nat @ B5 @ C2 ) )
= ( minus_minus_nat @ ( times_times_nat @ A3 @ B5 ) @ ( times_times_nat @ A3 @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_1185_right__diff__distrib_H,axiom,
! [A3: int,B5: int,C2: int] :
( ( times_times_int @ A3 @ ( minus_minus_int @ B5 @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ A3 @ B5 ) @ ( times_times_int @ A3 @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_1186_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: real,B5: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A3 ) @ ( times_times_real @ C2 @ B5 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1187_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: nat,B5: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B5 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1188_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: int,B5: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B5 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1189_zero__le__mult__iff,axiom,
! [A3: real,B5: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B5 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B5 ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B5 @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1190_zero__le__mult__iff,axiom,
! [A3: int,B5: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B5 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ zero_zero_int @ B5 ) )
| ( ( ord_less_eq_int @ A3 @ zero_zero_int )
& ( ord_less_eq_int @ B5 @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1191_mult__nonneg__nonpos2,axiom,
! [A3: real,B5: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B5 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B5 @ A3 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1192_mult__nonneg__nonpos2,axiom,
! [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B5 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B5 @ A3 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1193_mult__nonneg__nonpos2,axiom,
! [A3: int,B5: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B5 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B5 @ A3 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1194_mult__nonpos__nonneg,axiom,
! [A3: real,B5: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B5 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B5 ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1195_mult__nonpos__nonneg,axiom,
! [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B5 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B5 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1196_mult__nonpos__nonneg,axiom,
! [A3: int,B5: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B5 )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B5 ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1197_mult__nonneg__nonpos,axiom,
! [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B5 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B5 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1198_mult__nonneg__nonpos,axiom,
! [A3: int,B5: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B5 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A3 @ B5 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1199_sumset__subset__Un_I1_J,axiom,
! [A2: set_a,B2: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sumset_subset_Un(1)
thf(fact_1200_sumset__subset__Un_I2_J,axiom,
! [A2: set_a,B2: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ C ) @ B2 ) ) ).
% sumset_subset_Un(2)
thf(fact_1201_sumset__subset__Un1,axiom,
! [A2: set_a,A: set_a,B2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ A ) @ B2 )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B2 ) ) ) ).
% sumset_subset_Un1
thf(fact_1202_sumset__subset__Un2,axiom,
! [A2: set_a,B2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B2 @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ).
% sumset_subset_Un2
thf(fact_1203_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1204_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ord_less_nat @ X2 @ N ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1205_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1206_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1207_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1208_int__ops_I6_J,axiom,
! [A3: nat,B5: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B5 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A3 @ B5 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B5 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A3 @ B5 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1209_pos__int__cases,axiom,
! [K3: int] :
( ( ord_less_int @ zero_zero_int @ K3 )
=> ~ ! [N3: nat] :
( ( K3
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1210_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1211_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1212_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1213_int__ops_I7_J,axiom,
! [A3: nat,B5: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A3 @ B5 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ).
% int_ops(7)
thf(fact_1214_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K3: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K3 ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K3 ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1215_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1216_zero__less__imp__eq__int,axiom,
! [K3: int] :
( ( ord_less_int @ zero_zero_int @ K3 )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K3
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1217_Bolzano,axiom,
! [A3: real,B5: real,P: real > real > $o] :
( ( ord_less_eq_real @ A3 @ B5 )
=> ( ! [A4: real,B3: real,C5: real] :
( ( P @ A4 @ B3 )
=> ( ( P @ B3 @ C5 )
=> ( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C5 )
=> ( P @ A4 @ C5 ) ) ) ) )
=> ( ! [X2: real] :
( ( ord_less_eq_real @ A3 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ B5 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A4: real,B3: real] :
( ( ( ord_less_eq_real @ A4 @ X2 )
& ( ord_less_eq_real @ X2 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A4 ) @ D3 ) )
=> ( P @ A4 @ B3 ) ) ) ) )
=> ( P @ A3 @ B5 ) ) ) ) ).
% Bolzano
thf(fact_1218_additive__abelian__group__axioms,axiom,
pluenn1164192988769422572roup_a @ g @ addition @ zero ).
% additive_abelian_group_axioms
thf(fact_1219_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_1220_Ruzsa__triangle__ineq2,axiom,
! [U2: set_a,V3: set_a,W2: set_a] :
( ( finite_finite_a @ U2 )
=> ( ( ord_less_eq_set_a @ U2 @ g )
=> ( ( U2 != bot_bot_set_a )
=> ( ( finite_finite_a @ V3 )
=> ( ( ord_less_eq_set_a @ V3 @ g )
=> ( ( finite_finite_a @ W2 )
=> ( ( ord_less_eq_set_a @ W2 @ g )
=> ( ord_less_eq_real @ ( pluenn5761198478017115492ance_a @ g @ addition @ zero @ V3 @ W2 ) @ ( times_times_real @ ( pluenn5761198478017115492ance_a @ g @ addition @ zero @ V3 @ U2 ) @ ( pluenn5761198478017115492ance_a @ g @ addition @ zero @ U2 @ W2 ) ) ) ) ) ) ) ) ) ) ).
% Ruzsa_triangle_ineq2
thf(fact_1221_unit__closed,axiom,
member_a @ zero @ g ).
% unit_closed
thf(fact_1222_right__unit,axiom,
! [A3: a] :
( ( member_a @ A3 @ g )
=> ( ( addition @ A3 @ zero )
= A3 ) ) ).
% right_unit
thf(fact_1223_left__unit,axiom,
! [A3: a] :
( ( member_a @ A3 @ g )
=> ( ( addition @ zero @ A3 )
= A3 ) ) ).
% left_unit
thf(fact_1224_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_1225_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_1226_commutative__monoid__axioms,axiom,
group_4866109990395492029noid_a @ g @ addition @ zero ).
% commutative_monoid_axioms
thf(fact_1227_abelian__group__axioms,axiom,
group_201663378560352916roup_a @ g @ addition @ zero ).
% abelian_group_axioms
thf(fact_1228_sumset__iterated__empty,axiom,
! [R: nat] :
( ( ord_less_nat @ zero_zero_nat @ R )
=> ( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ bot_bot_set_a @ R )
= bot_bot_set_a ) ) ).
% sumset_iterated_empty
thf(fact_1229_finite__sumset__iterated,axiom,
! [A2: set_a,R: nat] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ R ) ) ) ).
% finite_sumset_iterated
thf(fact_1230_sumset__iterated__subset__carrier,axiom,
! [A2: set_a,K3: nat] : ( ord_less_eq_set_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K3 ) @ g ) ).
% sumset_iterated_subset_carrier
thf(fact_1231_sumset__iterated__0,axiom,
! [A2: set_a] :
( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ zero_zero_nat )
= ( insert_a @ zero @ bot_bot_set_a ) ) ).
% sumset_iterated_0
thf(fact_1232_that,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ a2 )
=> ( ( A != bot_bot_set_a )
=> ( ! [R2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ b @ R2 ) ) ) ) @ ( times_times_real @ ( power_power_real @ k @ R2 ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A ) ) ) )
=> thesis ) ) ) ).
% that
thf(fact_1233_sumset__iterated__r,axiom,
! [R: nat,A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ R )
=> ( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ R )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ ( minus_minus_nat @ R @ one_one_nat ) ) ) ) ) ).
% sumset_iterated_r
thf(fact_1234_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1235_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1236_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1237_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1238_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1239_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1240_realpow__pos__nth__unique,axiom,
! [N: nat,A3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ? [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
& ( ( power_power_real @ X2 @ N )
= A3 )
& ! [Y5: real] :
( ( ( ord_less_real @ zero_zero_real @ Y5 )
& ( ( power_power_real @ Y5 @ N )
= A3 ) )
=> ( Y5 = X2 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1241_realpow__pos__nth,axiom,
! [N: nat,A3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ N )
= A3 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1242_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1243_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1244_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1245_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1246_Ruzsa__triangle__ineq1,axiom,
! [U2: set_a,V3: set_a,W2: set_a] :
( ( finite_finite_a @ U2 )
=> ( ( ord_less_eq_set_a @ U2 @ g )
=> ( ( finite_finite_a @ V3 )
=> ( ( ord_less_eq_set_a @ V3 @ g )
=> ( ( finite_finite_a @ W2 )
=> ( ( ord_less_eq_set_a @ W2 @ g )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( finite_card_a @ U2 ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ V3 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ W2 ) ) ) ) @ ( times_times_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ U2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ V3 ) ) ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ U2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ W2 ) ) ) ) ) ) ) ) ) ) ) ).
% Ruzsa_triangle_ineq1
thf(fact_1247_minusset__distrib__sum,axiom,
! [A2: set_a,B2: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B2 ) ) ) ).
% minusset_distrib_sum
thf(fact_1248_finite__minusset,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) ) ).
% finite_minusset
thf(fact_1249_minusset__subset__carrier,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ g ) ).
% minusset_subset_carrier
thf(fact_1250_minusset__iterated__minusset,axiom,
! [A2: set_a,K3: nat] :
( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ K3 )
= ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K3 ) ) ) ).
% minusset_iterated_minusset
thf(fact_1251_card__differenceset__commute,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) )
= ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B2 ) ) ) ) ).
% card_differenceset_commute
thf(fact_1252_finite__differenceset,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B2 ) ) ) ) ) ).
% finite_differenceset
thf(fact_1253_card__minusset_H,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ g )
=> ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
= ( finite_card_a @ A2 ) ) ) ).
% card_minusset'
thf(fact_1254_card__sumset__iterated__minusset,axiom,
! [A2: set_a,K3: nat] :
( ( finite_card_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ K3 ) )
= ( finite_card_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K3 ) ) ) ).
% card_sumset_iterated_minusset
thf(fact_1255_differenceset__commute,axiom,
! [B2: set_a,A2: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B2 ) ) ) ).
% differenceset_commute
thf(fact_1256_minus__minusset,axiom,
! [A2: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
= ( inf_inf_set_a @ A2 @ g ) ) ).
% minus_minusset
thf(fact_1257_card__minusset,axiom,
! [A2: set_a] :
( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) ) ) ).
% card_minusset
thf(fact_1258_minusset__is__empty__iff,axiom,
! [A2: set_a] :
( ( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 )
= bot_bot_set_a )
= ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a ) ) ).
% minusset_is_empty_iff
thf(fact_1259_minusset__triv,axiom,
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( insert_a @ zero @ bot_bot_set_a ) )
= ( insert_a @ zero @ bot_bot_set_a ) ) ).
% minusset_triv
thf(fact_1260_group__axioms,axiom,
group_group_a @ g @ addition @ zero ).
% group_axioms
thf(fact_1261_group__of__Units,axiom,
group_group_a @ ( group_Units_a @ g @ addition @ zero ) @ addition @ zero ).
% group_of_Units
thf(fact_1262_minusset_Ocases,axiom,
! [A3: a,A2: set_a] :
( ( member_a @ A3 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
=> ~ ! [A4: a] :
( ( A3
= ( group_inverse_a @ g @ addition @ zero @ A4 ) )
=> ( ( member_a @ A4 @ A2 )
=> ~ ( member_a @ A4 @ g ) ) ) ) ).
% minusset.cases
thf(fact_1263_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_1264_inverse__closed,axiom,
! [X: a] :
( ( member_a @ X @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ X ) @ g ) ) ).
% inverse_closed
thf(fact_1265_minusset_Osimps,axiom,
! [A3: a,A2: set_a] :
( ( member_a @ A3 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
= ( ? [A5: a] :
( ( A3
= ( group_inverse_a @ g @ addition @ zero @ A5 ) )
& ( member_a @ A5 @ A2 )
& ( member_a @ A5 @ g ) ) ) ) ).
% minusset.simps
thf(fact_1266_minusset_OminussetI,axiom,
! [A3: a,A2: set_a] :
( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ g )
=> ( member_a @ ( group_inverse_a @ g @ addition @ zero @ A3 ) @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) ) ) ).
% minusset.minussetI
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_set_a @ a3 @ g ).
%------------------------------------------------------------------------------