TPTP Problem File: SLH0030^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Pluennecke_Ruzsa_Inequality/0003_Pluennecke_Ruzsa_Inequality/prob_00533_021289__12335790_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1350 ( 536 unt; 79 typ; 0 def)
% Number of atoms : 3704 (1190 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 11578 ( 411 ~; 74 |; 210 &;9152 @)
% ( 0 <=>;1731 =>; 0 <=; 0 <~>)
% Maximal formula depth : 35 ( 7 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 468 ( 468 >; 0 *; 0 +; 0 <<)
% Number of symbols : 73 ( 72 usr; 18 con; 0-5 aty)
% Number of variables : 3483 ( 170 ^;3225 !; 88 ?;3483 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:24:33.313
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
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_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_tf__a,type,
a: $tType ).
% Explicit typings (72)
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__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_Oinvertible_001tf__a,type,
group_invertible_a: set_a > ( a > a > a ) > a > a > $o ).
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__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__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__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__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__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_OSuc,type,
suc: nat > nat ).
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__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__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__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__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__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_001t__Real__Oreal,type,
pluenn1014277435162747966p_real: set_real > ( real > real > real ) > real > $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_001t__Real__Oreal,type,
pluenn7361685508355272389t_real: set_real > ( real > real > real ) > set_real > set_real > set_real ).
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_001t__Real__Oreal,type,
pluenn3384280056939765061p_real: set_real > ( real > real > real ) > ( real > $o ) > ( real > $o ) > real > $o ).
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__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__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_r____,type,
r: nat ).
thf(sy_v_thesis,type,
thesis: $o ).
thf(sy_v_zero,type,
zero: a ).
% Relevant facts (1270)
thf(fact_0_A_H_I4_J,axiom,
ord_less_eq_real @ k2 @ k ).
% A'(4)
thf(fact_1_assms_I3_J,axiom,
ord_less_eq_set_a @ a2 @ g ).
% assms(3)
thf(fact_2_False,axiom,
b != bot_bot_set_a ).
% False
thf(fact_3_A_H_I2_J,axiom,
a3 != bot_bot_set_a ).
% A'(2)
thf(fact_4_A_H_I1_J,axiom,
ord_less_eq_set_a @ a3 @ a2 ).
% A'(1)
thf(fact_5_assms_I6_J,axiom,
ord_less_eq_set_a @ b @ g ).
% assms(6)
thf(fact_6_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( addition @ X @ Y )
= ( addition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_7_sumset_Ocases,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( addition @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ g )
=> ( ( member_a @ B2 @ B )
=> ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).
% sumset.cases
thf(fact_8_sumset_Osimps,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= ( ? [A4: a,B3: a] :
( ( A
= ( addition @ A4 @ B3 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ g )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumset.simps
thf(fact_9_sumset_OsumsetI,axiom,
! [A: a,A2: set_a,B4: a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ B )
=> ( ( member_a @ B4 @ g )
=> ( member_a @ ( addition @ A @ B4 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ) ) ).
% sumset.sumsetI
thf(fact_10_sumset__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ C )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ C ) ) ) ).
% sumset_assoc
thf(fact_11_sumset__commute,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A2 ) ) ).
% sumset_commute
thf(fact_12_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_13__092_060open_062A_H_A_092_060subseteq_062_AG_092_060close_062,axiom,
ord_less_eq_set_a @ a3 @ g ).
% \<open>A' \<subseteq> G\<close>
thf(fact_14_sumset__mono,axiom,
! [A5: set_a,A2: set_a,B5: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ B5 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_mono
thf(fact_15_sumset__subset__carrier,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ g ) ).
% sumset_subset_carrier
thf(fact_16_assms_I5_J,axiom,
finite_finite_a @ b ).
% assms(5)
thf(fact_17_sumset__iterated__subset__carrier,axiom,
! [A2: set_a,K: nat] : ( ord_less_eq_set_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K ) @ g ) ).
% sumset_iterated_subset_carrier
thf(fact_18_associative,axiom,
! [A: a,B4: a,C2: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ g )
=> ( ( member_a @ C2 @ g )
=> ( ( addition @ ( addition @ A @ B4 ) @ C2 )
= ( addition @ A @ ( addition @ B4 @ C2 ) ) ) ) ) ) ).
% associative
thf(fact_19_composition__closed,axiom,
! [A: a,B4: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ g )
=> ( member_a @ ( addition @ A @ B4 ) @ g ) ) ) ).
% composition_closed
thf(fact_20_unit__closed,axiom,
member_a @ zero @ g ).
% unit_closed
thf(fact_21_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_22_left__unit,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( addition @ zero @ A )
= A ) ) ).
% left_unit
thf(fact_23_right__unit,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( addition @ A @ zero )
= A ) ) ).
% right_unit
thf(fact_24_sumset__iterated__Suc,axiom,
! [A2: set_a,K: nat] :
( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ ( suc @ K ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K ) ) ) ).
% sumset_iterated_Suc
thf(fact_25_additive__abelian__group__axioms,axiom,
pluenn1164192988769422572roup_a @ g @ addition @ zero ).
% additive_abelian_group_axioms
thf(fact_26_commutative__monoid__axioms,axiom,
group_4866109990395492029noid_a @ g @ addition @ zero ).
% commutative_monoid_axioms
thf(fact_27_that,axiom,
! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ a2 )
=> ( ( A5 != bot_bot_set_a )
=> ( ! [R: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ b @ R ) ) ) ) @ ( times_times_real @ ( power_power_real @ k @ R ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A5 ) ) ) )
=> thesis ) ) ) ).
% that
thf(fact_28_additive__abelian__group_Osumset_Ocong,axiom,
pluenn3038260743871226533mset_a = pluenn3038260743871226533mset_a ).
% additive_abelian_group.sumset.cong
thf(fact_29_additive__abelian__group_Osumset__iterated_Ocong,axiom,
pluenn1960970773371692859ated_a = pluenn1960970773371692859ated_a ).
% additive_abelian_group.sumset_iterated.cong
thf(fact_30_abelian__group__axioms,axiom,
group_201663378560352916roup_a @ g @ addition @ zero ).
% abelian_group_axioms
thf(fact_31_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_32_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_33_sumsetp_Ocases,axiom,
! [A2: a > $o,B: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ A )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( addition @ A3 @ B2 ) )
=> ( ( A2 @ A3 )
=> ( ( member_a @ A3 @ g )
=> ( ( B @ B2 )
=> ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).
% sumsetp.cases
thf(fact_34_sumsetp_Osimps,axiom,
! [A2: a > $o,B: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ A )
= ( ? [A4: a,B3: a] :
( ( A
= ( addition @ A4 @ B3 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ g )
& ( B @ B3 )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumsetp.simps
thf(fact_35_sumsetp_OsumsetI,axiom,
! [A2: a > $o,A: a,B: a > $o,B4: a] :
( ( A2 @ A )
=> ( ( member_a @ A @ g )
=> ( ( B @ B4 )
=> ( ( member_a @ B4 @ g )
=> ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ ( addition @ A @ B4 ) ) ) ) ) ) ).
% sumsetp.sumsetI
thf(fact_36_assms_I2_J,axiom,
finite_finite_a @ a2 ).
% assms(2)
thf(fact_37_group__axioms,axiom,
group_group_a @ g @ addition @ zero ).
% group_axioms
thf(fact_38_finite__sumset__iterated,axiom,
! [A2: set_a,R2: nat] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ R2 ) ) ) ).
% finite_sumset_iterated
thf(fact_39_card__sumset__iterated__minusset,axiom,
! [A2: set_a,K: nat] :
( ( finite_card_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ K ) )
= ( finite_card_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K ) ) ) ).
% card_sumset_iterated_minusset
thf(fact_40_card__differenceset__commute,axiom,
! [B: set_a,A2: set_a] :
( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) )
= ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ) ).
% card_differenceset_commute
thf(fact_41_assms_I4_J,axiom,
a2 != bot_bot_set_a ).
% assms(4)
thf(fact_42_finite__sumset,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% finite_sumset
thf(fact_43_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_minusset__distrib__sum,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ).
% minusset_distrib_sum
thf(fact_48_finite__minusset,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) ) ).
% finite_minusset
thf(fact_49_minusset__subset__carrier,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ g ) ).
% minusset_subset_carrier
thf(fact_50_minusset__iterated__minusset,axiom,
! [A2: set_a,K: nat] :
( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ K )
= ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K ) ) ) ).
% minusset_iterated_minusset
thf(fact_51_finite__differenceset,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ) ) ).
% finite_differenceset
thf(fact_52_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_53_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_54_differenceset__commute,axiom,
! [B: set_a,A2: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ).
% differenceset_commute
thf(fact_55_Ruzsa__triangle__ineq2,axiom,
! [U2: set_a,V3: set_a,W: 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 @ W )
=> ( ( ord_less_eq_set_a @ W @ g )
=> ( ord_less_eq_real @ ( pluenn5761198478017115492ance_a @ g @ addition @ zero @ V3 @ W ) @ ( times_times_real @ ( pluenn5761198478017115492ance_a @ g @ addition @ zero @ V3 @ U2 ) @ ( pluenn5761198478017115492ance_a @ g @ addition @ zero @ U2 @ W ) ) ) ) ) ) ) ) ) ) ).
% Ruzsa_triangle_ineq2
thf(fact_56_additive__abelian__group_Ointro,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( group_201663378560352916roup_a @ G @ Addition @ Zero )
=> ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero ) ) ).
% additive_abelian_group.intro
thf(fact_57_additive__abelian__group_Oaxioms,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( group_201663378560352916roup_a @ G @ Addition @ Zero ) ) ).
% additive_abelian_group.axioms
thf(fact_58_additive__abelian__group_Ominusset_Ocong,axiom,
pluenn2534204936789923946sset_a = pluenn2534204936789923946sset_a ).
% additive_abelian_group.minusset.cong
thf(fact_59_additive__abelian__group_Osumsetp_Ocases,axiom,
! [G: set_real,Addition: real > real > real,Zero: real,A2: real > $o,B: real > $o,A: real] :
( ( pluenn1014277435162747966p_real @ G @ Addition @ Zero )
=> ( ( pluenn3384280056939765061p_real @ G @ Addition @ A2 @ B @ A )
=> ~ ! [A3: real,B2: real] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( A2 @ A3 )
=> ( ( member_real @ A3 @ G )
=> ( ( B @ B2 )
=> ~ ( member_real @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.cases
thf(fact_60_additive__abelian__group_Osumsetp_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,B: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ A )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( A2 @ A3 )
=> ( ( member_a @ A3 @ G )
=> ( ( B @ B2 )
=> ~ ( member_a @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.cases
thf(fact_61_additive__abelian__group_Osumsetp_Osimps,axiom,
! [G: set_real,Addition: real > real > real,Zero: real,A2: real > $o,B: real > $o,A: real] :
( ( pluenn1014277435162747966p_real @ G @ Addition @ Zero )
=> ( ( pluenn3384280056939765061p_real @ G @ Addition @ A2 @ B @ A )
= ( ? [A4: real,B3: real] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( A2 @ A4 )
& ( member_real @ A4 @ G )
& ( B @ B3 )
& ( member_real @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumsetp.simps
thf(fact_62_additive__abelian__group_Osumsetp_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,B: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ A )
= ( ? [A4: a,B3: a] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ G )
& ( B @ B3 )
& ( member_a @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumsetp.simps
thf(fact_63_additive__abelian__group_Ofinite__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) ) ) ) ).
% additive_abelian_group.finite_minusset
thf(fact_64_additive__abelian__group_Osumsetp_OsumsetI,axiom,
! [G: set_real,Addition: real > real > real,Zero: real,A2: real > $o,A: real,B: real > $o,B4: real] :
( ( pluenn1014277435162747966p_real @ G @ Addition @ Zero )
=> ( ( A2 @ A )
=> ( ( member_real @ A @ G )
=> ( ( B @ B4 )
=> ( ( member_real @ B4 @ G )
=> ( pluenn3384280056939765061p_real @ G @ Addition @ A2 @ B @ ( Addition @ A @ B4 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.sumsetI
thf(fact_65_additive__abelian__group_Osumsetp_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,A: a,B: a > $o,B4: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( A2 @ A )
=> ( ( member_a @ A @ G )
=> ( ( B @ B4 )
=> ( ( member_a @ B4 @ G )
=> ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ ( Addition @ A @ B4 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.sumsetI
thf(fact_66_additive__abelian__group_Ofinite__differenceset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ) ) ).
% additive_abelian_group.finite_differenceset
thf(fact_67_additive__abelian__group_Ominusset__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) @ G ) ) ).
% additive_abelian_group.minusset_subset_carrier
thf(fact_68_additive__abelian__group__def,axiom,
pluenn1164192988769422572roup_a = group_201663378560352916roup_a ).
% additive_abelian_group_def
thf(fact_69_additive__abelian__group_Ocard__minusset_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A2 @ G )
=> ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% additive_abelian_group.card_minusset'
thf(fact_70_additive__abelian__group_Ofinite__sumset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset
thf(fact_71_additive__abelian__group_Osumsetp_Ocong,axiom,
pluenn895083305082786853setp_a = pluenn895083305082786853setp_a ).
% additive_abelian_group.sumsetp.cong
thf(fact_72_additive__abelian__group_Odifferenceset__commute,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ).
% additive_abelian_group.differenceset_commute
thf(fact_73_additive__abelian__group_Ominusset__distrib__sum,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ).
% additive_abelian_group.minusset_distrib_sum
thf(fact_74_additive__abelian__group_Odiff__minus__set,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) )
= ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ).
% additive_abelian_group.diff_minus_set
thf(fact_75_additive__abelian__group_Ofinite__sumset__iterated,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,R2: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ R2 ) ) ) ) ).
% additive_abelian_group.finite_sumset_iterated
thf(fact_76_additive__abelian__group_Ominusset__iterated__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,K: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) @ K )
= ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ K ) ) ) ) ).
% additive_abelian_group.minusset_iterated_minusset
thf(fact_77_additive__abelian__group_Osumset__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ G ) ) ).
% additive_abelian_group.sumset_subset_carrier
thf(fact_78_additive__abelian__group_Osumset__mono,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A5: set_a,A2: set_a,B5: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ B5 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_mono
thf(fact_79_additive__abelian__group_Osumset__empty_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ) ).
% additive_abelian_group.sumset_empty(1)
thf(fact_80_additive__abelian__group_Osumset__empty_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ) ).
% additive_abelian_group.sumset_empty(2)
thf(fact_81_additive__abelian__group_Osumset__iterated__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,K: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ K ) @ G ) ) ).
% additive_abelian_group.sumset_iterated_subset_carrier
thf(fact_82_additive__abelian__group_Ocard__differenceset__commute,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) ) )
= ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ) ).
% additive_abelian_group.card_differenceset_commute
thf(fact_83_additive__abelian__group_Ocard__sumset__iterated__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,K: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_card_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) @ K ) )
= ( finite_card_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ K ) ) ) ) ).
% additive_abelian_group.card_sumset_iterated_minusset
thf(fact_84_additive__abelian__group_Osumset__commute,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ A2 ) ) ) ).
% additive_abelian_group.sumset_commute
thf(fact_85_additive__abelian__group_Osumset_OsumsetI,axiom,
! [G: set_real,Addition: real > real > real,Zero: real,A: real,A2: set_real,B4: real,B: set_real] :
( ( pluenn1014277435162747966p_real @ G @ Addition @ Zero )
=> ( ( member_real @ A @ A2 )
=> ( ( member_real @ A @ G )
=> ( ( member_real @ B4 @ B )
=> ( ( member_real @ B4 @ G )
=> ( member_real @ ( Addition @ A @ B4 ) @ ( pluenn7361685508355272389t_real @ G @ Addition @ A2 @ B ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.sumsetI
thf(fact_86_additive__abelian__group_Osumset_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B4: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ G )
=> ( ( member_a @ B4 @ B )
=> ( ( member_a @ B4 @ G )
=> ( member_a @ ( Addition @ A @ B4 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.sumsetI
thf(fact_87_additive__abelian__group_Osumset__assoc,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ C )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ C ) ) ) ) ).
% additive_abelian_group.sumset_assoc
thf(fact_88_additive__abelian__group_Osumset_Osimps,axiom,
! [G: set_real,Addition: real > real > real,Zero: real,A: real,A2: set_real,B: set_real] :
( ( pluenn1014277435162747966p_real @ G @ Addition @ Zero )
=> ( ( member_real @ A @ ( pluenn7361685508355272389t_real @ G @ Addition @ A2 @ B ) )
= ( ? [A4: real,B3: real] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( member_real @ A4 @ A2 )
& ( member_real @ A4 @ G )
& ( member_real @ B3 @ B )
& ( member_real @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset.simps
thf(fact_89_additive__abelian__group_Osumset_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= ( ? [A4: a,B3: a] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ G )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset.simps
thf(fact_90_additive__abelian__group_Osumset_Ocases,axiom,
! [G: set_real,Addition: real > real > real,Zero: real,A: real,A2: set_real,B: set_real] :
( ( pluenn1014277435162747966p_real @ G @ Addition @ Zero )
=> ( ( member_real @ A @ ( pluenn7361685508355272389t_real @ G @ Addition @ A2 @ B ) )
=> ~ ! [A3: real,B2: real] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( member_real @ A3 @ A2 )
=> ( ( member_real @ A3 @ G )
=> ( ( member_real @ B2 @ B )
=> ~ ( member_real @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.cases
thf(fact_91_additive__abelian__group_Osumset_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ G )
=> ( ( member_a @ B2 @ B )
=> ~ ( member_a @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.cases
thf(fact_92_additive__abelian__group_Osumset__iterated__Suc,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,K: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ ( suc @ K ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ K ) ) ) ) ).
% additive_abelian_group.sumset_iterated_Suc
thf(fact_93_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B4: nat,W2: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B4 ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B4 @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_94_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B4: nat,W2: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B4 ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B4 @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_95_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B4: nat,W2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B4 ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B4 @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_96_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B4: nat,W2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B4 ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B4 @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_97_card__le__sumset,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ g )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ) ) ) ) ).
% card_le_sumset
thf(fact_98_card__sumset__0__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ g )
=> ( ( ord_less_eq_set_a @ B @ g )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
| ( ( finite_card_a @ B )
= zero_zero_nat ) ) ) ) ) ).
% card_sumset_0_iff
thf(fact_99_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B4: nat,W2: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B4 ) @ W2 ) )
= ( X
= ( power_power_nat @ B4 @ W2 ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_100_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B4: nat,W2: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B4 ) @ W2 ) )
= ( X
= ( power_power_nat @ B4 @ W2 ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_101_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B4: nat,W2: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B4 ) @ W2 )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B4 @ W2 )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_102_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B4: nat,W2: nat,X: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B4 ) @ W2 )
= ( semiri5074537144036343181t_real @ X ) )
= ( ( power_power_nat @ B4 @ W2 )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_103_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_104_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% of_nat_power
thf(fact_105_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_106_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_107_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_108_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_109_infinite__sumset__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) )
= ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
& ( ( inf_inf_set_a @ B @ g )
!= bot_bot_set_a ) )
| ( ( ( inf_inf_set_a @ A2 @ g )
!= bot_bot_set_a )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ B @ g ) ) ) ) ) ).
% infinite_sumset_iff
thf(fact_110_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_111_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_112_all__not__in__conv,axiom,
! [A2: set_real] :
( ( ! [X2: real] :
~ ( member_real @ X2 @ A2 ) )
= ( A2 = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_113_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_114_empty__iff,axiom,
! [C2: real] :
~ ( member_real @ C2 @ bot_bot_set_real ) ).
% empty_iff
thf(fact_115_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_116_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_117_subsetI,axiom,
! [A2: set_real,B: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ X3 @ B ) )
=> ( ord_less_eq_set_real @ A2 @ B ) ) ).
% subsetI
thf(fact_118_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_119_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_120_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_121_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_122_Int__iff,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B ) )
= ( ( member_real @ C2 @ A2 )
& ( member_real @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_123_Int__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
& ( member_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_124_IntI,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ A2 )
=> ( ( member_real @ C2 @ B )
=> ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_125_IntI,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_126_sumset__empty_H_I2_J,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= bot_bot_set_a ) ) ).
% sumset_empty'(2)
thf(fact_127_sumset__empty_H_I1_J,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A2 )
= bot_bot_set_a ) ) ).
% sumset_empty'(1)
thf(fact_128_finite__sumset_H,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B @ g ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% finite_sumset'
thf(fact_129_card__sumset__0__iff_H,axiom,
! [A2: set_a,B: set_a] :
( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) )
= zero_zero_nat )
| ( ( finite_card_a @ ( inf_inf_set_a @ B @ g ) )
= zero_zero_nat ) ) ) ).
% card_sumset_0_iff'
thf(fact_130_infinite__sumset__aux,axiom,
! [A2: set_a,B: set_a] :
( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) )
= ( ( inf_inf_set_a @ B @ g )
!= bot_bot_set_a ) ) ) ).
% infinite_sumset_aux
thf(fact_131_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_132_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_133_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_134_Int__subset__iff,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_135_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_136_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_137_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_138_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_139_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_140_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_141_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_142_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_143_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_144_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_145_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_146_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_147_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_148_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_149_power__Suc0__right,axiom,
! [A: real] :
( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_150_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_151_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_152_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_153_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_154_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_155_Ruzsa__triangle__ineq1,axiom,
! [U2: set_a,V3: set_a,W: 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 @ W )
=> ( ( ord_less_eq_set_a @ W @ 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 @ W ) ) ) ) @ ( 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 @ W ) ) ) ) ) ) ) ) ) ) ) ).
% Ruzsa_triangle_ineq1
thf(fact_156_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_157_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_158_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_159_sumset__Int__carrier__eq_I2_J,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( inf_inf_set_a @ A2 @ g ) @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ).
% sumset_Int_carrier_eq(2)
thf(fact_160_sumset__Int__carrier__eq_I1_J,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( inf_inf_set_a @ B @ g ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ).
% sumset_Int_carrier_eq(1)
thf(fact_161_sumset__Int__carrier,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ g )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ).
% sumset_Int_carrier
thf(fact_162_sumset__is__empty__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ g )
= bot_bot_set_a ) ) ) ).
% sumset_is_empty_iff
thf(fact_163_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_164_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_165_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_166_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_167_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_168_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_169_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_170_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_171_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_172_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_173_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_174_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_175_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_176_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_177_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_178_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_179_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_180_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_181_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_182_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_183_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B4: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B4 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_184_Int__left__commute,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_left_commute
thf(fact_185_Int__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_186_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] : ( inf_inf_set_a @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_187_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_188_Int__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_assoc
thf(fact_189_IntD2,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B ) )
=> ( member_real @ C2 @ B ) ) ).
% IntD2
thf(fact_190_IntD2,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ B ) ) ).
% IntD2
thf(fact_191_IntD1,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B ) )
=> ( member_real @ C2 @ A2 ) ) ).
% IntD1
thf(fact_192_IntD1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_193_IntE,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B ) )
=> ~ ( ( member_real @ C2 @ A2 )
=> ~ ( member_real @ C2 @ B ) ) ) ).
% IntE
thf(fact_194_IntE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ~ ( member_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_195_disjoint__iff__not__equal,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y5: a] :
( ( member_a @ Y5 @ B )
=> ( X2 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_196_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_197_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_198_disjoint__iff,axiom,
! [A2: set_real,B: set_real] :
( ( ( inf_inf_set_real @ A2 @ B )
= bot_bot_set_real )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ~ ( member_real @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_199_disjoint__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_200_Int__emptyI,axiom,
! [A2: set_real,B: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ~ ( member_real @ X3 @ B ) )
=> ( ( inf_inf_set_real @ A2 @ B )
= bot_bot_set_real ) ) ).
% Int_emptyI
thf(fact_201_Int__emptyI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_202_Int__Collect__mono,axiom,
! [A2: set_real,B: set_real,P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_real @ ( inf_inf_set_real @ A2 @ ( collect_real @ P ) ) @ ( inf_inf_set_real @ B @ ( collect_real @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_203_Int__Collect__mono,axiom,
! [A2: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_204_Int__greatest,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_205_Int__absorb2,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_206_Int__absorb1,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_207_Int__lower2,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_208_Int__lower1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_209_Int__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% Int_mono
thf(fact_210_additive__abelian__group_ORuzsa__distance_Ocong,axiom,
pluenn5761198478017115492ance_a = pluenn5761198478017115492ance_a ).
% additive_abelian_group.Ruzsa_distance.cong
thf(fact_211_power__mult,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_212_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_213_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_214_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_215_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_216_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_217_Suc__le__D,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M2 )
=> ? [M3: nat] :
( M2
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_218_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_219_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_220_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_221_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_222_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_223_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( R3 @ X3 @ Y3 )
=> ( ( R3 @ Y3 @ Z )
=> ( R3 @ X3 @ Z ) ) )
=> ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
=> ( R3 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_224_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_225_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_226_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_227_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_228_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_229_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_230_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_231_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_232_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_233_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_234_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_235_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_236_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_237_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_238_zero__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_power
thf(fact_239_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_240_power__mono,axiom,
! [A: real,B4: real,N: nat] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B4 @ N ) ) ) ) ).
% power_mono
thf(fact_241_power__mono,axiom,
! [A: nat,B4: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B4 @ N ) ) ) ) ).
% power_mono
thf(fact_242_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_243_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_244_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_245_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_246_additive__abelian__group_Ocard__sumset__0__iff_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ ( inf_inf_set_a @ A2 @ G ) )
= zero_zero_nat )
| ( ( finite_card_a @ ( inf_inf_set_a @ B @ G ) )
= zero_zero_nat ) ) ) ) ).
% additive_abelian_group.card_sumset_0_iff'
thf(fact_247_power__inject__base,axiom,
! [A: real,N: nat,B4: real] :
( ( ( power_power_real @ A @ ( suc @ N ) )
= ( power_power_real @ B4 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( A = B4 ) ) ) ) ).
% power_inject_base
thf(fact_248_power__inject__base,axiom,
! [A: nat,N: nat,B4: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B4 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( A = B4 ) ) ) ) ).
% power_inject_base
thf(fact_249_power__le__imp__le__base,axiom,
! [A: real,N: nat,B4: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B4 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ord_less_eq_real @ A @ B4 ) ) ) ).
% power_le_imp_le_base
thf(fact_250_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B4: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B4 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ord_less_eq_nat @ A @ B4 ) ) ) ).
% power_le_imp_le_base
thf(fact_251_additive__abelian__group_Osumset__Int__carrier__eq_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( inf_inf_set_a @ A2 @ G ) @ B )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier_eq(2)
thf(fact_252_additive__abelian__group_Osumset__Int__carrier__eq_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( inf_inf_set_a @ B @ G ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier_eq(1)
thf(fact_253_additive__abelian__group_Osumset__Int__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ G )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier
thf(fact_254_additive__abelian__group_Ominus__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) )
= ( inf_inf_set_a @ A2 @ G ) ) ) ).
% additive_abelian_group.minus_minusset
thf(fact_255_additive__abelian__group_ORuzsa__triangle__ineq1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,U2: set_a,V3: set_a,W: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( 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 @ W )
=> ( ( ord_less_eq_set_a @ W @ 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 @ W ) ) ) ) @ ( 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 @ W ) ) ) ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.Ruzsa_triangle_ineq1
thf(fact_256_additive__abelian__group_Osumset__empty_H_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_empty'(2)
thf(fact_257_additive__abelian__group_Osumset__empty_H_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_empty'(1)
thf(fact_258_additive__abelian__group_Osumset__is__empty__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ G )
= bot_bot_set_a ) ) ) ) ).
% additive_abelian_group.sumset_is_empty_iff
thf(fact_259_additive__abelian__group_Ofinite__sumset_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ G ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B @ G ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset'
thf(fact_260_ex__in__conv,axiom,
! [A2: set_real] :
( ( ? [X2: real] : ( member_real @ X2 @ A2 ) )
= ( A2 != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_261_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_262_equals0I,axiom,
! [A2: set_real] :
( ! [Y3: real] :
~ ( member_real @ Y3 @ A2 )
=> ( A2 = bot_bot_set_real ) ) ).
% equals0I
thf(fact_263_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_264_equals0D,axiom,
! [A2: set_real,A: real] :
( ( A2 = bot_bot_set_real )
=> ~ ( member_real @ A @ A2 ) ) ).
% equals0D
thf(fact_265_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_266_emptyE,axiom,
! [A: real] :
~ ( member_real @ A @ bot_bot_set_real ) ).
% emptyE
thf(fact_267_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_268_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_269_set__eq__subset,axiom,
( ( ^ [Y6: set_a,Z2: set_a] : ( Y6 = Z2 ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_270_subset__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_271_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_272_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_273_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [T: real] :
( ( member_real @ T @ A6 )
=> ( member_real @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_274_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T: a] :
( ( member_a @ T @ A6 )
=> ( member_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_275_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_276_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_277_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [X2: real] :
( ( member_real @ X2 @ A6 )
=> ( member_real @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_278_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( member_a @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_279_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_280_subsetD,axiom,
! [A2: set_real,B: set_real,C2: real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ B ) ) ) ).
% subsetD
thf(fact_281_subsetD,axiom,
! [A2: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_282_in__mono,axiom,
! [A2: set_real,B: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B ) ) ) ).
% in_mono
thf(fact_283_in__mono,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_284_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_285_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_286_additive__abelian__group_Ominusset__is__empty__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 )
= bot_bot_set_a )
= ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.minusset_is_empty_iff
thf(fact_287_additive__abelian__group_Ocard__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ G ) ) ) ) ).
% additive_abelian_group.card_minusset
thf(fact_288_additive__abelian__group_Oinfinite__sumset__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) )
= ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ G ) )
& ( ( inf_inf_set_a @ B @ G )
!= bot_bot_set_a ) )
| ( ( ( inf_inf_set_a @ A2 @ G )
!= bot_bot_set_a )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ B @ G ) ) ) ) ) ) ).
% additive_abelian_group.infinite_sumset_iff
thf(fact_289_additive__abelian__group_Oinfinite__sumset__aux,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ G ) )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) )
= ( ( inf_inf_set_a @ B @ G )
!= bot_bot_set_a ) ) ) ) ).
% additive_abelian_group.infinite_sumset_aux
thf(fact_290_additive__abelian__group_Ocard__sumset__0__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A2 @ G )
=> ( ( ord_less_eq_set_a @ B @ G )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
| ( ( finite_card_a @ B )
= zero_zero_nat ) ) ) ) ) ) ).
% additive_abelian_group.card_sumset_0_iff
thf(fact_291_additive__abelian__group_Ocard__le__sumset,axiom,
! [G: set_real,Addition: real > real > real,Zero: real,A2: set_real,A: real,B: set_real] :
( ( pluenn1014277435162747966p_real @ G @ Addition @ Zero )
=> ( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ( ( member_real @ A @ G )
=> ( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ B @ G )
=> ( ord_less_eq_nat @ ( finite_card_real @ B ) @ ( finite_card_real @ ( pluenn7361685508355272389t_real @ G @ Addition @ A2 @ B ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.card_le_sumset
thf(fact_292_additive__abelian__group_Ocard__le__sumset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ G )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.card_le_sumset
thf(fact_293_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_294_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_295_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_296_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_297_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_298_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_299_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_300_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_301_power__commuting__commutes,axiom,
! [X: real,Y: real,N: nat] :
( ( ( times_times_real @ X @ Y )
= ( times_times_real @ Y @ X ) )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y )
= ( times_times_real @ Y @ ( power_power_real @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_302_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_303_power__mult__distrib,axiom,
! [A: real,B4: real,N: nat] :
( ( power_power_real @ ( times_times_real @ A @ B4 ) @ N )
= ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B4 @ N ) ) ) ).
% power_mult_distrib
thf(fact_304_power__mult__distrib,axiom,
! [A: nat,B4: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B4 ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B4 @ N ) ) ) ).
% power_mult_distrib
thf(fact_305_power__commutes,axiom,
! [A: real,N: nat] :
( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_commutes
thf(fact_306_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_307_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_308_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_309_power__Suc,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_Suc
thf(fact_310_power__Suc,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_Suc
thf(fact_311_power__Suc2,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_312_power__Suc2,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_313_additive__abelian__group_ORuzsa__triangle__ineq2,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,U2: set_a,V3: set_a,W: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( 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 @ W )
=> ( ( ord_less_eq_set_a @ W @ G )
=> ( ord_less_eq_real @ ( pluenn5761198478017115492ance_a @ G @ Addition @ Zero @ V3 @ W ) @ ( times_times_real @ ( pluenn5761198478017115492ance_a @ G @ Addition @ Zero @ V3 @ U2 ) @ ( pluenn5761198478017115492ance_a @ G @ Addition @ Zero @ U2 @ W ) ) ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.Ruzsa_triangle_ineq2
thf(fact_314_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_315_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_316_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_317_card__sumset__singleton__eq,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ A @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) ) ) )
& ( ~ ( member_a @ A @ g )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= zero_zero_nat ) ) ) ) ).
% card_sumset_singleton_eq
thf(fact_318_sumset__iterated__empty,axiom,
! [R2: nat] :
( ( ord_less_nat @ zero_zero_nat @ R2 )
=> ( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ bot_bot_set_a @ R2 )
= bot_bot_set_a ) ) ).
% sumset_iterated_empty
thf(fact_319_card__sumset__le,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ).
% card_sumset_le
thf(fact_320_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_321_card__le__Suc0__iff__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y5: a] :
( ( member_a @ Y5 @ A2 )
=> ( X2 = Y5 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_322_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_323_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_324_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_325_insert__iff,axiom,
! [A: a,B4: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B4 @ A2 ) )
= ( ( A = B4 )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_326_insert__iff,axiom,
! [A: real,B4: real,A2: set_real] :
( ( member_real @ A @ ( insert_real @ B4 @ A2 ) )
= ( ( A = B4 )
| ( member_real @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_327_insertCI,axiom,
! [A: a,B: set_a,B4: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = B4 ) )
=> ( member_a @ A @ ( insert_a @ B4 @ B ) ) ) ).
% insertCI
thf(fact_328_insertCI,axiom,
! [A: real,B: set_real,B4: real] :
( ( ~ ( member_real @ A @ B )
=> ( A = B4 ) )
=> ( member_real @ A @ ( insert_real @ B4 @ B ) ) ) ).
% insertCI
thf(fact_329_sumset__subset__insert_I1_J,axiom,
! [A2: set_a,B: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B ) ) ) ).
% sumset_subset_insert(1)
thf(fact_330_sumset__subset__insert_I2_J,axiom,
! [A2: set_a,B: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B ) ) ).
% sumset_subset_insert(2)
thf(fact_331_singletonI,axiom,
! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).
% singletonI
thf(fact_332_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_333_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_334_insert__subset,axiom,
! [X: real,A2: set_real,B: set_real] :
( ( ord_less_eq_set_real @ ( insert_real @ X @ A2 ) @ B )
= ( ( member_real @ X @ B )
& ( ord_less_eq_set_real @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_335_insert__subset,axiom,
! [X: a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_336_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_337_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_338_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_339_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_340_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_341_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_342_Int__insert__left__if0,axiom,
! [A: real,C: set_real,B: set_real] :
( ~ ( member_real @ A @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A @ B ) @ C )
= ( inf_inf_set_real @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_343_Int__insert__left__if0,axiom,
! [A: a,C: set_a,B: set_a] :
( ~ ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_344_Int__insert__left__if1,axiom,
! [A: real,C: set_real,B: set_real] :
( ( member_real @ A @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A @ B ) @ C )
= ( insert_real @ A @ ( inf_inf_set_real @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_345_Int__insert__left__if1,axiom,
! [A: a,C: set_a,B: set_a] :
( ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_346_insert__inter__insert,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_347_Int__insert__right__if0,axiom,
! [A: real,A2: set_real,B: set_real] :
( ~ ( member_real @ A @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B ) )
= ( inf_inf_set_real @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_348_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_349_Int__insert__right__if1,axiom,
! [A: real,A2: set_real,B: set_real] :
( ( member_real @ A @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B ) )
= ( insert_real @ A @ ( inf_inf_set_real @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_350_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_351_A_H_I3_J,axiom,
ord_less_real @ zero_zero_real @ k2 ).
% A'(3)
thf(fact_352_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_353_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_354_singleton__insert__inj__eq,axiom,
! [B4: a,A: a,A2: set_a] :
( ( ( insert_a @ B4 @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B4 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_355_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B4: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B4 @ bot_bot_set_a ) )
= ( ( A = B4 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_356_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_357_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_358_insert__disjoint_I1_J,axiom,
! [A: real,A2: set_real,B: set_real] :
( ( ( inf_inf_set_real @ ( insert_real @ A @ A2 ) @ B )
= bot_bot_set_real )
= ( ~ ( member_real @ A @ B )
& ( ( inf_inf_set_real @ A2 @ B )
= bot_bot_set_real ) ) ) ).
% insert_disjoint(1)
thf(fact_359_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_360_insert__disjoint_I2_J,axiom,
! [A: real,A2: set_real,B: set_real] :
( ( bot_bot_set_real
= ( inf_inf_set_real @ ( insert_real @ A @ A2 ) @ B ) )
= ( ~ ( member_real @ A @ B )
& ( bot_bot_set_real
= ( inf_inf_set_real @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_361_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B ) )
= ( ~ ( member_a @ A @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_362_disjoint__insert_I1_J,axiom,
! [B: set_real,A: real,A2: set_real] :
( ( ( inf_inf_set_real @ B @ ( insert_real @ A @ A2 ) )
= bot_bot_set_real )
= ( ~ ( member_real @ A @ B )
& ( ( inf_inf_set_real @ B @ A2 )
= bot_bot_set_real ) ) ) ).
% disjoint_insert(1)
thf(fact_363_disjoint__insert_I1_J,axiom,
! [B: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_364_disjoint__insert_I2_J,axiom,
! [A2: set_real,B4: real,B: set_real] :
( ( bot_bot_set_real
= ( inf_inf_set_real @ A2 @ ( insert_real @ B4 @ B ) ) )
= ( ~ ( member_real @ B4 @ A2 )
& ( bot_bot_set_real
= ( inf_inf_set_real @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_365_disjoint__insert_I2_J,axiom,
! [A2: set_a,B4: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B4 @ B ) ) )
= ( ~ ( member_a @ B4 @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_366_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_367_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_368_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_369_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( power_power_real @ A @ N )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_370_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_371_card__insert__disjoint,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ~ ( member_real @ X @ A2 )
=> ( ( finite_card_real @ ( insert_real @ X @ A2 ) )
= ( suc @ ( finite_card_real @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_372_card__insert__disjoint,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( suc @ ( finite_card_a @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_373_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_374_power__mono__iff,axiom,
! [A: real,B4: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B4 @ N ) )
= ( ord_less_eq_real @ A @ B4 ) ) ) ) ) ).
% power_mono_iff
thf(fact_375_power__mono__iff,axiom,
! [A: nat,B4: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B4 @ N ) )
= ( ord_less_eq_nat @ A @ B4 ) ) ) ) ) ).
% power_mono_iff
thf(fact_376_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_377_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_378_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B4: nat,W2: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B4 ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B4 @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_379_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B4: nat,W2: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B4 ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B4 @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_380_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B4: nat,W2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B4 ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B4 @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_381_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B4: nat,W2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B4 ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B4 @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_382_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_383_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_384_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_385_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_386_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_387_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_388_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B7: set_a] :
( ( A2
= ( insert_a @ A @ B7 ) )
& ~ ( member_a @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_389_mk__disjoint__insert,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ? [B7: set_real] :
( ( A2
= ( insert_real @ A @ B7 ) )
& ~ ( member_real @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_390_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_391_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_392_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_393_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_394_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_395_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_396_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_397_insert__eq__iff,axiom,
! [A: a,A2: set_a,B4: a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B4 @ B )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B4 @ B ) )
= ( ( ( A = B4 )
=> ( A2 = B ) )
& ( ( A != B4 )
=> ? [C3: set_a] :
( ( A2
= ( insert_a @ B4 @ C3 ) )
& ~ ( member_a @ B4 @ C3 )
& ( B
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_398_insert__eq__iff,axiom,
! [A: real,A2: set_real,B4: real,B: set_real] :
( ~ ( member_real @ A @ A2 )
=> ( ~ ( member_real @ B4 @ B )
=> ( ( ( insert_real @ A @ A2 )
= ( insert_real @ B4 @ B ) )
= ( ( ( A = B4 )
=> ( A2 = B ) )
& ( ( A != B4 )
=> ? [C3: set_real] :
( ( A2
= ( insert_real @ B4 @ C3 ) )
& ~ ( member_real @ B4 @ C3 )
& ( B
= ( insert_real @ A @ C3 ) )
& ~ ( member_real @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_399_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_400_insert__absorb,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ( ( insert_real @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_401_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_402_insert__ident,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_403_insert__ident,axiom,
! [X: real,A2: set_real,B: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ~ ( member_real @ X @ B )
=> ( ( ( insert_real @ X @ A2 )
= ( insert_real @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_404_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_405_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_406_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_407_insertI2,axiom,
! [A: a,B: set_a,B4: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ B4 @ B ) ) ) ).
% insertI2
thf(fact_408_insertI2,axiom,
! [A: real,B: set_real,B4: real] :
( ( member_real @ A @ B )
=> ( member_real @ A @ ( insert_real @ B4 @ B ) ) ) ).
% insertI2
thf(fact_409_insertI1,axiom,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).
% insertI1
thf(fact_410_insertI1,axiom,
! [A: real,B: set_real] : ( member_real @ A @ ( insert_real @ A @ B ) ) ).
% insertI1
thf(fact_411_insertE,axiom,
! [A: a,B4: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B4 @ A2 ) )
=> ( ( A != B4 )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_412_insertE,axiom,
! [A: real,B4: real,A2: set_real] :
( ( member_real @ A @ ( insert_real @ B4 @ A2 ) )
=> ( ( A != B4 )
=> ( member_real @ A @ A2 ) ) ) ).
% insertE
thf(fact_413_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_414_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_415_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_416_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_417_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_418_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_419_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_420_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_421_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_422_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_423_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_424_singletonD,axiom,
! [B4: real,A: real] :
( ( member_real @ B4 @ ( insert_real @ A @ bot_bot_set_real ) )
=> ( B4 = A ) ) ).
% singletonD
thf(fact_425_singletonD,axiom,
! [B4: a,A: a] :
( ( member_a @ B4 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B4 = A ) ) ).
% singletonD
thf(fact_426_singleton__iff,axiom,
! [B4: real,A: real] :
( ( member_real @ B4 @ ( insert_real @ A @ bot_bot_set_real ) )
= ( B4 = A ) ) ).
% singleton_iff
thf(fact_427_singleton__iff,axiom,
! [B4: a,A: a] :
( ( member_a @ B4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B4 = A ) ) ).
% singleton_iff
thf(fact_428_doubleton__eq__iff,axiom,
! [A: a,B4: a,C2: a,D2: a] :
( ( ( insert_a @ A @ ( insert_a @ B4 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C2 )
& ( B4 = D2 ) )
| ( ( A = D2 )
& ( B4 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_429_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_430_singleton__inject,axiom,
! [A: a,B4: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B4 @ bot_bot_set_a ) )
=> ( A = B4 ) ) ).
% singleton_inject
thf(fact_431_insert__mono,axiom,
! [C: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_432_subset__insert,axiom,
! [X: real,A2: set_real,B: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B ) )
= ( ord_less_eq_set_real @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_433_subset__insert,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_434_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).
% subset_insertI
thf(fact_435_subset__insertI2,axiom,
! [A2: set_a,B: set_a,B4: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ B ) ) ) ).
% subset_insertI2
thf(fact_436_Int__insert__left,axiom,
! [A: real,C: set_real,B: set_real] :
( ( ( member_real @ A @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A @ B ) @ C )
= ( insert_real @ A @ ( inf_inf_set_real @ B @ C ) ) ) )
& ( ~ ( member_real @ A @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A @ B ) @ C )
= ( inf_inf_set_real @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_437_Int__insert__left,axiom,
! [A: a,C: set_a,B: set_a] :
( ( ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) )
& ( ~ ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_438_Int__insert__right,axiom,
! [A: real,A2: set_real,B: set_real] :
( ( ( member_real @ A @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B ) )
= ( insert_real @ A @ ( inf_inf_set_real @ A2 @ B ) ) ) )
& ( ~ ( member_real @ A @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B ) )
= ( inf_inf_set_real @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_439_Int__insert__right,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_440_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_441_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_442_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_443_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_444_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_445_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_446_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_447_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_448_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_449_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_450_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_451_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_452_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_453_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_454_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_455_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_456_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_457_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_458_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_459_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_460_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_461_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_462_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_463_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_464_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_465_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_466_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_467_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N4: nat] :
( ( ord_less_nat @ M6 @ N4 )
| ( M6 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_468_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_469_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N4: nat] :
( ( ord_less_eq_nat @ M6 @ N4 )
& ( M6 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_470_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A7: set_a] :
( ? [A3: a] :
( A
= ( insert_a @ A3 @ A7 ) )
=> ~ ( finite_finite_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_471_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A6: set_a,B3: a] :
( ( A4
= ( insert_a @ B3 @ A6 ) )
& ( finite_finite_a @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_472_finite__induct,axiom,
! [F2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ~ ( member_real @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_473_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_474_finite__ne__induct,axiom,
! [F2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( F2 != bot_bot_set_real )
=> ( ! [X3: real] : ( P @ ( insert_real @ X3 @ bot_bot_set_real ) )
=> ( ! [X3: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( F3 != bot_bot_set_real )
=> ( ~ ( member_real @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_475_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_476_infinite__finite__induct,axiom,
! [P: set_real > $o,A2: set_real] :
( ! [A7: set_real] :
( ~ ( finite_finite_real @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ~ ( member_real @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_477_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A7: set_a] :
( ~ ( finite_finite_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_478_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_479_power__strict__mono,axiom,
! [A: real,B4: real,N: nat] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B4 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_480_power__strict__mono,axiom,
! [A: nat,B4: nat,N: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B4 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_481_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 )
=> ( ! [A3: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( member_real @ A3 @ A2 )
=> ( ~ ( member_real @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_482_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 )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A2 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_483_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 )
=> ( ! [A3: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( member_real @ A3 @ A2 )
=> ( ( ord_less_eq_set_real @ F3 @ A2 )
=> ( ~ ( member_real @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_484_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 )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_485_card__Suc__eq__finite,axiom,
! [A2: set_real,K: nat] :
( ( ( finite_card_real @ A2 )
= ( suc @ K ) )
= ( ? [B3: real,B6: set_real] :
( ( A2
= ( insert_real @ B3 @ B6 ) )
& ~ ( member_real @ B3 @ B6 )
& ( ( finite_card_real @ B6 )
= K )
& ( finite_finite_real @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_486_card__Suc__eq__finite,axiom,
! [A2: set_a,K: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K ) )
= ( ? [B3: a,B6: set_a] :
( ( A2
= ( insert_a @ B3 @ B6 ) )
& ~ ( member_a @ B3 @ B6 )
& ( ( finite_card_a @ B6 )
= K )
& ( finite_finite_a @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_487_card__insert__if,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( ( member_real @ X @ A2 )
=> ( ( finite_card_real @ ( insert_real @ X @ A2 ) )
= ( finite_card_real @ A2 ) ) )
& ( ~ ( member_real @ X @ A2 )
=> ( ( finite_card_real @ ( insert_real @ X @ A2 ) )
= ( suc @ ( finite_card_real @ A2 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_488_card__insert__if,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( finite_card_a @ A2 ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( suc @ ( finite_card_a @ A2 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_489_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_490_subset__singleton__iff,axiom,
! [X4: set_a,A: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_491_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_492_zero__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_493_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_494_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_495_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_496_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_497_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_498_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_499_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_500_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_501_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_502_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_503_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_504_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_505_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_506_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_507_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_508_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_509_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_510_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_511_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_512_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_513_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_514_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_515_card__1__singleton__iff,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: a] :
( A2
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_516_card__eq__SucD,axiom,
! [A2: set_real,K: nat] :
( ( ( finite_card_real @ A2 )
= ( suc @ K ) )
=> ? [B2: real,B7: set_real] :
( ( A2
= ( insert_real @ B2 @ B7 ) )
& ~ ( member_real @ B2 @ B7 )
& ( ( finite_card_real @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_real ) ) ) ) ).
% card_eq_SucD
thf(fact_517_card__eq__SucD,axiom,
! [A2: set_a,K: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K ) )
=> ? [B2: a,B7: set_a] :
( ( A2
= ( insert_a @ B2 @ B7 ) )
& ~ ( member_a @ B2 @ B7 )
& ( ( finite_card_a @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_518_card__Suc__eq,axiom,
! [A2: set_real,K: nat] :
( ( ( finite_card_real @ A2 )
= ( suc @ K ) )
= ( ? [B3: real,B6: set_real] :
( ( A2
= ( insert_real @ B3 @ B6 ) )
& ~ ( member_real @ B3 @ B6 )
& ( ( finite_card_real @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_real ) ) ) ) ) ).
% card_Suc_eq
thf(fact_519_card__Suc__eq,axiom,
! [A2: set_a,K: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K ) )
= ( ? [B3: a,B6: set_a] :
( ( A2
= ( insert_a @ B3 @ B6 ) )
& ~ ( member_a @ B3 @ B6 )
& ( ( finite_card_a @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_520_card__le__Suc__iff,axiom,
! [N: nat,A2: set_real] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_real @ A2 ) )
= ( ? [A4: real,B6: set_real] :
( ( A2
= ( insert_real @ A4 @ B6 ) )
& ~ ( member_real @ A4 @ B6 )
& ( ord_less_eq_nat @ N @ ( finite_card_real @ B6 ) )
& ( finite_finite_real @ B6 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_521_card__le__Suc__iff,axiom,
! [N: nat,A2: set_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_a @ A2 ) )
= ( ? [A4: a,B6: set_a] :
( ( A2
= ( insert_a @ A4 @ B6 ) )
& ~ ( member_a @ A4 @ B6 )
& ( ord_less_eq_nat @ N @ ( finite_card_a @ B6 ) )
& ( finite_finite_a @ B6 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_522_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_523_power__less__imp__less__base,axiom,
! [A: real,N: nat,B4: real] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B4 @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ord_less_real @ A @ B4 ) ) ) ).
% power_less_imp_less_base
thf(fact_524_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B4: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B4 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ord_less_nat @ A @ B4 ) ) ) ).
% power_less_imp_less_base
thf(fact_525_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_526_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_527_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_528_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_529_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_530_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_531_additive__abelian__group_Osumset__subset__insert_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ B ) ) ) ) ).
% additive_abelian_group.sumset_subset_insert(1)
thf(fact_532_additive__abelian__group_Osumset__subset__insert_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A2 ) @ B ) ) ) ).
% additive_abelian_group.sumset_subset_insert(2)
thf(fact_533_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_534_additive__abelian__group_Ominusset__triv,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( insert_a @ Zero @ bot_bot_set_a ) )
= ( insert_a @ Zero @ bot_bot_set_a ) ) ) ).
% additive_abelian_group.minusset_triv
thf(fact_535_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_a,K: set_a,A: set_a,B4: set_a] :
( ( A2
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A2 @ B4 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_536_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B4: set_a,A: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B4 ) )
=> ( ( inf_inf_set_a @ A @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_537_power__eq__imp__eq__base,axiom,
! [A: real,N: nat,B4: real] :
( ( ( power_power_real @ A @ N )
= ( power_power_real @ B4 @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B4 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_538_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B4: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B4 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B4 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_539_power__eq__iff__eq__base,axiom,
! [N: nat,A: real,B4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ( ( power_power_real @ A @ N )
= ( power_power_real @ B4 @ N ) )
= ( A = B4 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_540_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B4 @ N ) )
= ( A = B4 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_541_additive__abelian__group_Osumset__D_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ Zero @ bot_bot_set_a ) @ A2 )
= ( inf_inf_set_a @ A2 @ G ) ) ) ).
% additive_abelian_group.sumset_D(2)
thf(fact_542_additive__abelian__group_Osumset__D_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ Zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A2 @ G ) ) ) ).
% additive_abelian_group.sumset_D(1)
thf(fact_543_additive__abelian__group_Osumset__iterated__0,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ zero_zero_nat )
= ( insert_a @ Zero @ bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_iterated_0
thf(fact_544_additive__abelian__group_Osumset__iterated__empty,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,R2: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_nat @ zero_zero_nat @ R2 )
=> ( ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ bot_bot_set_a @ R2 )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_iterated_empty
thf(fact_545_additive__abelian__group_Ocard__sumset__le,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ).
% additive_abelian_group.card_sumset_le
thf(fact_546_finite__has__maximal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ord_less_eq_real @ A @ X3 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_547_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_548_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_549_finite__has__minimal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ord_less_eq_real @ X3 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_550_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_551_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_552_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_553_infinite__imp__nonempty,axiom,
! [S2: set_a] :
( ~ ( finite_finite_a @ S2 )
=> ( S2 != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_554_finite__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_555_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_556_rev__finite__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_557_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
! [G: set_real,Addition: real > real > real,Zero: real,A2: set_real,A: real] :
( ( pluenn1014277435162747966p_real @ G @ Addition @ Zero )
=> ( ( finite_finite_real @ A2 )
=> ( ( ( member_real @ A @ G )
=> ( ( finite_card_real @ ( pluenn7361685508355272389t_real @ G @ Addition @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( finite_card_real @ ( inf_inf_set_real @ A2 @ G ) ) ) )
& ( ~ ( member_real @ A @ G )
=> ( ( finite_card_real @ ( pluenn7361685508355272389t_real @ G @ Addition @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= zero_zero_nat ) ) ) ) ) ).
% additive_abelian_group.card_sumset_singleton_eq
thf(fact_558_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ A @ G )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ G ) ) ) )
& ( ~ ( member_a @ A @ G )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= zero_zero_nat ) ) ) ) ) ).
% additive_abelian_group.card_sumset_singleton_eq
thf(fact_559_finite__has__maximal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_560_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_561_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_562_finite__has__minimal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_563_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_564_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_565_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_566_card__subset__eq,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_567_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_568_card__mono,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ).
% card_mono
thf(fact_569_card__seteq,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_570_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_571_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_572_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_573__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 )
=> ! [K3: real] :
( ( ord_less_real @ zero_zero_real @ K3 )
=> ( ( ord_less_eq_real @ K3 @ k )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ g )
=> ( ( finite_finite_a @ C4 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A8 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ C4 ) ) ) ) @ ( times_times_real @ K3 @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A8 @ C4 ) ) ) ) ) ) ) ) ) ) ) ).
% \<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_574_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_575_Plu__2__2,axiom,
! [A0: set_a,B: set_a,K0: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A0 @ B ) ) ) @ ( 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 @ B )
=> ( ( ord_less_eq_set_a @ B @ g )
=> ( ( B != bot_bot_set_a )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A0 )
=> ( ( A7 != bot_bot_set_a )
=> ! [K4: real] :
( ( ord_less_real @ zero_zero_real @ K4 )
=> ( ( ord_less_eq_real @ K4 @ K0 )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ g )
=> ( ( finite_finite_a @ C4 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A7 @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ C4 ) ) ) ) @ ( times_times_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A7 @ C4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Plu_2_2
thf(fact_576_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_577_sumsetdiff__sing,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( minus_minus_set_a @ A2 @ B ) @ ( 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 @ B @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% sumsetdiff_sing
thf(fact_578_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 )
=> ( ! [M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X ) @ C2 ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_579_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_580_psubsetI,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_a @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_581_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_582_inf_Oright__idem,axiom,
! [A: set_a,B4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B4 ) @ B4 )
= ( inf_inf_set_a @ A @ B4 ) ) ).
% inf.right_idem
thf(fact_583_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_584_inf_Oleft__idem,axiom,
! [A: set_a,B4: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B4 ) )
= ( inf_inf_set_a @ A @ B4 ) ) ).
% inf.left_idem
thf(fact_585_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_586_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_587_DiffI,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ A2 )
=> ( ~ ( member_real @ C2 @ B )
=> ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_588_DiffI,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_589_Diff__iff,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B ) )
= ( ( member_real @ C2 @ A2 )
& ~ ( member_real @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_590_Diff__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
& ~ ( member_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_591_Diff__idemp,axiom,
! [A2: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_592_inf_Obounded__iff,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B4 @ C2 ) )
= ( ( ord_less_eq_real @ A @ B4 )
& ( ord_less_eq_real @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_593_inf_Obounded__iff,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) )
= ( ( ord_less_eq_set_a @ A @ B4 )
& ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_594_inf_Obounded__iff,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B4 )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_595_le__inf__iff,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z3 ) )
= ( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_596_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z3 ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_597_le__inf__iff,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_598_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_599_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_600_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_601_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_602_finite__Diff2,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_603_finite__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_604_Diff__insert0,axiom,
! [X: real,A2: set_real,B: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ( minus_minus_set_real @ A2 @ ( insert_real @ X @ B ) )
= ( minus_minus_set_real @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_605_Diff__insert0,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_606_insert__Diff1,axiom,
! [X: real,B: set_real,A2: set_real] :
( ( member_real @ X @ B )
=> ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B )
= ( minus_minus_set_real @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_607_insert__Diff1,axiom,
! [X: a,B: set_a,A2: set_a] :
( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_608_Diff__eq__empty__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_609_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_610_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_611_Diff__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_612_psubset__imp__ex__mem,axiom,
! [A2: set_real,B: set_real] :
( ( ord_less_set_real @ A2 @ B )
=> ? [B2: real] : ( member_real @ B2 @ ( minus_minus_set_real @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_613_psubset__imp__ex__mem,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ? [B2: a] : ( member_a @ B2 @ ( minus_minus_set_a @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_614_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_615_DiffE,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B ) )
=> ~ ( ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ B ) ) ) ).
% DiffE
thf(fact_616_DiffE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_617_DiffD1,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B ) )
=> ( member_real @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_618_DiffD1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_619_DiffD2,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B ) )
=> ~ ( member_real @ C2 @ B ) ) ).
% DiffD2
thf(fact_620_DiffD2,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( member_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_621_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_622_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_623_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_624_double__diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_625_Diff__subset,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_626_Diff__mono,axiom,
! [A2: set_a,C: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_627_insert__Diff__if,axiom,
! [X: real,B: set_real,A2: set_real] :
( ( ( member_real @ X @ B )
=> ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B )
= ( minus_minus_set_real @ A2 @ B ) ) )
& ( ~ ( member_real @ X @ B )
=> ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B )
= ( insert_real @ X @ ( minus_minus_set_real @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_628_insert__Diff__if,axiom,
! [X: a,B: set_a,A2: set_a] :
( ( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) )
& ( ~ ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_629_not__psubset__empty,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_630_finite__psubset__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ! [B8: set_a] :
( ( ord_less_set_a @ B8 @ A7 )
=> ( P @ B8 ) )
=> ( P @ A7 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_631_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_set_a @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_632_subset__psubset__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_633_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ~ ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_634_psubset__subset__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_635_psubset__imp__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% psubset_imp_subset
thf(fact_636_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_637_psubsetE,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% psubsetE
thf(fact_638_Diff__Int__distrib2,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_639_Diff__Int__distrib,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C @ A2 ) @ ( inf_inf_set_a @ C @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_640_Diff__Diff__Int,axiom,
! [A2: set_a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_641_Diff__Int2,axiom,
! [A2: set_a,C: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B @ C ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ B ) ) ).
% Diff_Int2
thf(fact_642_Int__Diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Int_Diff
thf(fact_643_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y4: real] :
? [N2: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_644_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 )
=> ? [X5: a] :
( ( member_a @ X5 @ ( minus_minus_set_a @ S2 @ T4 ) )
& ( P @ ( insert_a @ X5 @ T4 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_645_psubset__insert__iff,axiom,
! [A2: set_real,X: real,B: set_real] :
( ( ord_less_set_real @ A2 @ ( insert_real @ X @ B ) )
= ( ( ( member_real @ X @ B )
=> ( ord_less_set_real @ A2 @ B ) )
& ( ~ ( member_real @ X @ B )
=> ( ( ( member_real @ X @ A2 )
=> ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B ) )
& ( ~ ( member_real @ X @ A2 )
=> ( ord_less_eq_set_real @ A2 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_646_psubset__insert__iff,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ B )
=> ( ord_less_set_a @ A2 @ B ) )
& ( ~ ( member_a @ X @ B )
=> ( ( ( member_a @ X @ A2 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_647_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_648_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_649_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_650_Diff__insert2,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_651_insert__Diff,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_652_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_653_Diff__insert,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_654_subset__Diff__insert,axiom,
! [A2: set_real,B: set_real,X: real,C: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B @ ( insert_real @ X @ C ) ) )
= ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B @ C ) )
& ~ ( member_real @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_655_subset__Diff__insert,axiom,
! [A2: set_a,B: set_a,X: a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ ( insert_a @ X @ C ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ C ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_656_Diff__triv,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_657_Int__Diff__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_658_finite__empty__induct,axiom,
! [A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: real,A7: set_real] :
( ( finite_finite_real @ A7 )
=> ( ( member_real @ A3 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ A3 @ bot_bot_set_real ) ) ) ) ) )
=> ( P @ bot_bot_set_real ) ) ) ) ).
% finite_empty_induct
thf(fact_659_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: a,A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( member_a @ A3 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_660_infinite__coinduct,axiom,
! [X4: set_a > $o,A2: set_a] :
( ( X4 @ A2 )
=> ( ! [A7: set_a] :
( ( X4 @ A7 )
=> ? [X5: a] :
( ( member_a @ X5 @ A7 )
& ( ( X4 @ ( minus_minus_set_a @ A7 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A7 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_661_infinite__remove,axiom,
! [S2: set_a,A: a] :
( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_662_Diff__single__insert,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_663_subset__insert__iff,axiom,
! [A2: set_real,X: real,B: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B ) )
= ( ( ( member_real @ X @ A2 )
=> ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B ) )
& ( ~ ( member_real @ X @ A2 )
=> ( ord_less_eq_set_real @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_664_subset__insert__iff,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_665_card__less__sym__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_666_card__le__sym__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_667_psubset__card__mono,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_set_a @ A2 @ B )
=> ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_668_complete__real,axiom,
! [S2: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S2 )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y3: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S2 )
=> ( ord_less_eq_real @ X5 @ Y3 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_669_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z3 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z3 ) ) ) ).
% inf_left_commute
thf(fact_670_inf_Oleft__commute,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( inf_inf_set_a @ B4 @ ( inf_inf_set_a @ A @ C2 ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_671_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y5: set_a] : ( inf_inf_set_a @ Y5 @ X2 ) ) ) ).
% inf_commute
thf(fact_672_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_673_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z3 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z3 ) ) ) ).
% inf_assoc
thf(fact_674_inf_Oassoc,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% inf.assoc
thf(fact_675_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y5: set_a] : ( inf_inf_set_a @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_676_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z3 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z3 ) ) ) ).
% inf_sup_aci(2)
thf(fact_677_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z3 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z3 ) ) ) ).
% inf_sup_aci(3)
thf(fact_678_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_679_finite__remove__induct,axiom,
! [B: set_real,P: set_real > $o] :
( ( finite_finite_real @ B )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A7: set_real] :
( ( finite_finite_real @ A7 )
=> ( ( A7 != bot_bot_set_real )
=> ( ( ord_less_eq_set_real @ A7 @ B )
=> ( ! [X5: real] :
( ( member_real @ X5 @ A7 )
=> ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_680_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( A7 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A7 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_681_remove__induct,axiom,
! [P: set_real > $o,B: set_real] :
( ( P @ bot_bot_set_real )
=> ( ( ~ ( finite_finite_real @ B )
=> ( P @ B ) )
=> ( ! [A7: set_real] :
( ( finite_finite_real @ A7 )
=> ( ( A7 != bot_bot_set_real )
=> ( ( ord_less_eq_set_real @ A7 @ B )
=> ( ! [X5: real] :
( ( member_real @ X5 @ A7 )
=> ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_682_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( A7 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A7 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_683_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_684_card__psubset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) )
=> ( ord_less_set_a @ A2 @ B ) ) ) ) ).
% card_psubset
thf(fact_685_additive__abelian__group_Osumsetdiff__sing,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( minus_minus_set_a @ A2 @ B ) @ ( 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 @ B @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% additive_abelian_group.sumsetdiff_sing
thf(fact_686_card_Oremove,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ X @ A2 )
=> ( ( finite_card_real @ A2 )
= ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).
% card.remove
thf(fact_687_card_Oremove,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ A2 )
= ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ) ).
% card.remove
thf(fact_688_card_Oinsert__remove,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_689_card__Suc__Diff1,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ X @ A2 )
=> ( ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) )
= ( finite_card_real @ A2 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_690_card__Suc__Diff1,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_691_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_692_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_693_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_694_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_695_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_696_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_697_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_698_inf_OcoboundedI2,axiom,
! [B4: real,C2: real,A: real] :
( ( ord_less_eq_real @ B4 @ C2 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_699_inf_OcoboundedI2,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_700_inf_OcoboundedI2,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_701_inf_OcoboundedI1,axiom,
! [A: real,C2: real,B4: real] :
( ( ord_less_eq_real @ A @ C2 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_702_inf_OcoboundedI1,axiom,
! [A: set_a,C2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_703_inf_OcoboundedI1,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_704_inf_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( inf_inf_real @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_705_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_706_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_707_inf_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( inf_inf_real @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_708_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_709_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_710_inf_Ocobounded2,axiom,
! [A: real,B4: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_711_inf_Ocobounded2,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_712_inf_Ocobounded2,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_713_inf_Ocobounded1,axiom,
! [A: real,B4: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_714_inf_Ocobounded1,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_715_inf_Ocobounded1,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_716_inf_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( A4
= ( inf_inf_real @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_717_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_718_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( A4
= ( inf_inf_nat @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_719_inf__greatest,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Z3 )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_720_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z3 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_721_inf__greatest,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z3 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_722_inf_OboundedI,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ A @ C2 )
=> ( ord_less_eq_real @ A @ ( inf_inf_real @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_723_inf_OboundedI,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_724_inf_OboundedI,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_725_inf_OboundedE,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_real @ A @ B4 )
=> ~ ( ord_less_eq_real @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_726_inf_OboundedE,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B4 )
=> ~ ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_727_inf_OboundedE,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A @ B4 )
=> ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_728_inf__absorb2,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( inf_inf_real @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_729_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_730_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_731_inf__absorb1,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( inf_inf_real @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_732_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_733_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_734_inf_Oabsorb2,axiom,
! [B4: real,A: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( inf_inf_real @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_735_inf_Oabsorb2,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( inf_inf_set_a @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_736_inf_Oabsorb2,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( inf_inf_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_737_inf_Oabsorb1,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( inf_inf_real @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_738_inf_Oabsorb1,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( inf_inf_set_a @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_739_inf_Oabsorb1,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( inf_inf_nat @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_740_le__iff__inf,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( inf_inf_real @ X2 @ Y5 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_741_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y5: set_a] :
( ( inf_inf_set_a @ X2 @ Y5 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_742_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( inf_inf_nat @ X2 @ Y5 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_743_inf__unique,axiom,
! [F: real > real > real,X: real,Y: real] :
( ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: real,Y3: real,Z: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ( ord_less_eq_real @ X3 @ Z )
=> ( ord_less_eq_real @ X3 @ ( F @ Y3 @ Z ) ) ) )
=> ( ( inf_inf_real @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_744_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ X3 @ Z )
=> ( ord_less_eq_set_a @ X3 @ ( F @ Y3 @ Z ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_745_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ X3 @ Z )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y3 @ Z ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_746_inf_OorderI,axiom,
! [A: real,B4: real] :
( ( A
= ( inf_inf_real @ A @ B4 ) )
=> ( ord_less_eq_real @ A @ B4 ) ) ).
% inf.orderI
thf(fact_747_inf_OorderI,axiom,
! [A: set_a,B4: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B4 ) )
=> ( ord_less_eq_set_a @ A @ B4 ) ) ).
% inf.orderI
thf(fact_748_inf_OorderI,axiom,
! [A: nat,B4: nat] :
( ( A
= ( inf_inf_nat @ A @ B4 ) )
=> ( ord_less_eq_nat @ A @ B4 ) ) ).
% inf.orderI
thf(fact_749_inf_OorderE,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( A
= ( inf_inf_real @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_750_inf_OorderE,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( A
= ( inf_inf_set_a @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_751_inf_OorderE,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( A
= ( inf_inf_nat @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_752_le__infI2,axiom,
! [B4: real,X: real,A: real] :
( ( ord_less_eq_real @ B4 @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_753_le__infI2,axiom,
! [B4: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_754_le__infI2,axiom,
! [B4: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_755_le__infI1,axiom,
! [A: real,X: real,B4: real] :
( ( ord_less_eq_real @ A @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_756_le__infI1,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_757_le__infI1,axiom,
! [A: nat,X: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_758_inf__mono,axiom,
! [A: real,C2: real,B4: real,D2: real] :
( ( ord_less_eq_real @ A @ C2 )
=> ( ( ord_less_eq_real @ B4 @ D2 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ ( inf_inf_real @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_759_inf__mono,axiom,
! [A: set_a,C2: set_a,B4: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_760_inf__mono,axiom,
! [A: nat,C2: nat,B4: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ ( inf_inf_nat @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_761_le__infI,axiom,
! [X: real,A: real,B4: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ X @ B4 )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_762_le__infI,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B4 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_763_le__infI,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B4 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_764_le__infE,axiom,
! [X: real,A: real,B4: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ A @ B4 ) )
=> ~ ( ( ord_less_eq_real @ X @ A )
=> ~ ( ord_less_eq_real @ X @ B4 ) ) ) ).
% le_infE
thf(fact_765_le__infE,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B4 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B4 ) ) ) ).
% le_infE
thf(fact_766_le__infE,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B4 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B4 ) ) ) ).
% le_infE
thf(fact_767_inf__le2,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_768_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_769_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_770_inf__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_771_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_772_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_773_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_774_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_775_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_776_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_777_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_778_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_779_inf_Ostrict__coboundedI2,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_780_inf_Ostrict__coboundedI2,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_nat @ B4 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_781_inf_Ostrict__coboundedI2,axiom,
! [B4: real,C2: real,A: real] :
( ( ord_less_real @ B4 @ C2 )
=> ( ord_less_real @ ( inf_inf_real @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_782_inf_Ostrict__coboundedI1,axiom,
! [A: set_a,C2: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ C2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_783_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_nat @ A @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_784_inf_Ostrict__coboundedI1,axiom,
! [A: real,C2: real,B4: real] :
( ( ord_less_real @ A @ C2 )
=> ( ord_less_real @ ( inf_inf_real @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_785_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( A4
= ( inf_inf_set_a @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_786_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( A4
= ( inf_inf_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_787_inf_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( A4
= ( inf_inf_real @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_788_inf_Ostrict__boundedE,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) )
=> ~ ( ( ord_less_set_a @ A @ B4 )
=> ~ ( ord_less_set_a @ A @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_789_inf_Ostrict__boundedE,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) )
=> ~ ( ( ord_less_nat @ A @ B4 )
=> ~ ( ord_less_nat @ A @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_790_inf_Ostrict__boundedE,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_real @ A @ ( inf_inf_real @ B4 @ C2 ) )
=> ~ ( ( ord_less_real @ A @ B4 )
=> ~ ( ord_less_real @ A @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_791_inf_Oabsorb4,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ A )
=> ( ( inf_inf_set_a @ A @ B4 )
= B4 ) ) ).
% inf.absorb4
thf(fact_792_inf_Oabsorb4,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ( inf_inf_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb4
thf(fact_793_inf_Oabsorb4,axiom,
! [B4: real,A: real] :
( ( ord_less_real @ B4 @ A )
=> ( ( inf_inf_real @ A @ B4 )
= B4 ) ) ).
% inf.absorb4
thf(fact_794_inf_Oabsorb3,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ B4 )
=> ( ( inf_inf_set_a @ A @ B4 )
= A ) ) ).
% inf.absorb3
thf(fact_795_inf_Oabsorb3,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( inf_inf_nat @ A @ B4 )
= A ) ) ).
% inf.absorb3
thf(fact_796_inf_Oabsorb3,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( inf_inf_real @ A @ B4 )
= A ) ) ).
% inf.absorb3
thf(fact_797_less__infI2,axiom,
! [B4: set_a,X: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% less_infI2
thf(fact_798_less__infI2,axiom,
! [B4: nat,X: nat,A: nat] :
( ( ord_less_nat @ B4 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% less_infI2
thf(fact_799_less__infI2,axiom,
! [B4: real,X: real,A: real] :
( ( ord_less_real @ B4 @ X )
=> ( ord_less_real @ ( inf_inf_real @ A @ B4 ) @ X ) ) ).
% less_infI2
thf(fact_800_less__infI1,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% less_infI1
thf(fact_801_less__infI1,axiom,
! [A: nat,X: nat,B4: nat] :
( ( ord_less_nat @ A @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% less_infI1
thf(fact_802_less__infI1,axiom,
! [A: real,X: real,B4: real] :
( ( ord_less_real @ A @ X )
=> ( ord_less_real @ ( inf_inf_real @ A @ B4 ) @ X ) ) ).
% less_infI1
thf(fact_803_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_804_additive__abelian__group_OPlu__2__2,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A0: set_a,B: set_a,K0: real] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A0 @ B ) ) ) @ ( 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 @ B )
=> ( ( ord_less_eq_set_a @ B @ G )
=> ( ( B != bot_bot_set_a )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A0 )
=> ( ( A7 != bot_bot_set_a )
=> ! [K4: real] :
( ( ord_less_real @ zero_zero_real @ K4 )
=> ( ( ord_less_eq_real @ K4 @ K0 )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ G )
=> ( ( finite_finite_a @ C4 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A7 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ C4 ) ) ) ) @ ( times_times_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A7 @ C4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.Plu_2_2
thf(fact_805_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_806_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_807_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_808_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_809_diff__gt__0__iff__gt,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B4 ) )
= ( ord_less_real @ B4 @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_810_diff__ge__0__iff__ge,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B4 ) )
= ( ord_less_eq_real @ B4 @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_811_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_812_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_813_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_814_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_815_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_816_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_817_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_818_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_819_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_820_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_821_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_822_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_823_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_824_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_825_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_826_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_827_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_828_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_829_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_830_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_831_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_832_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_833_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_834_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_835_psubsetD,axiom,
! [A2: set_a,B: set_a,C2: a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_836_psubsetD,axiom,
! [A2: set_real,B: set_real,C2: real] :
( ( ord_less_set_real @ A2 @ B )
=> ( ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_837_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_838_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_839_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_840_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_841_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_842_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_843_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_844_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_845_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_846_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_847_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_848_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B4 ) )
= ( ord_less_eq_nat @ B4 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_849_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_850_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_851_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_852_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_853_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_854_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_855_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_856_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_857_diff__less__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B4 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_858_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_859_nle__le,axiom,
! [A: real,B4: real] :
( ( ~ ( ord_less_eq_real @ A @ B4 ) )
= ( ( ord_less_eq_real @ B4 @ A )
& ( B4 != A ) ) ) ).
% nle_le
thf(fact_860_nle__le,axiom,
! [A: nat,B4: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B4 ) )
= ( ( ord_less_eq_nat @ B4 @ A )
& ( B4 != A ) ) ) ).
% nle_le
thf(fact_861_le__cases3,axiom,
! [X: real,Y: real,Z3: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_862_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_863_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z2: real] : ( Y6 = Z2 ) )
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ( ord_less_eq_real @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_864_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_a,Z2: set_a] : ( Y6 = Z2 ) )
= ( ^ [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
& ( ord_less_eq_set_a @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_865_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_866_ord__eq__le__trans,axiom,
! [A: real,B4: real,C2: real] :
( ( A = B4 )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_867_ord__eq__le__trans,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( A = B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_868_ord__eq__le__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( A = B4 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_869_ord__le__eq__trans,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_870_ord__le__eq__trans,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_871_ord__le__eq__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_872_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_873_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_874_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_875_order_Otrans,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% order.trans
thf(fact_876_order_Otrans,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% order.trans
thf(fact_877_order_Otrans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_878_order__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_eq_real @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_879_order__trans,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z3 )
=> ( ord_less_eq_set_a @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_880_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_881_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B4: real] :
( ! [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: real,B2: real] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B4 ) ) ) ).
% linorder_wlog
thf(fact_882_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B4: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B4 ) ) ) ).
% linorder_wlog
thf(fact_883_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: real,Z2: real] : ( Y6 = Z2 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_884_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_a,Z2: set_a] : ( Y6 = Z2 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_885_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_886_dual__order_Oantisym,axiom,
! [B4: real,A: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( ord_less_eq_real @ A @ B4 )
=> ( A = B4 ) ) ) ).
% dual_order.antisym
thf(fact_887_dual__order_Oantisym,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ A @ B4 )
=> ( A = B4 ) ) ) ).
% dual_order.antisym
thf(fact_888_dual__order_Oantisym,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ A @ B4 )
=> ( A = B4 ) ) ) ).
% dual_order.antisym
thf(fact_889_dual__order_Otrans,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( ord_less_eq_real @ C2 @ B4 )
=> ( ord_less_eq_real @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_890_dual__order_Otrans,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B4 )
=> ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_891_dual__order_Otrans,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ C2 @ B4 )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_892_antisym,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ B4 @ A )
=> ( A = B4 ) ) ) ).
% antisym
thf(fact_893_antisym,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A )
=> ( A = B4 ) ) ) ).
% antisym
thf(fact_894_antisym,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ A )
=> ( A = B4 ) ) ) ).
% antisym
thf(fact_895_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z2: real] : ( Y6 = Z2 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_896_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_a,Z2: set_a] : ( Y6 = Z2 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_897_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_898_order__subst1,axiom,
! [A: real,F: real > real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_899_order__subst1,axiom,
! [A: real,F: set_a > real,B4: set_a,C2: set_a] :
( ( ord_less_eq_real @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_900_order__subst1,axiom,
! [A: real,F: nat > real,B4: nat,C2: nat] :
( ( ord_less_eq_real @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_901_order__subst1,axiom,
! [A: set_a,F: real > set_a,B4: real,C2: real] :
( ( ord_less_eq_set_a @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_902_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_903_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B4: nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_904_order__subst1,axiom,
! [A: nat,F: real > nat,B4: real,C2: real] :
( ( ord_less_eq_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_905_order__subst1,axiom,
! [A: nat,F: set_a > nat,B4: set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_906_order__subst1,axiom,
! [A: nat,F: nat > nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_907_order__subst2,axiom,
! [A: real,B4: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ ( F @ B4 ) @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_908_order__subst2,axiom,
! [A: real,B4: real,F: real > set_a,C2: set_a] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_set_a @ ( F @ B4 ) @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_909_order__subst2,axiom,
! [A: real,B4: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_910_order__subst2,axiom,
! [A: set_a,B4: set_a,F: set_a > real,C2: real] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_real @ ( F @ B4 ) @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_911_order__subst2,axiom,
! [A: set_a,B4: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ ( F @ B4 ) @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_912_order__subst2,axiom,
! [A: set_a,B4: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_913_order__subst2,axiom,
! [A: nat,B4: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_real @ ( F @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_914_order__subst2,axiom,
! [A: nat,B4: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_set_a @ ( F @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_915_order__subst2,axiom,
! [A: nat,B4: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_916_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_917_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_918_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_919_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_920_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_921_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B4: real,C2: real] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_922_ord__eq__le__subst,axiom,
! [A: set_a,F: real > set_a,B4: real,C2: real] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_923_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B4: real,C2: real] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_924_ord__eq__le__subst,axiom,
! [A: real,F: set_a > real,B4: set_a,C2: set_a] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_925_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B4: set_a,C2: set_a] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_926_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B4: set_a,C2: set_a] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_927_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B4: nat,C2: nat] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_928_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B4: nat,C2: nat] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_929_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B4: nat,C2: nat] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_930_ord__le__eq__subst,axiom,
! [A: real,B4: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_931_ord__le__eq__subst,axiom,
! [A: real,B4: real,F: real > set_a,C2: set_a] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_932_ord__le__eq__subst,axiom,
! [A: real,B4: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_933_ord__le__eq__subst,axiom,
! [A: set_a,B4: set_a,F: set_a > real,C2: real] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_934_ord__le__eq__subst,axiom,
! [A: set_a,B4: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_935_ord__le__eq__subst,axiom,
! [A: set_a,B4: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_936_ord__le__eq__subst,axiom,
! [A: nat,B4: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_937_ord__le__eq__subst,axiom,
! [A: nat,B4: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_938_ord__le__eq__subst,axiom,
! [A: nat,B4: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_939_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_940_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_941_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_942_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_943_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_944_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_945_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_946_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_947_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_948_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_949_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z: real] :
( ( ord_less_real @ X @ Z )
& ( ord_less_real @ Z @ Y ) ) ) ).
% dense
thf(fact_950_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_951_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_952_order_Oasym,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ~ ( ord_less_nat @ B4 @ A ) ) ).
% order.asym
thf(fact_953_order_Oasym,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ A @ B4 )
=> ~ ( ord_less_real @ B4 @ A ) ) ).
% order.asym
thf(fact_954_ord__eq__less__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( A = B4 )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_955_ord__eq__less__trans,axiom,
! [A: real,B4: real,C2: real] :
( ( A = B4 )
=> ( ( ord_less_real @ B4 @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_956_ord__less__eq__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_957_ord__less__eq__trans,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_958_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_959_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_960_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_961_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_962_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_963_dual__order_Oasym,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ~ ( ord_less_nat @ A @ B4 ) ) ).
% dual_order.asym
thf(fact_964_dual__order_Oasym,axiom,
! [B4: real,A: real] :
( ( ord_less_real @ B4 @ A )
=> ~ ( ord_less_real @ A @ B4 ) ) ).
% dual_order.asym
thf(fact_965_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_966_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_967_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N4: nat] :
( ( P3 @ N4 )
& ! [M6: nat] :
( ( ord_less_nat @ M6 @ N4 )
=> ~ ( P3 @ M6 ) ) ) ) ) ).
% exists_least_iff
thf(fact_968_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B4: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B4 ) ) ) ) ).
% linorder_less_wlog
thf(fact_969_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B4: real] :
( ! [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: real] : ( P @ A3 @ A3 )
=> ( ! [A3: real,B2: real] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B4 ) ) ) ) ).
% linorder_less_wlog
thf(fact_970_order_Ostrict__trans,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_971_order_Ostrict__trans,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_real @ B4 @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_972_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_973_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_974_dual__order_Ostrict__trans,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ( ord_less_nat @ C2 @ B4 )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_975_dual__order_Ostrict__trans,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_real @ B4 @ A )
=> ( ( ord_less_real @ C2 @ B4 )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_976_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( A != B4 ) ) ).
% order.strict_implies_not_eq
thf(fact_977_order_Ostrict__implies__not__eq,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ A @ B4 )
=> ( A != B4 ) ) ).
% order.strict_implies_not_eq
thf(fact_978_dual__order_Ostrict__implies__not__eq,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( A != B4 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_979_dual__order_Ostrict__implies__not__eq,axiom,
! [B4: real,A: real] :
( ( ord_less_real @ B4 @ A )
=> ( A != B4 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_980_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_981_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_982_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_983_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_984_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_985_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_986_order__less__asym_H,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ~ ( ord_less_nat @ B4 @ A ) ) ).
% order_less_asym'
thf(fact_987_order__less__asym_H,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ A @ B4 )
=> ~ ( ord_less_real @ B4 @ A ) ) ).
% order_less_asym'
thf(fact_988_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_989_order__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_990_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B4: nat,C2: nat] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_991_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B4: nat,C2: nat] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_992_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B4: real,C2: real] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_993_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B4: real,C2: real] :
( ( A
= ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_994_ord__less__eq__subst,axiom,
! [A: nat,B4: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_995_ord__less__eq__subst,axiom,
! [A: nat,B4: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_996_ord__less__eq__subst,axiom,
! [A: real,B4: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_997_ord__less__eq__subst,axiom,
! [A: real,B4: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_998_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_999_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_1000_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_1001_order__less__subst1,axiom,
! [A: nat,F: real > nat,B4: real,C2: real] :
( ( ord_less_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_1002_order__less__subst1,axiom,
! [A: real,F: nat > real,B4: nat,C2: nat] :
( ( ord_less_real @ A @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_1003_order__less__subst1,axiom,
! [A: real,F: real > real,B4: real,C2: real] :
( ( ord_less_real @ A @ ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_1004_order__less__subst2,axiom,
! [A: nat,B4: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_1005_order__less__subst2,axiom,
! [A: nat,B4: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_real @ ( F @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_1006_order__less__subst2,axiom,
! [A: real,B4: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_1007_order__less__subst2,axiom,
! [A: real,B4: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_real @ ( F @ B4 ) @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_1008_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_1009_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_1010_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_1011_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_1012_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_1013_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_1014_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_1015_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_1016_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_1017_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_1018_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_1019_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_1020_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B4: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B4 ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B4 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1021_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B4 ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B4 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1022_mult_Oassoc,axiom,
! [A: real,B4: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B4 ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B4 @ C2 ) ) ) ).
% mult.assoc
thf(fact_1023_mult_Oassoc,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B4 ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B4 @ C2 ) ) ) ).
% mult.assoc
thf(fact_1024_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B3: real] : ( times_times_real @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_1025_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_1026_mult_Oleft__commute,axiom,
! [B4: real,A: real,C2: real] :
( ( times_times_real @ B4 @ ( times_times_real @ A @ C2 ) )
= ( times_times_real @ A @ ( times_times_real @ B4 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_1027_mult_Oleft__commute,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( times_times_nat @ B4 @ ( times_times_nat @ A @ C2 ) )
= ( times_times_nat @ A @ ( times_times_nat @ B4 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_1028_diff__right__commute,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B4 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B4 ) @ C2 ) ) ).
% diff_right_commute
thf(fact_1029_card__Diff__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1030_diff__card__le__card__Diff,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1031_card__Diff__subset__Int,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ B ) )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_1032_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_1033_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_1034_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_1035_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_1036_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_1037_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_1038_nless__le,axiom,
! [A: real,B4: real] :
( ( ~ ( ord_less_real @ A @ B4 ) )
= ( ~ ( ord_less_eq_real @ A @ B4 )
| ( A = B4 ) ) ) ).
% nless_le
thf(fact_1039_nless__le,axiom,
! [A: set_a,B4: set_a] :
( ( ~ ( ord_less_set_a @ A @ B4 ) )
= ( ~ ( ord_less_eq_set_a @ A @ B4 )
| ( A = B4 ) ) ) ).
% nless_le
thf(fact_1040_nless__le,axiom,
! [A: nat,B4: nat] :
( ( ~ ( ord_less_nat @ A @ B4 ) )
= ( ~ ( ord_less_eq_nat @ A @ B4 )
| ( A = B4 ) ) ) ).
% nless_le
thf(fact_1041_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1042_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_1043_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1044_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1045_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_1046_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1047_dense__ge,axiom,
! [Z3: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_ge
thf(fact_1048_dense__le,axiom,
! [Y: real,Z3: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_le
thf(fact_1049_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1050_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
& ~ ( ord_less_eq_set_a @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1051_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1052_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_1053_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_1054_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1055_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_set_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1056_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1057_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1058_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1059_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1060_order_Ostrict__trans1,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_real @ B4 @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1061_order_Ostrict__trans1,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1062_order_Ostrict__trans1,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1063_order_Ostrict__trans2,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1064_order_Ostrict__trans2,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1065_order_Ostrict__trans2,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1066_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1067_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1068_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1069_dense__ge__bounded,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ Z3 @ X )
=> ( ! [W3: real] :
( ( ord_less_real @ Z3 @ W3 )
=> ( ( ord_less_real @ W3 @ X )
=> ( ord_less_eq_real @ Y @ W3 ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_1070_dense__le__bounded,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W3: real] :
( ( ord_less_real @ X @ W3 )
=> ( ( ord_less_real @ W3 @ Y )
=> ( ord_less_eq_real @ W3 @ Z3 ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_1071_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_real @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1072_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_set_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1073_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1074_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1075_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1076_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1077_dual__order_Ostrict__trans1,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( ord_less_real @ C2 @ B4 )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1078_dual__order_Ostrict__trans1,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_set_a @ C2 @ B4 )
=> ( ord_less_set_a @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1079_dual__order_Ostrict__trans1,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_nat @ C2 @ B4 )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1080_dual__order_Ostrict__trans2,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_real @ B4 @ A )
=> ( ( ord_less_eq_real @ C2 @ B4 )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1081_dual__order_Ostrict__trans2,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( ord_less_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B4 )
=> ( ord_less_set_a @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1082_dual__order_Ostrict__trans2,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ C2 @ B4 )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1083_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1084_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1085_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1086_order_Ostrict__implies__order,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ A @ B4 )
=> ( ord_less_eq_real @ A @ B4 ) ) ).
% order.strict_implies_order
thf(fact_1087_order_Ostrict__implies__order,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ B4 )
=> ( ord_less_eq_set_a @ A @ B4 ) ) ).
% order.strict_implies_order
thf(fact_1088_order_Ostrict__implies__order,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ord_less_eq_nat @ A @ B4 ) ) ).
% order.strict_implies_order
thf(fact_1089_dual__order_Ostrict__implies__order,axiom,
! [B4: real,A: real] :
( ( ord_less_real @ B4 @ A )
=> ( ord_less_eq_real @ B4 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1090_dual__order_Ostrict__implies__order,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ A )
=> ( ord_less_eq_set_a @ B4 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1091_dual__order_Ostrict__implies__order,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ord_less_eq_nat @ B4 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1092_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1093_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y5: set_a] :
( ( ord_less_set_a @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1094_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1095_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_eq_real @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1096_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1097_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1098_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1099_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1100_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1101_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1102_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_1103_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_1104_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_1105_order__le__neq__trans,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( A != B4 )
=> ( ord_less_real @ A @ B4 ) ) ) ).
% order_le_neq_trans
thf(fact_1106_order__le__neq__trans,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( A != B4 )
=> ( ord_less_set_a @ A @ B4 ) ) ) ).
% order_le_neq_trans
thf(fact_1107_order__le__neq__trans,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( A != B4 )
=> ( ord_less_nat @ A @ B4 ) ) ) ).
% order_le_neq_trans
thf(fact_1108_order__neq__le__trans,axiom,
! [A: real,B4: real] :
( ( A != B4 )
=> ( ( ord_less_eq_real @ A @ B4 )
=> ( ord_less_real @ A @ B4 ) ) ) ).
% order_neq_le_trans
thf(fact_1109_order__neq__le__trans,axiom,
! [A: set_a,B4: set_a] :
( ( A != B4 )
=> ( ( ord_less_eq_set_a @ A @ B4 )
=> ( ord_less_set_a @ A @ B4 ) ) ) ).
% order_neq_le_trans
thf(fact_1110_order__neq__le__trans,axiom,
! [A: nat,B4: nat] :
( ( A != B4 )
=> ( ( ord_less_eq_nat @ A @ B4 )
=> ( ord_less_nat @ A @ B4 ) ) ) ).
% order_neq_le_trans
thf(fact_1111_order__le__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1112_order__le__less__trans,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z3 )
=> ( ord_less_set_a @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1113_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1114_order__less__le__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1115_order__less__le__trans,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z3 )
=> ( ord_less_set_a @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1116_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1117_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B4: nat,C2: nat] :
( ( ord_less_eq_real @ A @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1118_order__le__less__subst1,axiom,
! [A: real,F: real > real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1119_order__le__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B4: nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1120_order__le__less__subst1,axiom,
! [A: set_a,F: real > set_a,B4: real,C2: real] :
( ( ord_less_eq_set_a @ A @ ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1121_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1122_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B4: real,C2: real] :
( ( ord_less_eq_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1123_order__le__less__subst2,axiom,
! [A: real,B4: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_real @ ( F @ B4 ) @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1124_order__le__less__subst2,axiom,
! [A: real,B4: real,F: real > set_a,C2: set_a] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_set_a @ ( F @ B4 ) @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1125_order__le__less__subst2,axiom,
! [A: real,B4: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1126_order__le__less__subst2,axiom,
! [A: set_a,B4: set_a,F: set_a > real,C2: real] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_real @ ( F @ B4 ) @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1127_order__le__less__subst2,axiom,
! [A: set_a,B4: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_set_a @ ( F @ B4 ) @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1128_order__le__less__subst2,axiom,
! [A: set_a,B4: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1129_order__le__less__subst2,axiom,
! [A: nat,B4: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_real @ ( F @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1130_order__le__less__subst2,axiom,
! [A: nat,B4: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_set_a @ ( F @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1131_order__le__less__subst2,axiom,
! [A: nat,B4: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1132_order__less__le__subst1,axiom,
! [A: real,F: real > real,B4: real,C2: real] :
( ( ord_less_real @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1133_order__less__le__subst1,axiom,
! [A: set_a,F: real > set_a,B4: real,C2: real] :
( ( ord_less_set_a @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1134_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B4: real,C2: real] :
( ( ord_less_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1135_order__less__le__subst1,axiom,
! [A: real,F: set_a > real,B4: set_a,C2: set_a] :
( ( ord_less_real @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1136_order__less__le__subst1,axiom,
! [A: set_a,F: set_a > set_a,B4: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1137_order__less__le__subst1,axiom,
! [A: nat,F: set_a > nat,B4: set_a,C2: set_a] :
( ( ord_less_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1138_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B4: nat,C2: nat] :
( ( ord_less_real @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1139_order__less__le__subst1,axiom,
! [A: set_a,F: nat > set_a,B4: nat,C2: nat] :
( ( ord_less_set_a @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1140_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1141_order__less__le__subst2,axiom,
! [A: nat,B4: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_real @ ( F @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1142_order__less__le__subst2,axiom,
! [A: real,B4: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_eq_real @ ( F @ B4 ) @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1143_order__less__le__subst2,axiom,
! [A: nat,B4: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_set_a @ ( F @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1144_order__less__le__subst2,axiom,
! [A: real,B4: real,F: real > set_a,C2: set_a] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_eq_set_a @ ( F @ B4 ) @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1145_order__less__le__subst2,axiom,
! [A: nat,B4: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1146_order__less__le__subst2,axiom,
! [A: real,B4: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C2 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1147_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_1148_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_1149_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_1150_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_1151_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_1152_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_1153_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1154_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1155_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_1156_diff__eq__diff__less__eq,axiom,
! [A: real,B4: real,C2: real,D2: real] :
( ( ( minus_minus_real @ A @ B4 )
= ( minus_minus_real @ C2 @ D2 ) )
=> ( ( ord_less_eq_real @ A @ B4 )
= ( ord_less_eq_real @ C2 @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1157_diff__right__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B4 @ C2 ) ) ) ).
% diff_right_mono
thf(fact_1158_diff__left__mono,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B4 ) ) ) ).
% diff_left_mono
thf(fact_1159_diff__mono,axiom,
! [A: real,B4: real,D2: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ D2 @ C2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B4 @ D2 ) ) ) ) ).
% diff_mono
thf(fact_1160_eq__iff__diff__eq__0,axiom,
( ( ^ [Y6: real,Z2: real] : ( Y6 = Z2 ) )
= ( ^ [A4: real,B3: real] :
( ( minus_minus_real @ A4 @ B3 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1161_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_1162_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_1163_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_1164_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1165_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_1166_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1167_diff__strict__right__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_real @ A @ B4 )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B4 @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_1168_diff__strict__left__mono,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_real @ B4 @ A )
=> ( ord_less_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B4 ) ) ) ).
% diff_strict_left_mono
thf(fact_1169_diff__eq__diff__less,axiom,
! [A: real,B4: real,C2: real,D2: real] :
( ( ( minus_minus_real @ A @ B4 )
= ( minus_minus_real @ C2 @ D2 ) )
=> ( ( ord_less_real @ A @ B4 )
= ( ord_less_real @ C2 @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_1170_diff__strict__mono,axiom,
! [A: real,B4: real,D2: real,C2: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_real @ D2 @ C2 )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B4 @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_1171_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_1172_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1173_bot_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1174_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1175_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B3 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_1176_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B3 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_1177_realpow__pos__nth2,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ ( suc @ N ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_1178_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1179_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A )
& ! [Y4: real] :
( ( ( ord_less_real @ zero_zero_real @ Y4 )
& ( ( power_power_real @ Y4 @ N )
= A ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1180_sumset__iterated__r,axiom,
! [R2: nat,A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ R2 )
=> ( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ R2 )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ ( minus_minus_nat @ R2 @ one_one_nat ) ) ) ) ) ).
% sumset_iterated_r
thf(fact_1181_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_1182_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1183_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_1184_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1185_mult__cancel__right,axiom,
! [A: real,C2: real,B4: real] :
( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B4 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A = B4 ) ) ) ).
% mult_cancel_right
thf(fact_1186_mult__cancel__right,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B4 @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B4 ) ) ) ).
% mult_cancel_right
thf(fact_1187_mult__cancel__left,axiom,
! [C2: real,A: real,B4: real] :
( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B4 ) )
= ( ( C2 = zero_zero_real )
| ( A = B4 ) ) ) ).
% mult_cancel_left
thf(fact_1188_mult__cancel__left,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B4 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B4 ) ) ) ).
% mult_cancel_left
thf(fact_1189_mult__eq__0__iff,axiom,
! [A: real,B4: real] :
( ( ( times_times_real @ A @ B4 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B4 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_1190_mult__eq__0__iff,axiom,
! [A: nat,B4: nat] :
( ( ( times_times_nat @ A @ B4 )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B4 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1191_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_1192_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1193_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_1194_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1195_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_1196_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_1197_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_1198_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_1199_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_1200_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_1201_mult__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ( times_times_real @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_1202_mult__cancel__right1,axiom,
! [C2: real,B4: real] :
( ( C2
= ( times_times_real @ B4 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B4 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_1203_mult__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ( times_times_real @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_1204_mult__cancel__left1,axiom,
! [C2: real,B4: real] :
( ( C2
= ( times_times_real @ C2 @ B4 ) )
= ( ( C2 = zero_zero_real )
| ( B4 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_1205_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_1206_power__inject__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M )
= ( power_power_real @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_1207_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_1208_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_1209_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1210_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1211_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1212_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1213_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1214_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1215_power__strict__increasing__iff,axiom,
! [B4: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B4 )
=> ( ( ord_less_nat @ ( power_power_nat @ B4 @ X ) @ ( power_power_nat @ B4 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_1216_power__strict__increasing__iff,axiom,
! [B4: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B4 )
=> ( ( ord_less_real @ ( power_power_real @ B4 @ X ) @ ( power_power_real @ B4 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_1217_power__strict__decreasing__iff,axiom,
! [B4: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_nat @ B4 @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B4 @ M ) @ ( power_power_nat @ B4 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1218_power__strict__decreasing__iff,axiom,
! [B4: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B4 )
=> ( ( ord_less_real @ B4 @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B4 @ M ) @ ( power_power_real @ B4 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1219_power__increasing__iff,axiom,
! [B4: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B4 )
=> ( ( ord_less_eq_real @ ( power_power_real @ B4 @ X ) @ ( power_power_real @ B4 @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_1220_power__increasing__iff,axiom,
! [B4: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B4 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B4 @ X ) @ ( power_power_nat @ B4 @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_1221_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1222_card__Diff__insert,axiom,
! [A: real,A2: set_real,B: set_real] :
( ( member_real @ A @ A2 )
=> ( ~ ( member_real @ A @ B )
=> ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B ) ) )
= ( minus_minus_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1223_card__Diff__insert,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ~ ( member_a @ A @ B )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1224_power__decreasing__iff,axiom,
! [B4: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B4 )
=> ( ( ord_less_real @ B4 @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B4 @ M ) @ ( power_power_real @ B4 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1225_power__decreasing__iff,axiom,
! [B4: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_nat @ B4 @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B4 @ M ) @ ( power_power_nat @ B4 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1226_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1227_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_1228_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1229_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_1230_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1231_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1232_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_1233_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_1234_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_1235_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_1236_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1237_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1238_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_1239_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1240_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1241_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1242_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_1243_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1244_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1245_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1246_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_1247_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1248_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1249_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_1250_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1251_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1252_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1253_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ).
% real_arch_pow
thf(fact_1254_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1255_group__of__Units,axiom,
group_group_a @ ( group_Units_a @ g @ addition @ zero ) @ addition @ zero ).
% group_of_Units
thf(fact_1256_mem__UnitsD,axiom,
! [U: a] :
( ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
& ( member_a @ U @ g ) ) ) ).
% mem_UnitsD
thf(fact_1257_mem__UnitsI,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) ) ) ) ).
% mem_UnitsI
thf(fact_1258_unit__invertible,axiom,
group_invertible_a @ g @ addition @ zero @ zero ).
% unit_invertible
thf(fact_1259_invertible__def,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
= ( ? [X2: a] :
( ( member_a @ X2 @ g )
& ( ( addition @ U @ X2 )
= zero )
& ( ( addition @ X2 @ U )
= zero ) ) ) ) ) ).
% invertible_def
thf(fact_1260_invertibleE,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ! [V4: a] :
( ( ( ( addition @ U @ V4 )
= zero )
& ( ( addition @ V4 @ U )
= zero ) )
=> ~ ( member_a @ V4 @ g ) )
=> ~ ( member_a @ U @ g ) ) ) ).
% invertibleE
thf(fact_1261_invertible__right__cancel,axiom,
! [X: a,Y: a,Z3: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z3 @ g )
=> ( ( ( addition @ Y @ X )
= ( addition @ Z3 @ X ) )
= ( Y = Z3 ) ) ) ) ) ) ).
% invertible_right_cancel
thf(fact_1262_invertible__left__cancel,axiom,
! [X: a,Y: a,Z3: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z3 @ g )
=> ( ( ( addition @ X @ Y )
= ( addition @ X @ Z3 ) )
= ( Y = Z3 ) ) ) ) ) ) ).
% invertible_left_cancel
thf(fact_1263_invertibleI,axiom,
! [U: a,V2: a] :
( ( ( addition @ U @ V2 )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ) ) ) ).
% invertibleI
thf(fact_1264_invertible,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ).
% invertible
thf(fact_1265_composition__invertible,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( group_invertible_a @ g @ addition @ zero @ Y )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( group_invertible_a @ g @ addition @ zero @ ( addition @ X @ Y ) ) ) ) ) ) ).
% composition_invertible
thf(fact_1266_sumset__subset__Un_I1_J,axiom,
! [A2: set_a,B: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sumset_subset_Un(1)
thf(fact_1267_sumset__subset__Un_I2_J,axiom,
! [A2: set_a,B: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ C ) @ B ) ) ).
% sumset_subset_Un(2)
thf(fact_1268_sumset__subset__Un1,axiom,
! [A2: set_a,A5: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ A5 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B ) ) ) ).
% sumset_subset_Un1
thf(fact_1269_sumset__subset__Un2,axiom,
! [A2: set_a,B: set_a,B5: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B @ B5 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B5 ) ) ) ).
% sumset_subset_Un2
% Conjectures (1)
thf(conj_0,conjecture,
( ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ b @ ( suc @ r ) ) ) ) )
= ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ b @ r ) ) ) ) ) ) ).
%------------------------------------------------------------------------------