TPTP Problem File: SLH0783^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Pluennecke_Ruzsa_Inequality/0003_Pluennecke_Ruzsa_Inequality/prob_00446_016747__12272046_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1386 ( 622 unt; 113 typ; 0 def)
% Number of atoms : 3471 (1350 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 10236 ( 412 ~; 94 |; 245 &;8051 @)
% ( 0 <=>;1434 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 292 ( 292 >; 0 *; 0 +; 0 <<)
% Number of symbols : 107 ( 104 usr; 19 con; 0-5 aty)
% Number of variables : 3115 ( 141 ^;2897 !; 77 ?;3115 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:24:15.588
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (104)
thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
finite_card_int: set_int > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Real__Oreal,type,
finite_card_real: set_real > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J,type,
finite_card_set_a: set_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
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_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_HOL_ONO__MATCH_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_Itf__a_J,type,
nO_MAT3201932972334532047_set_a: set_real > set_a > $o ).
thf(sy_c_HOL_ONO__MATCH_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_Itf__a_J,type,
nO_MAT8518843428946182283_set_a: set_set_a > set_a > $o ).
thf(sy_c_HOL_ONO__MATCH_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
nO_MATCH_set_a_set_a: set_a > set_a > $o ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
inf_inf_int: int > int > int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Real__Oreal,type,
inf_inf_real: real > real > real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Int__Oint_J,type,
inf_inf_set_int: set_int > set_int > set_int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Real__Oreal_J,type,
inf_inf_set_real: set_real > set_real > set_real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
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__Int__Oint,type,
sup_sup_int: int > int > int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Real__Oreal,type,
sup_sup_real: real > real > real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Int__Oint_J,type,
sup_sup_set_int: set_int > set_int > set_int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Real__Oreal_J,type,
sup_sup_set_real: set_real > set_real > set_real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_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_Lattices__Big_Olinorder__class_OMin_001t__Int__Oint,type,
lattic8718645017227715691in_int: set_int > int ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
lattic8721135487736765967in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Real__Oreal,type,
lattic3629708407755379051n_real: set_real > real ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_Eo_J,type,
bot_bot_int_o: int > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
bot_bot_real_o: real > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
bot_bot_set_a_o: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $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_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_Osumsetp_001tf__a,type,
pluenn895083305082786853setp_a: set_a > ( a > a > a ) > ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OPow_001tf__a,type,
pow_a: set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Real__Oreal,type,
image_set_a_real: ( set_a > real ) > set_set_a > set_real ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_fChoice_001t__Set__Oset_Itf__a_J,type,
fChoice_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A0,type,
a0: set_a ).
thf(sy_v_A____,type,
a2: set_a ).
thf(sy_v_B,type,
b: set_a ).
thf(sy_v_C____,type,
c: set_a ).
thf(sy_v_G,type,
g: set_a ).
thf(sy_v_K0,type,
k0: real ).
thf(sy_v_KS____,type,
ks: set_real ).
thf(sy_v_K____,type,
k: real ).
thf(sy_v_addition,type,
addition: a > a > a ).
% Relevant facts (1268)
thf(fact_0_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( addition @ X @ Y )
= ( addition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_1_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_2_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_3_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_4_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_5_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_6_assms_I7_J,axiom,
b != bot_bot_set_a ).
% assms(7)
thf(fact_7_assms_I5_J,axiom,
finite_finite_a @ b ).
% assms(5)
thf(fact_8__092_060open_062K_A_092_060le_062_AK0_092_060close_062,axiom,
ord_less_eq_real @ k @ k0 ).
% \<open>K \<le> K0\<close>
thf(fact_9_assms_I6_J,axiom,
ord_less_eq_set_a @ b @ g ).
% assms(6)
thf(fact_10_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_11_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_12_that_I1_J,axiom,
finite_finite_a @ c ).
% that(1)
thf(fact_13_K__cardA,axiom,
( ( times_times_real @ k @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a2 ) ) )
= ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ b ) ) ) ) ).
% K_cardA
thf(fact_14_that_I2_J,axiom,
ord_less_eq_set_a @ c @ g ).
% that(2)
thf(fact_15_Keq,axiom,
( k
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a2 ) ) ) ) ).
% Keq
thf(fact_16_additive__abelian__group_Osumset_Ocong,axiom,
pluenn3038260743871226533mset_a = pluenn3038260743871226533mset_a ).
% additive_abelian_group.sumset.cong
thf(fact_17_K0,axiom,
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 ) ) ) ).
% K0
thf(fact_18_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_19_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_20_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_21_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_22_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_23_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_24_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_25_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_26_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_27__092_060open_062K_A_092_060in_062_AKS_092_060close_062,axiom,
member_real @ k @ ks ).
% \<open>K \<in> KS\<close>
thf(fact_28_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_29_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_30_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_31_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_32_assms_I4_J,axiom,
a0 != bot_bot_set_a ).
% assms(4)
thf(fact_33_assms_I2_J,axiom,
finite_finite_a @ a0 ).
% assms(2)
thf(fact_34_assms_I3_J,axiom,
ord_less_eq_set_a @ a0 @ g ).
% assms(3)
thf(fact_35_A_I2_J,axiom,
a2 != bot_bot_set_a ).
% A(2)
thf(fact_36__092_060open_062finite_AA_092_060close_062,axiom,
finite_finite_a @ a2 ).
% \<open>finite A\<close>
thf(fact_37_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_38_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_39__092_060open_062A_A_092_060subseteq_062_AG_092_060close_062,axiom,
ord_less_eq_set_a @ a2 @ g ).
% \<open>A \<subseteq> G\<close>
thf(fact_40_A_I1_J,axiom,
ord_less_eq_set_a @ a2 @ a0 ).
% A(1)
thf(fact_41_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
thf(fact_42_KS_I2_J,axiom,
ks != bot_bot_set_real ).
% KS(2)
thf(fact_43_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_44_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_45_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_46_KS_I1_J,axiom,
finite_finite_real @ ks ).
% KS(1)
thf(fact_47__092_060open_062real_A_Icard_A_Isumset_AA0_AB_J_J_A_P_Areal_A_Icard_AA0_J_A_092_060in_062_AKS_092_060close_062,axiom,
member_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a0 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a0 ) ) ) @ ks ).
% \<open>real (card (sumset A0 B)) / real (card A0) \<in> KS\<close>
thf(fact_48_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_50_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_51_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_53_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_58_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_59_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_60_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_61_K__def,axiom,
( k
= ( lattic3629708407755379051n_real @ ks ) ) ).
% K_def
thf(fact_62_K__le,axiom,
! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ a2 )
=> ( ( A5 != bot_bot_set_a )
=> ( ord_less_eq_real @ k @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A5 ) ) ) ) ) ) ).
% K_le
thf(fact_63_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_64_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_65_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_66_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_67_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_68_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_69_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_70_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_71_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_72_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_73_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_74_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_75_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B4: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B4 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_76_additive__abelian__group_Osumsetp_Ocong,axiom,
pluenn895083305082786853setp_a = pluenn895083305082786853setp_a ).
% additive_abelian_group.sumsetp.cong
thf(fact_77_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_78_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_79_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_80_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_81_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_82_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_83_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_84_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_85_real__divide__square__eq,axiom,
! [R: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
= ( divide_divide_real @ A @ R ) ) ).
% real_divide_square_eq
thf(fact_86_Un__subset__iff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( ( ord_less_eq_set_a @ A2 @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_87_finite__Un,axiom,
! [F: set_real,G: set_real] :
( ( finite_finite_real @ ( sup_sup_set_real @ F @ G ) )
= ( ( finite_finite_real @ F )
& ( finite_finite_real @ G ) ) ) ).
% finite_Un
thf(fact_88_finite__Un,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) )
= ( ( finite_finite_nat @ F )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_89_finite__Un,axiom,
! [F: set_int,G: set_int] :
( ( finite_finite_int @ ( sup_sup_set_int @ F @ G ) )
= ( ( finite_finite_int @ F )
& ( finite_finite_int @ G ) ) ) ).
% finite_Un
thf(fact_90_finite__Un,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) )
= ( ( finite_finite_a @ F )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_91_Un__empty,axiom,
! [A2: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_92_Un__empty,axiom,
! [A2: set_real,B: set_real] :
( ( ( sup_sup_set_real @ A2 @ B )
= bot_bot_set_real )
= ( ( A2 = bot_bot_set_real )
& ( B = bot_bot_set_real ) ) ) ).
% Un_empty
thf(fact_93_Un__empty,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( sup_sup_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ( A2 = bot_bot_set_set_a )
& ( B = bot_bot_set_set_a ) ) ) ).
% Un_empty
thf(fact_94_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_95_sup__bot__left,axiom,
! [X: set_real] :
( ( sup_sup_set_real @ bot_bot_set_real @ X )
= X ) ).
% sup_bot_left
thf(fact_96_sup__bot__left,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_97_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_98_sup__bot__right,axiom,
! [X: set_real] :
( ( sup_sup_set_real @ X @ bot_bot_set_real )
= X ) ).
% sup_bot_right
thf(fact_99_sup__bot__right,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ bot_bot_set_set_a )
= X ) ).
% sup_bot_right
thf(fact_100_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_101_bot__eq__sup__iff,axiom,
! [X: set_real,Y: set_real] :
( ( bot_bot_set_real
= ( sup_sup_set_real @ X @ Y ) )
= ( ( X = bot_bot_set_real )
& ( Y = bot_bot_set_real ) ) ) ).
% bot_eq_sup_iff
thf(fact_102_bot__eq__sup__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_set_a )
& ( Y = bot_bot_set_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_103_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_104_sup__eq__bot__iff,axiom,
! [X: set_real,Y: set_real] :
( ( ( sup_sup_set_real @ X @ Y )
= bot_bot_set_real )
= ( ( X = bot_bot_set_real )
& ( Y = bot_bot_set_real ) ) ) ).
% sup_eq_bot_iff
thf(fact_105_sup__eq__bot__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( sup_sup_set_set_a @ X @ Y )
= bot_bot_set_set_a )
= ( ( X = bot_bot_set_set_a )
& ( Y = bot_bot_set_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_106_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B4: set_a] :
( ( ( sup_sup_set_a @ A @ B4 )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_107_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_real,B4: set_real] :
( ( ( sup_sup_set_real @ A @ B4 )
= bot_bot_set_real )
= ( ( A = bot_bot_set_real )
& ( B4 = bot_bot_set_real ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_108_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_set_a,B4: set_set_a] :
( ( ( sup_sup_set_set_a @ A @ B4 )
= bot_bot_set_set_a )
= ( ( A = bot_bot_set_set_a )
& ( B4 = bot_bot_set_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_109_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_110_empty__Collect__eq,axiom,
! [P: int > $o] :
( ( bot_bot_set_int
= ( collect_int @ P ) )
= ( ! [X2: int] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_111_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_112_empty__Collect__eq,axiom,
! [P: real > $o] :
( ( bot_bot_set_real
= ( collect_real @ P ) )
= ( ! [X2: real] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_113_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X2: set_a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_114_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_115_Collect__empty__eq,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( ! [X2: int] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_116_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_117_Collect__empty__eq,axiom,
! [P: real > $o] :
( ( ( collect_real @ P )
= bot_bot_set_real )
= ( ! [X2: real] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_118_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_119_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_120_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_121_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_122_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_123_empty__iff,axiom,
! [C2: real] :
~ ( member_real @ C2 @ bot_bot_set_real ) ).
% empty_iff
thf(fact_124_empty__iff,axiom,
! [C2: set_a] :
~ ( member_set_a @ C2 @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_125_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_126_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_127_subsetI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_128_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_129_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_130_insert__absorb2,axiom,
! [X: set_a,A2: set_set_a] :
( ( insert_set_a @ X @ ( insert_set_a @ X @ A2 ) )
= ( insert_set_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_131_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_132_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_133_insert__iff,axiom,
! [A: set_a,B4: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B4 @ A2 ) )
= ( ( A = B4 )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_134_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_135_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_136_insertCI,axiom,
! [A: set_a,B: set_set_a,B4: set_a] :
( ( ~ ( member_set_a @ A @ B )
=> ( A = B4 ) )
=> ( member_set_a @ A @ ( insert_set_a @ B4 @ B ) ) ) ).
% insertCI
thf(fact_137_sup_Oright__idem,axiom,
! [A: set_a,B4: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B4 ) @ B4 )
= ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.right_idem
thf(fact_138_sup__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_139_sup_Oleft__idem,axiom,
! [A: set_a,B4: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B4 ) )
= ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.left_idem
thf(fact_140_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_141_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_142_Un__iff,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( sup_sup_set_real @ A2 @ B ) )
= ( ( member_real @ C2 @ A2 )
| ( member_real @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_143_Un__iff,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( ( member_set_a @ C2 @ A2 )
| ( member_set_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_144_Un__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
| ( member_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_145_UnCI,axiom,
! [C2: real,B: set_real,A2: set_real] :
( ( ~ ( member_real @ C2 @ B )
=> ( member_real @ C2 @ A2 ) )
=> ( member_real @ C2 @ ( sup_sup_set_real @ A2 @ B ) ) ) ).
% UnCI
thf(fact_146_UnCI,axiom,
! [C2: set_a,B: set_set_a,A2: set_set_a] :
( ( ~ ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ A2 ) )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_147_UnCI,axiom,
! [C2: a,B: set_a,A2: set_a] :
( ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ A2 ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_148__092_060open_062_092_060And_062thesis_O_A_I_092_060lbrakk_062finite_AKS_059_AKS_A_092_060noteq_062_A_123_125_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ( ( finite_finite_real @ ks )
=> ( ks = bot_bot_set_real ) ) ).
% \<open>\<And>thesis. (\<lbrakk>finite KS; KS \<noteq> {}\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_149_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_150_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_151_le__sup__iff,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ X @ Y ) @ Z )
= ( ( ord_less_eq_real @ X @ Z )
& ( ord_less_eq_real @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_152_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_153_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_154_le__sup__iff,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z )
= ( ( ord_less_eq_int @ X @ Z )
& ( ord_less_eq_int @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_155_sup_Obounded__iff,axiom,
! [B4: real,C2: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B4 @ C2 ) @ A )
= ( ( ord_less_eq_real @ B4 @ A )
& ( ord_less_eq_real @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_156_sup_Obounded__iff,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A )
= ( ( ord_less_eq_set_a @ B4 @ A )
& ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_157_sup_Obounded__iff,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A )
= ( ( ord_less_eq_nat @ B4 @ A )
& ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_158_sup_Obounded__iff,axiom,
! [B4: int,C2: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B4 @ C2 ) @ A )
= ( ( ord_less_eq_int @ B4 @ A )
& ( ord_less_eq_int @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_159_empty__subsetI,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% empty_subsetI
thf(fact_160_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_161_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_162_subset__empty,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
= ( A2 = bot_bot_set_real ) ) ).
% subset_empty
thf(fact_163_subset__empty,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_164_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_165_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_166_singletonI,axiom,
! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).
% singletonI
thf(fact_167_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_168_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_169_sup__bot_Oright__neutral,axiom,
! [A: set_real] :
( ( sup_sup_set_real @ A @ bot_bot_set_real )
= A ) ).
% sup_bot.right_neutral
thf(fact_170_sup__bot_Oright__neutral,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_171_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B4: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B4 ) )
= ( ( A = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_172_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_real,B4: set_real] :
( ( bot_bot_set_real
= ( sup_sup_set_real @ A @ B4 ) )
= ( ( A = bot_bot_set_real )
& ( B4 = bot_bot_set_real ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_173_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_set_a,B4: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ A @ B4 ) )
= ( ( A = bot_bot_set_set_a )
& ( B4 = bot_bot_set_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_174_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_175_sup__bot_Oleft__neutral,axiom,
! [A: set_real] :
( ( sup_sup_set_real @ bot_bot_set_real @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_176_sup__bot_Oleft__neutral,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_177_finite__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) )
= ( finite_finite_set_a @ A2 ) ) ).
% finite_insert
thf(fact_178_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_179_finite__insert,axiom,
! [A: real,A2: set_real] :
( ( finite_finite_real @ ( insert_real @ A @ A2 ) )
= ( finite_finite_real @ A2 ) ) ).
% finite_insert
thf(fact_180_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_181_finite__insert,axiom,
! [A: int,A2: set_int] :
( ( finite_finite_int @ ( insert_int @ A @ A2 ) )
= ( finite_finite_int @ A2 ) ) ).
% finite_insert
thf(fact_182_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_183_insert__subset,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A2 ) @ B )
= ( ( member_set_a @ X @ B )
& ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_184_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_185_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_186_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_187_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_188_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_189_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_190_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_191_Un__insert__right,axiom,
! [A2: set_set_a,A: set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_192_Un__insert__right,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_193_Un__insert__left,axiom,
! [A: set_a,B: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ ( insert_set_a @ A @ B ) @ C )
= ( insert_set_a @ A @ ( sup_sup_set_set_a @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_194_Un__insert__left,axiom,
! [A: a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_195_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_196_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_197_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_198_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_199_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_200_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_201_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_202_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_203_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_204_singleton__insert__inj__eq_H,axiom,
! [A: real,A2: set_real,B4: real] :
( ( ( insert_real @ A @ A2 )
= ( insert_real @ B4 @ bot_bot_set_real ) )
= ( ( A = B4 )
& ( ord_less_eq_set_real @ A2 @ ( insert_real @ B4 @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_205_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A2: set_set_a,B4: set_a] :
( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B4 @ bot_bot_set_set_a ) )
= ( ( A = B4 )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B4 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_206_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_207_singleton__insert__inj__eq,axiom,
! [B4: real,A: real,A2: set_real] :
( ( ( insert_real @ B4 @ bot_bot_set_real )
= ( insert_real @ A @ A2 ) )
= ( ( A = B4 )
& ( ord_less_eq_set_real @ A2 @ ( insert_real @ B4 @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_208_singleton__insert__inj__eq,axiom,
! [B4: set_a,A: set_a,A2: set_set_a] :
( ( ( insert_set_a @ B4 @ bot_bot_set_set_a )
= ( insert_set_a @ A @ A2 ) )
= ( ( A = B4 )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B4 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_209_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_210_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_211_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_212_card_Oempty,axiom,
( ( finite_card_real @ bot_bot_set_real )
= zero_zero_nat ) ).
% card.empty
thf(fact_213_card_Oempty,axiom,
( ( finite_card_set_a @ bot_bot_set_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_214_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_215_card_Oinfinite,axiom,
! [A2: set_real] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite_card_real @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_216_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_217_card_Oinfinite,axiom,
! [A2: set_int] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_card_int @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_218_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_219_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_220_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_221_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_222_card__0__eq,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( ( finite_card_int @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_int ) ) ) ).
% card_0_eq
thf(fact_223_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_224_card__0__eq,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( ( finite_card_real @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_real ) ) ) ).
% card_0_eq
thf(fact_225_card__0__eq,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( finite_card_set_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_set_a ) ) ) ).
% card_0_eq
thf(fact_226_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_227_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B6: set_a] :
( ( A2
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_228_mk__disjoint__insert,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ? [B6: set_real] :
( ( A2
= ( insert_real @ A @ B6 ) )
& ~ ( member_real @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_229_mk__disjoint__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ? [B6: set_set_a] :
( ( A2
= ( insert_set_a @ A @ B6 ) )
& ~ ( member_set_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_230_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_231_insert__commute,axiom,
! [X: set_a,Y: set_a,A2: set_set_a] :
( ( insert_set_a @ X @ ( insert_set_a @ Y @ A2 ) )
= ( insert_set_a @ Y @ ( insert_set_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_232_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_233_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_234_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B4: set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B4 @ B )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B4 @ B ) )
= ( ( ( A = B4 )
=> ( A2 = B ) )
& ( ( A != B4 )
=> ? [C3: set_set_a] :
( ( A2
= ( insert_set_a @ B4 @ C3 ) )
& ~ ( member_set_a @ B4 @ C3 )
& ( B
= ( insert_set_a @ A @ C3 ) )
& ~ ( member_set_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_235_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_236_insert__absorb,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ( ( insert_real @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_237_insert__absorb,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_238_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_239_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_240_insert__ident,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ~ ( member_set_a @ X @ B )
=> ( ( ( insert_set_a @ X @ A2 )
= ( insert_set_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_241_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B6: set_a] :
( ( A2
= ( insert_a @ X @ B6 ) )
=> ( member_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_242_Set_Oset__insert,axiom,
! [X: real,A2: set_real] :
( ( member_real @ X @ A2 )
=> ~ ! [B6: set_real] :
( ( A2
= ( insert_real @ X @ B6 ) )
=> ( member_real @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_243_Set_Oset__insert,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ~ ! [B6: set_set_a] :
( ( A2
= ( insert_set_a @ X @ B6 ) )
=> ( member_set_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_244_insertI2,axiom,
! [A: a,B: set_a,B4: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ B4 @ B ) ) ) ).
% insertI2
thf(fact_245_insertI2,axiom,
! [A: real,B: set_real,B4: real] :
( ( member_real @ A @ B )
=> ( member_real @ A @ ( insert_real @ B4 @ B ) ) ) ).
% insertI2
thf(fact_246_insertI2,axiom,
! [A: set_a,B: set_set_a,B4: set_a] :
( ( member_set_a @ A @ B )
=> ( member_set_a @ A @ ( insert_set_a @ B4 @ B ) ) ) ).
% insertI2
thf(fact_247_insertI1,axiom,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).
% insertI1
thf(fact_248_insertI1,axiom,
! [A: real,B: set_real] : ( member_real @ A @ ( insert_real @ A @ B ) ) ).
% insertI1
thf(fact_249_insertI1,axiom,
! [A: set_a,B: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B ) ) ).
% insertI1
thf(fact_250_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_251_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_252_insertE,axiom,
! [A: set_a,B4: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B4 @ A2 ) )
=> ( ( A != B4 )
=> ( member_set_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_253_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_254_bot__set__def,axiom,
( bot_bot_set_int
= ( collect_int @ bot_bot_int_o ) ) ).
% bot_set_def
thf(fact_255_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_256_bot__set__def,axiom,
( bot_bot_set_real
= ( collect_real @ bot_bot_real_o ) ) ).
% bot_set_def
thf(fact_257_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_258_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_259_singleton__inject,axiom,
! [A: real,B4: real] :
( ( ( insert_real @ A @ bot_bot_set_real )
= ( insert_real @ B4 @ bot_bot_set_real ) )
=> ( A = B4 ) ) ).
% singleton_inject
thf(fact_260_singleton__inject,axiom,
! [A: set_a,B4: set_a] :
( ( ( insert_set_a @ A @ bot_bot_set_set_a )
= ( insert_set_a @ B4 @ bot_bot_set_set_a ) )
=> ( A = B4 ) ) ).
% singleton_inject
thf(fact_261_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_262_insert__not__empty,axiom,
! [A: real,A2: set_real] :
( ( insert_real @ A @ A2 )
!= bot_bot_set_real ) ).
% insert_not_empty
thf(fact_263_insert__not__empty,axiom,
! [A: set_a,A2: set_set_a] :
( ( insert_set_a @ A @ A2 )
!= bot_bot_set_set_a ) ).
% insert_not_empty
thf(fact_264_doubleton__eq__iff,axiom,
! [A: a,B4: a,C2: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B4 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C2 )
& ( B4 = D ) )
| ( ( A = D )
& ( B4 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_265_doubleton__eq__iff,axiom,
! [A: real,B4: real,C2: real,D: real] :
( ( ( insert_real @ A @ ( insert_real @ B4 @ bot_bot_set_real ) )
= ( insert_real @ C2 @ ( insert_real @ D @ bot_bot_set_real ) ) )
= ( ( ( A = C2 )
& ( B4 = D ) )
| ( ( A = D )
& ( B4 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_266_doubleton__eq__iff,axiom,
! [A: set_a,B4: set_a,C2: set_a,D: set_a] :
( ( ( insert_set_a @ A @ ( insert_set_a @ B4 @ bot_bot_set_set_a ) )
= ( insert_set_a @ C2 @ ( insert_set_a @ D @ bot_bot_set_set_a ) ) )
= ( ( ( A = C2 )
& ( B4 = D ) )
| ( ( A = D )
& ( B4 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_267_singleton__iff,axiom,
! [B4: a,A: a] :
( ( member_a @ B4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B4 = A ) ) ).
% singleton_iff
thf(fact_268_singleton__iff,axiom,
! [B4: real,A: real] :
( ( member_real @ B4 @ ( insert_real @ A @ bot_bot_set_real ) )
= ( B4 = A ) ) ).
% singleton_iff
thf(fact_269_singleton__iff,axiom,
! [B4: set_a,A: set_a] :
( ( member_set_a @ B4 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B4 = A ) ) ).
% singleton_iff
thf(fact_270_singletonD,axiom,
! [B4: a,A: a] :
( ( member_a @ B4 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B4 = A ) ) ).
% singletonD
thf(fact_271_singletonD,axiom,
! [B4: real,A: real] :
( ( member_real @ B4 @ ( insert_real @ A @ bot_bot_set_real ) )
=> ( B4 = A ) ) ).
% singletonD
thf(fact_272_singletonD,axiom,
! [B4: set_a,A: set_a] :
( ( member_set_a @ B4 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B4 = A ) ) ).
% singletonD
thf(fact_273_finite_OinsertI,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_274_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_275_finite_OinsertI,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( finite_finite_real @ ( insert_real @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_276_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_277_finite_OinsertI,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( finite_finite_int @ ( insert_int @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_278_subset__insertI2,axiom,
! [A2: set_set_a,B: set_set_a,B4: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B4 @ B ) ) ) ).
% subset_insertI2
thf(fact_279_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_280_subset__insertI,axiom,
! [B: set_set_a,A: set_a] : ( ord_le3724670747650509150_set_a @ B @ ( insert_set_a @ A @ B ) ) ).
% subset_insertI
thf(fact_281_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).
% subset_insertI
thf(fact_282_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_283_subset__insert,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_284_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_285_insert__mono,axiom,
! [C: set_set_a,D2: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ C @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A @ C ) @ ( insert_set_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_286_insert__mono,axiom,
! [C: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_287_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_288_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_289_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_290_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_291_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ X3 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_292_infinite__finite__induct,axiom,
! [P: set_int > $o,A2: set_int] :
( ! [A6: set_int] :
( ~ ( finite_finite_int @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X3: int,F2: set_int] :
( ( finite_finite_int @ F2 )
=> ( ~ ( member_int @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_int @ X3 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_293_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A6: set_a] :
( ~ ( finite_finite_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X3 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_294_infinite__finite__induct,axiom,
! [P: set_real > $o,A2: set_real] :
( ! [A6: set_real] :
( ~ ( finite_finite_real @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,F2: set_real] :
( ( finite_finite_real @ F2 )
=> ( ~ ( member_real @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_real @ X3 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_295_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A2: set_set_a] :
( ! [A6: set_set_a] :
( ~ ( finite_finite_set_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ~ ( member_set_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ X3 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_296_finite__ne__induct,axiom,
! [F: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( F != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ X3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_297_finite__ne__induct,axiom,
! [F: set_int,P: set_int > $o] :
( ( finite_finite_int @ F )
=> ( ( F != bot_bot_set_int )
=> ( ! [X3: int] : ( P @ ( insert_int @ X3 @ bot_bot_set_int ) )
=> ( ! [X3: int,F2: set_int] :
( ( finite_finite_int @ F2 )
=> ( ( F2 != bot_bot_set_int )
=> ( ~ ( member_int @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_int @ X3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_298_finite__ne__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( F != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_299_finite__ne__induct,axiom,
! [F: set_real,P: set_real > $o] :
( ( finite_finite_real @ F )
=> ( ( F != bot_bot_set_real )
=> ( ! [X3: real] : ( P @ ( insert_real @ X3 @ bot_bot_set_real ) )
=> ( ! [X3: real,F2: set_real] :
( ( finite_finite_real @ F2 )
=> ( ( F2 != bot_bot_set_real )
=> ( ~ ( member_real @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_real @ X3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_300_finite__ne__induct,axiom,
! [F: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( F != bot_bot_set_set_a )
=> ( ! [X3: set_a] : ( P @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
=> ( ! [X3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( F2 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ X3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_301_finite__induct,axiom,
! [F: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ X3 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_302_finite__induct,axiom,
! [F: set_int,P: set_int > $o] :
( ( finite_finite_int @ F )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X3: int,F2: set_int] :
( ( finite_finite_int @ F2 )
=> ( ~ ( member_int @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_int @ X3 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_303_finite__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X3 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_304_finite__induct,axiom,
! [F: set_real,P: set_real > $o] :
( ( finite_finite_real @ F )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,F2: set_real] :
( ( finite_finite_real @ F2 )
=> ( ~ ( member_real @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_real @ X3 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_305_finite__induct,axiom,
! [F: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ~ ( member_set_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ X3 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_306_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A7: set_nat,B3: nat] :
( ( A4
= ( insert_nat @ B3 @ A7 ) )
& ( finite_finite_nat @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_307_finite_Osimps,axiom,
( finite_finite_int
= ( ^ [A4: set_int] :
( ( A4 = bot_bot_set_int )
| ? [A7: set_int,B3: int] :
( ( A4
= ( insert_int @ B3 @ A7 ) )
& ( finite_finite_int @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_308_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A7: set_a,B3: a] :
( ( A4
= ( insert_a @ B3 @ A7 ) )
& ( finite_finite_a @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_309_finite_Osimps,axiom,
( finite_finite_real
= ( ^ [A4: set_real] :
( ( A4 = bot_bot_set_real )
| ? [A7: set_real,B3: real] :
( ( A4
= ( insert_real @ B3 @ A7 ) )
& ( finite_finite_real @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_310_finite_Osimps,axiom,
( finite_finite_set_a
= ( ^ [A4: set_set_a] :
( ( A4 = bot_bot_set_set_a )
| ? [A7: set_set_a,B3: set_a] :
( ( A4
= ( insert_set_a @ B3 @ A7 ) )
& ( finite_finite_set_a @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_311_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A3: nat] :
( A
= ( insert_nat @ A3 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_312_finite_Ocases,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ~ ! [A6: set_int] :
( ? [A3: int] :
( A
= ( insert_int @ A3 @ A6 ) )
=> ~ ( finite_finite_int @ A6 ) ) ) ) ).
% finite.cases
thf(fact_313_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A6: set_a] :
( ? [A3: a] :
( A
= ( insert_a @ A3 @ A6 ) )
=> ~ ( finite_finite_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_314_finite_Ocases,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ~ ! [A6: set_real] :
( ? [A3: real] :
( A
= ( insert_real @ A3 @ A6 ) )
=> ~ ( finite_finite_real @ A6 ) ) ) ) ).
% finite.cases
thf(fact_315_finite_Ocases,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ~ ! [A6: set_set_a] :
( ? [A3: set_a] :
( A
= ( insert_set_a @ A3 @ A6 ) )
=> ~ ( finite_finite_set_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_316_subset__singleton__iff,axiom,
! [X4: set_real,A: real] :
( ( ord_less_eq_set_real @ X4 @ ( insert_real @ A @ bot_bot_set_real ) )
= ( ( X4 = bot_bot_set_real )
| ( X4
= ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% subset_singleton_iff
thf(fact_317_subset__singleton__iff,axiom,
! [X4: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( ( X4 = bot_bot_set_set_a )
| ( X4
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_318_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_319_subset__singletonD,axiom,
! [A2: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) )
=> ( ( A2 = bot_bot_set_real )
| ( A2
= ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).
% subset_singletonD
thf(fact_320_subset__singletonD,axiom,
! [A2: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
=> ( ( A2 = bot_bot_set_set_a )
| ( A2
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_321_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_322_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_323_singleton__Un__iff,axiom,
! [X: a,A2: set_a,B: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_324_singleton__Un__iff,axiom,
! [X: real,A2: set_real,B: set_real] :
( ( ( insert_real @ X @ bot_bot_set_real )
= ( sup_sup_set_real @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_real )
& ( B
= ( insert_real @ X @ bot_bot_set_real ) ) )
| ( ( A2
= ( insert_real @ X @ bot_bot_set_real ) )
& ( B = bot_bot_set_real ) )
| ( ( A2
= ( insert_real @ X @ bot_bot_set_real ) )
& ( B
= ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_325_singleton__Un__iff,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ( ( insert_set_a @ X @ bot_bot_set_set_a )
= ( sup_sup_set_set_a @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_set_a )
& ( B
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
| ( ( A2
= ( insert_set_a @ X @ bot_bot_set_set_a ) )
& ( B = bot_bot_set_set_a ) )
| ( ( A2
= ( insert_set_a @ X @ bot_bot_set_set_a ) )
& ( B
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_326_Un__singleton__iff,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ( sup_sup_set_a @ A2 @ B )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_327_Un__singleton__iff,axiom,
! [A2: set_real,B: set_real,X: real] :
( ( ( sup_sup_set_real @ A2 @ B )
= ( insert_real @ X @ bot_bot_set_real ) )
= ( ( ( A2 = bot_bot_set_real )
& ( B
= ( insert_real @ X @ bot_bot_set_real ) ) )
| ( ( A2
= ( insert_real @ X @ bot_bot_set_real ) )
& ( B = bot_bot_set_real ) )
| ( ( A2
= ( insert_real @ X @ bot_bot_set_real ) )
& ( B
= ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_328_Un__singleton__iff,axiom,
! [A2: set_set_a,B: set_set_a,X: set_a] :
( ( ( sup_sup_set_set_a @ A2 @ B )
= ( insert_set_a @ X @ bot_bot_set_set_a ) )
= ( ( ( A2 = bot_bot_set_set_a )
& ( B
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
| ( ( A2
= ( insert_set_a @ X @ bot_bot_set_set_a ) )
& ( B = bot_bot_set_set_a ) )
| ( ( A2
= ( insert_set_a @ X @ bot_bot_set_set_a ) )
& ( B
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_329_insert__is__Un,axiom,
( insert_a
= ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_330_insert__is__Un,axiom,
( insert_real
= ( ^ [A4: real] : ( sup_sup_set_real @ ( insert_real @ A4 @ bot_bot_set_real ) ) ) ) ).
% insert_is_Un
thf(fact_331_insert__is__Un,axiom,
( insert_set_a
= ( ^ [A4: set_a] : ( sup_sup_set_set_a @ ( insert_set_a @ A4 @ bot_bot_set_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_332_card__insert__le,axiom,
! [A2: set_set_a,X: set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ ( insert_set_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_333_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_334_card__insert__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_335_finite__subset__induct_H,axiom,
! [F: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( member_nat @ A3 @ A2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ~ ( member_nat @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_336_finite__subset__induct_H,axiom,
! [F: set_int,A2: set_int,P: set_int > $o] :
( ( finite_finite_int @ F )
=> ( ( ord_less_eq_set_int @ F @ A2 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A3: int,F2: set_int] :
( ( finite_finite_int @ F2 )
=> ( ( member_int @ A3 @ A2 )
=> ( ( ord_less_eq_set_int @ F2 @ A2 )
=> ( ~ ( member_int @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_int @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_337_finite__subset__induct_H,axiom,
! [F: set_real,A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F )
=> ( ( ord_less_eq_set_real @ F @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A3: real,F2: set_real] :
( ( finite_finite_real @ F2 )
=> ( ( member_real @ A3 @ A2 )
=> ( ( ord_less_eq_set_real @ F2 @ A2 )
=> ( ~ ( member_real @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_real @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_338_finite__subset__induct_H,axiom,
! [F: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( member_set_a @ A3 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ~ ( member_set_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_339_finite__subset__induct_H,axiom,
! [F: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A3 @ A2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ~ ( member_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_340_finite__subset__induct,axiom,
! [F: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( member_nat @ A3 @ A2 )
=> ( ~ ( member_nat @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_341_finite__subset__induct,axiom,
! [F: set_int,A2: set_int,P: set_int > $o] :
( ( finite_finite_int @ F )
=> ( ( ord_less_eq_set_int @ F @ A2 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A3: int,F2: set_int] :
( ( finite_finite_int @ F2 )
=> ( ( member_int @ A3 @ A2 )
=> ( ~ ( member_int @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_int @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_342_finite__subset__induct,axiom,
! [F: set_real,A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F )
=> ( ( ord_less_eq_set_real @ F @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A3: real,F2: set_real] :
( ( finite_finite_real @ F2 )
=> ( ( member_real @ A3 @ A2 )
=> ( ~ ( member_real @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_real @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_343_finite__subset__induct,axiom,
! [F: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( member_set_a @ A3 @ A2 )
=> ( ~ ( member_set_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_344_finite__subset__induct,axiom,
! [F: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A3 @ A2 )
=> ( ~ ( member_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_345_card__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_346_card__eq__0__iff,axiom,
! [A2: set_int] :
( ( ( finite_card_int @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_int )
| ~ ( finite_finite_int @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_347_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_348_card__eq__0__iff,axiom,
! [A2: set_real] :
( ( ( finite_card_real @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_real )
| ~ ( finite_finite_real @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_349_card__eq__0__iff,axiom,
! [A2: set_set_a] :
( ( ( finite_card_set_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_set_a )
| ~ ( finite_finite_set_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_350_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_351_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_352_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_353_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_354_ex__in__conv,axiom,
! [A2: set_real] :
( ( ? [X2: real] : ( member_real @ X2 @ A2 ) )
= ( A2 != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_355_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_356_equals0I,axiom,
! [A2: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_357_equals0I,axiom,
! [A2: set_real] :
( ! [Y2: real] :
~ ( member_real @ Y2 @ A2 )
=> ( A2 = bot_bot_set_real ) ) ).
% equals0I
thf(fact_358_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_359_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_360_equals0D,axiom,
! [A2: set_real,A: real] :
( ( A2 = bot_bot_set_real )
=> ~ ( member_real @ A @ A2 ) ) ).
% equals0D
thf(fact_361_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_362_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_363_emptyE,axiom,
! [A: real] :
~ ( member_real @ A @ bot_bot_set_real ) ).
% emptyE
thf(fact_364_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_365_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_366_Collect__mono__iff,axiom,
! [P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
= ( ! [X2: int] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_367_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_368_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A7: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ A7 @ B7 )
& ( ord_less_eq_set_a @ B7 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_369_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_370_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_371_Collect__mono,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_372_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_373_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_374_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A7: set_real,B7: set_real] :
! [T: real] :
( ( member_real @ T @ A7 )
=> ( member_real @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_375_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A7: set_set_a,B7: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A7 )
=> ( member_set_a @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_376_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B7: set_a] :
! [T: a] :
( ( member_a @ T @ A7 )
=> ( member_a @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_377_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_378_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_379_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A7: set_real,B7: set_real] :
! [X2: real] :
( ( member_real @ X2 @ A7 )
=> ( member_real @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_380_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A7: set_set_a,B7: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A7 )
=> ( member_set_a @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_381_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B7: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A7 )
=> ( member_a @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_382_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_383_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_384_subsetD,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_385_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_386_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_387_in__mono,axiom,
! [A2: set_set_a,B: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_388_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_389_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_390_sup_Oleft__commute,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( sup_sup_set_a @ B4 @ ( sup_sup_set_a @ A @ C2 ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_391_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y5: set_a] : ( sup_sup_set_a @ Y5 @ X2 ) ) ) ).
% sup_commute
thf(fact_392_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A4 ) ) ) ).
% sup.commute
thf(fact_393_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_394_sup_Oassoc,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B4 ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% sup.assoc
thf(fact_395_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y5: set_a] : ( sup_sup_set_a @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_396_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_397_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_398_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_399_Un__left__commute,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_400_Un__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_401_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A7: set_a,B7: set_a] : ( sup_sup_set_a @ B7 @ A7 ) ) ) ).
% Un_commute
thf(fact_402_Un__absorb,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_403_Un__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_assoc
thf(fact_404_ball__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_405_bex__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: a] :
( ( member_a @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_406_UnI2,axiom,
! [C2: real,B: set_real,A2: set_real] :
( ( member_real @ C2 @ B )
=> ( member_real @ C2 @ ( sup_sup_set_real @ A2 @ B ) ) ) ).
% UnI2
thf(fact_407_UnI2,axiom,
! [C2: set_a,B: set_set_a,A2: set_set_a] :
( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_408_UnI2,axiom,
! [C2: a,B: set_a,A2: set_a] :
( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_409_UnI1,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ ( sup_sup_set_real @ A2 @ B ) ) ) ).
% UnI1
thf(fact_410_UnI1,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_411_UnI1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_412_UnE,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( sup_sup_set_real @ A2 @ B ) )
=> ( ~ ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ B ) ) ) ).
% UnE
thf(fact_413_UnE,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) )
=> ( ~ ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_414_UnE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( ~ ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_415_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_416_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_417_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_418_finite__has__maximal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ord_less_eq_int @ A @ X3 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_419_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_420_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_421_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_422_finite__has__minimal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ord_less_eq_int @ X3 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_423_inf__sup__ord_I4_J,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_424_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_425_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_426_inf__sup__ord_I4_J,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_427_inf__sup__ord_I3_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sup_sup_real @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_428_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_429_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_430_inf__sup__ord_I3_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_431_le__supE,axiom,
! [A: real,B4: real,X: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ A @ B4 ) @ X )
=> ~ ( ( ord_less_eq_real @ A @ X )
=> ~ ( ord_less_eq_real @ B4 @ X ) ) ) ).
% le_supE
thf(fact_432_le__supE,axiom,
! [A: set_a,B4: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A @ X )
=> ~ ( ord_less_eq_set_a @ B4 @ X ) ) ) ).
% le_supE
thf(fact_433_le__supE,axiom,
! [A: nat,B4: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B4 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B4 @ X ) ) ) ).
% le_supE
thf(fact_434_le__supE,axiom,
! [A: int,B4: int,X: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ A @ B4 ) @ X )
=> ~ ( ( ord_less_eq_int @ A @ X )
=> ~ ( ord_less_eq_int @ B4 @ X ) ) ) ).
% le_supE
thf(fact_435_le__supI,axiom,
! [A: real,X: real,B4: real] :
( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ B4 @ X )
=> ( ord_less_eq_real @ ( sup_sup_real @ A @ B4 ) @ X ) ) ) ).
% le_supI
thf(fact_436_le__supI,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ( ord_less_eq_set_a @ B4 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ X ) ) ) ).
% le_supI
thf(fact_437_le__supI,axiom,
! [A: nat,X: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B4 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B4 ) @ X ) ) ) ).
% le_supI
thf(fact_438_le__supI,axiom,
! [A: int,X: int,B4: int] :
( ( ord_less_eq_int @ A @ X )
=> ( ( ord_less_eq_int @ B4 @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ A @ B4 ) @ X ) ) ) ).
% le_supI
thf(fact_439_sup__ge1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sup_sup_real @ X @ Y ) ) ).
% sup_ge1
thf(fact_440_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_441_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_442_sup__ge1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge1
thf(fact_443_sup__ge2,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X @ Y ) ) ).
% sup_ge2
thf(fact_444_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_445_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_446_sup__ge2,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge2
thf(fact_447_le__supI1,axiom,
! [X: real,A: real,B4: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ord_less_eq_real @ X @ ( sup_sup_real @ A @ B4 ) ) ) ).
% le_supI1
thf(fact_448_le__supI1,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% le_supI1
thf(fact_449_le__supI1,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% le_supI1
thf(fact_450_le__supI1,axiom,
! [X: int,A: int,B4: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B4 ) ) ) ).
% le_supI1
thf(fact_451_le__supI2,axiom,
! [X: real,B4: real,A: real] :
( ( ord_less_eq_real @ X @ B4 )
=> ( ord_less_eq_real @ X @ ( sup_sup_real @ A @ B4 ) ) ) ).
% le_supI2
thf(fact_452_le__supI2,axiom,
! [X: set_a,B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ B4 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% le_supI2
thf(fact_453_le__supI2,axiom,
! [X: nat,B4: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B4 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% le_supI2
thf(fact_454_le__supI2,axiom,
! [X: int,B4: int,A: int] :
( ( ord_less_eq_int @ X @ B4 )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B4 ) ) ) ).
% le_supI2
thf(fact_455_sup_Omono,axiom,
! [C2: real,A: real,D: real,B4: real] :
( ( ord_less_eq_real @ C2 @ A )
=> ( ( ord_less_eq_real @ D @ B4 )
=> ( ord_less_eq_real @ ( sup_sup_real @ C2 @ D ) @ ( sup_sup_real @ A @ B4 ) ) ) ) ).
% sup.mono
thf(fact_456_sup_Omono,axiom,
! [C2: set_a,A: set_a,D: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ D @ B4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C2 @ D ) @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% sup.mono
thf(fact_457_sup_Omono,axiom,
! [C2: nat,A: nat,D: nat,B4: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ( ord_less_eq_nat @ D @ B4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A @ B4 ) ) ) ) ).
% sup.mono
thf(fact_458_sup_Omono,axiom,
! [C2: int,A: int,D: int,B4: int] :
( ( ord_less_eq_int @ C2 @ A )
=> ( ( ord_less_eq_int @ D @ B4 )
=> ( ord_less_eq_int @ ( sup_sup_int @ C2 @ D ) @ ( sup_sup_int @ A @ B4 ) ) ) ) ).
% sup.mono
thf(fact_459_sup__mono,axiom,
! [A: real,C2: real,B4: real,D: real] :
( ( ord_less_eq_real @ A @ C2 )
=> ( ( ord_less_eq_real @ B4 @ D )
=> ( ord_less_eq_real @ ( sup_sup_real @ A @ B4 ) @ ( sup_sup_real @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_460_sup__mono,axiom,
! [A: set_a,C2: set_a,B4: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_461_sup__mono,axiom,
! [A: nat,C2: nat,B4: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B4 ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_462_sup__mono,axiom,
! [A: int,C2: int,B4: int,D: int] :
( ( ord_less_eq_int @ A @ C2 )
=> ( ( ord_less_eq_int @ B4 @ D )
=> ( ord_less_eq_int @ ( sup_sup_int @ A @ B4 ) @ ( sup_sup_int @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_463_sup__least,axiom,
! [Y: real,X: real,Z: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ Z @ X )
=> ( ord_less_eq_real @ ( sup_sup_real @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_464_sup__least,axiom,
! [Y: set_a,X: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_465_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_466_sup__least,axiom,
! [Y: int,X: int,Z: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ Z @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_467_le__iff__sup,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( sup_sup_real @ X2 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_468_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y5: set_a] :
( ( sup_sup_set_a @ X2 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_469_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( sup_sup_nat @ X2 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_470_le__iff__sup,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y5: int] :
( ( sup_sup_int @ X2 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_471_sup_OorderE,axiom,
! [B4: real,A: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( A
= ( sup_sup_real @ A @ B4 ) ) ) ).
% sup.orderE
thf(fact_472_sup_OorderE,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( A
= ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.orderE
thf(fact_473_sup_OorderE,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( A
= ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.orderE
thf(fact_474_sup_OorderE,axiom,
! [B4: int,A: int] :
( ( ord_less_eq_int @ B4 @ A )
=> ( A
= ( sup_sup_int @ A @ B4 ) ) ) ).
% sup.orderE
thf(fact_475_sup_OorderI,axiom,
! [A: real,B4: real] :
( ( A
= ( sup_sup_real @ A @ B4 ) )
=> ( ord_less_eq_real @ B4 @ A ) ) ).
% sup.orderI
thf(fact_476_sup_OorderI,axiom,
! [A: set_a,B4: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B4 ) )
=> ( ord_less_eq_set_a @ B4 @ A ) ) ).
% sup.orderI
thf(fact_477_sup_OorderI,axiom,
! [A: nat,B4: nat] :
( ( A
= ( sup_sup_nat @ A @ B4 ) )
=> ( ord_less_eq_nat @ B4 @ A ) ) ).
% sup.orderI
thf(fact_478_sup_OorderI,axiom,
! [A: int,B4: int] :
( ( A
= ( sup_sup_int @ A @ B4 ) )
=> ( ord_less_eq_int @ B4 @ A ) ) ).
% sup.orderI
thf(fact_479_sup__unique,axiom,
! [F3: real > real > real,X: real,Y: real] :
( ! [X3: real,Y2: real] : ( ord_less_eq_real @ X3 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: real,Y2: real] : ( ord_less_eq_real @ Y2 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ Y2 @ X3 )
=> ( ( ord_less_eq_real @ Z3 @ X3 )
=> ( ord_less_eq_real @ ( F3 @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_real @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_480_sup__unique,axiom,
! [F3: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X3 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ( ( ord_less_eq_set_a @ Z3 @ X3 )
=> ( ord_less_eq_set_a @ ( F3 @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_481_sup__unique,axiom,
! [F3: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ ( F3 @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_482_sup__unique,axiom,
! [F3: int > int > int,X: int,Y: int] :
( ! [X3: int,Y2: int] : ( ord_less_eq_int @ X3 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: int,Y2: int] : ( ord_less_eq_int @ Y2 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ Y2 @ X3 )
=> ( ( ord_less_eq_int @ Z3 @ X3 )
=> ( ord_less_eq_int @ ( F3 @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_int @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_483_sup_Oabsorb1,axiom,
! [B4: real,A: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( sup_sup_real @ A @ B4 )
= A ) ) ).
% sup.absorb1
thf(fact_484_sup_Oabsorb1,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( sup_sup_set_a @ A @ B4 )
= A ) ) ).
% sup.absorb1
thf(fact_485_sup_Oabsorb1,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( sup_sup_nat @ A @ B4 )
= A ) ) ).
% sup.absorb1
thf(fact_486_sup_Oabsorb1,axiom,
! [B4: int,A: int] :
( ( ord_less_eq_int @ B4 @ A )
=> ( ( sup_sup_int @ A @ B4 )
= A ) ) ).
% sup.absorb1
thf(fact_487_sup_Oabsorb2,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( sup_sup_real @ A @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_488_sup_Oabsorb2,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( sup_sup_set_a @ A @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_489_sup_Oabsorb2,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( sup_sup_nat @ A @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_490_sup_Oabsorb2,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( sup_sup_int @ A @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_491_sup__absorb1,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( sup_sup_real @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_492_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_493_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_494_sup__absorb1,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( sup_sup_int @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_495_sup__absorb2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( sup_sup_real @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_496_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_497_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_498_sup__absorb2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( sup_sup_int @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_499_sup_OboundedE,axiom,
! [B4: real,C2: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B4 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_real @ B4 @ A )
=> ~ ( ord_less_eq_real @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_500_sup_OboundedE,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B4 @ A )
=> ~ ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_501_sup_OboundedE,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_nat @ B4 @ A )
=> ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_502_sup_OboundedE,axiom,
! [B4: int,C2: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B4 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_int @ B4 @ A )
=> ~ ( ord_less_eq_int @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_503_sup_OboundedI,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( ord_less_eq_real @ C2 @ A )
=> ( ord_less_eq_real @ ( sup_sup_real @ B4 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_504_sup_OboundedI,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_505_sup_OboundedI,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_506_sup_OboundedI,axiom,
! [B4: int,A: int,C2: int] :
( ( ord_less_eq_int @ B4 @ A )
=> ( ( ord_less_eq_int @ C2 @ A )
=> ( ord_less_eq_int @ ( sup_sup_int @ B4 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_507_sup_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( A4
= ( sup_sup_real @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_508_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_509_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_510_sup_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( A4
= ( sup_sup_int @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_511_sup_Ocobounded1,axiom,
! [A: real,B4: real] : ( ord_less_eq_real @ A @ ( sup_sup_real @ A @ B4 ) ) ).
% sup.cobounded1
thf(fact_512_sup_Ocobounded1,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.cobounded1
thf(fact_513_sup_Ocobounded1,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B4 ) ) ).
% sup.cobounded1
thf(fact_514_sup_Ocobounded1,axiom,
! [A: int,B4: int] : ( ord_less_eq_int @ A @ ( sup_sup_int @ A @ B4 ) ) ).
% sup.cobounded1
thf(fact_515_sup_Ocobounded2,axiom,
! [B4: real,A: real] : ( ord_less_eq_real @ B4 @ ( sup_sup_real @ A @ B4 ) ) ).
% sup.cobounded2
thf(fact_516_sup_Ocobounded2,axiom,
! [B4: set_a,A: set_a] : ( ord_less_eq_set_a @ B4 @ ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.cobounded2
thf(fact_517_sup_Ocobounded2,axiom,
! [B4: nat,A: nat] : ( ord_less_eq_nat @ B4 @ ( sup_sup_nat @ A @ B4 ) ) ).
% sup.cobounded2
thf(fact_518_sup_Ocobounded2,axiom,
! [B4: int,A: int] : ( ord_less_eq_int @ B4 @ ( sup_sup_int @ A @ B4 ) ) ).
% sup.cobounded2
thf(fact_519_sup_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( sup_sup_real @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_520_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_521_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_522_sup_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( sup_sup_int @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_523_sup_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( sup_sup_real @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_524_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_525_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_526_sup_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( sup_sup_int @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_527_sup_OcoboundedI1,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_eq_real @ C2 @ A )
=> ( ord_less_eq_real @ C2 @ ( sup_sup_real @ A @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_528_sup_OcoboundedI1,axiom,
! [C2: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_529_sup_OcoboundedI1,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_530_sup_OcoboundedI1,axiom,
! [C2: int,A: int,B4: int] :
( ( ord_less_eq_int @ C2 @ A )
=> ( ord_less_eq_int @ C2 @ ( sup_sup_int @ A @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_531_sup_OcoboundedI2,axiom,
! [C2: real,B4: real,A: real] :
( ( ord_less_eq_real @ C2 @ B4 )
=> ( ord_less_eq_real @ C2 @ ( sup_sup_real @ A @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_532_sup_OcoboundedI2,axiom,
! [C2: set_a,B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C2 @ B4 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_533_sup_OcoboundedI2,axiom,
! [C2: nat,B4: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ B4 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_534_sup_OcoboundedI2,axiom,
! [C2: int,B4: int,A: int] :
( ( ord_less_eq_int @ C2 @ B4 )
=> ( ord_less_eq_int @ C2 @ ( sup_sup_int @ A @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_535_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_536_infinite__imp__nonempty,axiom,
! [S: set_int] :
( ~ ( finite_finite_int @ S )
=> ( S != bot_bot_set_int ) ) ).
% infinite_imp_nonempty
thf(fact_537_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_538_infinite__imp__nonempty,axiom,
! [S: set_real] :
( ~ ( finite_finite_real @ S )
=> ( S != bot_bot_set_real ) ) ).
% infinite_imp_nonempty
thf(fact_539_infinite__imp__nonempty,axiom,
! [S: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ( S != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_540_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_541_finite_OemptyI,axiom,
finite_finite_int @ bot_bot_set_int ).
% finite.emptyI
thf(fact_542_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_543_finite_OemptyI,axiom,
finite_finite_real @ bot_bot_set_real ).
% finite.emptyI
thf(fact_544_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_545_rev__finite__subset,axiom,
! [B: set_real,A2: set_real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A2 @ B )
=> ( finite_finite_real @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_546_rev__finite__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_547_rev__finite__subset,axiom,
! [B: set_int,A2: set_int] :
( ( finite_finite_int @ B )
=> ( ( ord_less_eq_set_int @ A2 @ B )
=> ( finite_finite_int @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_548_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_549_infinite__super,axiom,
! [S: set_real,T2: set_real] :
( ( ord_less_eq_set_real @ S @ T2 )
=> ( ~ ( finite_finite_real @ S )
=> ~ ( finite_finite_real @ T2 ) ) ) ).
% infinite_super
thf(fact_550_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_551_infinite__super,axiom,
! [S: set_int,T2: set_int] :
( ( ord_less_eq_set_int @ S @ T2 )
=> ( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ T2 ) ) ) ).
% infinite_super
thf(fact_552_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_553_finite__subset,axiom,
! [A2: set_real,B: set_real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( finite_finite_real @ B )
=> ( finite_finite_real @ A2 ) ) ) ).
% finite_subset
thf(fact_554_finite__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_555_finite__subset,axiom,
! [A2: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A2 @ B )
=> ( ( finite_finite_int @ B )
=> ( finite_finite_int @ A2 ) ) ) ).
% finite_subset
thf(fact_556_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_557_Un__empty__right,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_558_Un__empty__right,axiom,
! [A2: set_real] :
( ( sup_sup_set_real @ A2 @ bot_bot_set_real )
= A2 ) ).
% Un_empty_right
thf(fact_559_Un__empty__right,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ bot_bot_set_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_560_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_561_Un__empty__left,axiom,
! [B: set_real] :
( ( sup_sup_set_real @ bot_bot_set_real @ B )
= B ) ).
% Un_empty_left
thf(fact_562_Un__empty__left,axiom,
! [B: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_563_infinite__Un,axiom,
! [S: set_real,T2: set_real] :
( ( ~ ( finite_finite_real @ ( sup_sup_set_real @ S @ T2 ) ) )
= ( ~ ( finite_finite_real @ S )
| ~ ( finite_finite_real @ T2 ) ) ) ).
% infinite_Un
thf(fact_564_infinite__Un,axiom,
! [S: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_565_infinite__Un,axiom,
! [S: set_int,T2: set_int] :
( ( ~ ( finite_finite_int @ ( sup_sup_set_int @ S @ T2 ) ) )
= ( ~ ( finite_finite_int @ S )
| ~ ( finite_finite_int @ T2 ) ) ) ).
% infinite_Un
thf(fact_566_infinite__Un,axiom,
! [S: set_a,T2: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_567_Un__infinite,axiom,
! [S: set_real,T2: set_real] :
( ~ ( finite_finite_real @ S )
=> ~ ( finite_finite_real @ ( sup_sup_set_real @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_568_Un__infinite,axiom,
! [S: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_569_Un__infinite,axiom,
! [S: set_int,T2: set_int] :
( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ ( sup_sup_set_int @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_570_Un__infinite,axiom,
! [S: set_a,T2: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_571_finite__UnI,axiom,
! [F: set_real,G: set_real] :
( ( finite_finite_real @ F )
=> ( ( finite_finite_real @ G )
=> ( finite_finite_real @ ( sup_sup_set_real @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_572_finite__UnI,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ F )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_573_finite__UnI,axiom,
! [F: set_int,G: set_int] :
( ( finite_finite_int @ F )
=> ( ( finite_finite_int @ G )
=> ( finite_finite_int @ ( sup_sup_set_int @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_574_finite__UnI,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ F )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_575_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B7: set_a] :
( ( sup_sup_set_a @ A7 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_576_subset__UnE,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
=> ~ ! [A8: set_a] :
( ( ord_less_eq_set_a @ A8 @ A2 )
=> ! [B8: set_a] :
( ( ord_less_eq_set_a @ B8 @ B )
=> ( C
!= ( sup_sup_set_a @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_577_Un__absorb2,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_578_Un__absorb1,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_579_Un__upper2,axiom,
! [B: set_a,A2: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper2
thf(fact_580_Un__upper1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper1
thf(fact_581_Un__least,axiom,
! [A2: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_582_Un__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_583_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_584_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_585_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_586_finite__has__minimal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_587_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_588_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_589_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_590_finite__has__maximal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_591_infinite__arbitrarily__large,axiom,
! [A2: set_real,N: nat] :
( ~ ( finite_finite_real @ A2 )
=> ? [B6: set_real] :
( ( finite_finite_real @ B6 )
& ( ( finite_card_real @ B6 )
= N )
& ( ord_less_eq_set_real @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_592_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ( finite_card_nat @ B6 )
= N )
& ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_593_infinite__arbitrarily__large,axiom,
! [A2: set_int,N: nat] :
( ~ ( finite_finite_int @ A2 )
=> ? [B6: set_int] :
( ( finite_finite_int @ B6 )
& ( ( finite_card_int @ B6 )
= N )
& ( ord_less_eq_set_int @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_594_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ( finite_card_a @ B6 )
= N )
& ( ord_less_eq_set_a @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_595_card__subset__eq,axiom,
! [B: set_real,A2: set_real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( ( finite_card_real @ A2 )
= ( finite_card_real @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_596_card__subset__eq,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_597_card__subset__eq,axiom,
! [B: set_int,A2: set_int] :
( ( finite_finite_int @ B )
=> ( ( ord_less_eq_set_int @ A2 @ B )
=> ( ( ( finite_card_int @ A2 )
= ( finite_card_int @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_598_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_599_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_real,C: nat] :
( ! [G2: set_real] :
( ( ord_less_eq_set_real @ G2 @ F )
=> ( ( finite_finite_real @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_real @ G2 ) @ C ) ) )
=> ( ( finite_finite_real @ F )
& ( ord_less_eq_nat @ ( finite_card_real @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_600_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_nat,C: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C ) ) )
=> ( ( finite_finite_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_601_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_int,C: nat] :
( ! [G2: set_int] :
( ( ord_less_eq_set_int @ G2 @ F )
=> ( ( finite_finite_int @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_int @ G2 ) @ C ) ) )
=> ( ( finite_finite_int @ F )
& ( ord_less_eq_nat @ ( finite_card_int @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_602_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_a,C: nat] :
( ! [G2: set_a] :
( ( ord_less_eq_set_a @ G2 @ F )
=> ( ( finite_finite_a @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G2 ) @ C ) ) )
=> ( ( finite_finite_a @ F )
& ( ord_less_eq_nat @ ( finite_card_a @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_603_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_real] :
( ( ord_less_eq_nat @ N @ ( finite_card_real @ S ) )
=> ~ ! [T3: set_real] :
( ( ord_less_eq_set_real @ T3 @ S )
=> ( ( ( finite_card_real @ T3 )
= N )
=> ~ ( finite_finite_real @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_604_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_605_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_int] :
( ( ord_less_eq_nat @ N @ ( finite_card_int @ S ) )
=> ~ ! [T3: set_int] :
( ( ord_less_eq_set_int @ T3 @ S )
=> ( ( ( finite_card_int @ T3 )
= N )
=> ~ ( finite_finite_int @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_606_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S )
=> ( ( ( finite_card_a @ T3 )
= N )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_607_exists__subset__between,axiom,
! [A2: set_real,N: nat,C: set_real] :
( ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_real @ C ) )
=> ( ( ord_less_eq_set_real @ A2 @ C )
=> ( ( finite_finite_real @ C )
=> ? [B6: set_real] :
( ( ord_less_eq_set_real @ A2 @ B6 )
& ( ord_less_eq_set_real @ B6 @ C )
& ( ( finite_card_real @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_608_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B6: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ C )
& ( ( finite_card_nat @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_609_exists__subset__between,axiom,
! [A2: set_int,N: nat,C: set_int] :
( ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_int @ C ) )
=> ( ( ord_less_eq_set_int @ A2 @ C )
=> ( ( finite_finite_int @ C )
=> ? [B6: set_int] :
( ( ord_less_eq_set_int @ A2 @ B6 )
& ( ord_less_eq_set_int @ B6 @ C )
& ( ( finite_card_int @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_610_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 )
=> ? [B6: set_a] :
( ( ord_less_eq_set_a @ A2 @ B6 )
& ( ord_less_eq_set_a @ B6 @ C )
& ( ( finite_card_a @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_611_card__seteq,axiom,
! [B: set_real,A2: set_real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_real @ B ) @ ( finite_card_real @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_612_card__seteq,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_613_card__seteq,axiom,
! [B: set_int,A2: set_int] :
( ( finite_finite_int @ B )
=> ( ( ord_less_eq_set_int @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ B ) @ ( finite_card_int @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_614_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_615_card__mono,axiom,
! [B: set_real,A2: set_real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B ) ) ) ) ).
% card_mono
thf(fact_616_card__mono,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_617_card__mono,axiom,
! [B: set_int,A2: set_int] :
( ( finite_finite_int @ B )
=> ( ( ord_less_eq_set_int @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B ) ) ) ) ).
% card_mono
thf(fact_618_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_619_sumset__insert1,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ A2 )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_insert1
thf(fact_620_sumset__insert1,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( nO_MAT3201932972334532047_set_a @ bot_bot_set_real @ A2 )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_insert1
thf(fact_621_sumset__insert1,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( nO_MAT8518843428946182283_set_a @ bot_bot_set_set_a @ A2 )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ bot_bot_set_a ) @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_insert1
thf(fact_622_sumset__insert2,axiom,
! [B: set_a,A2: set_a,X: a] :
( ( nO_MATCH_set_a_set_a @ bot_bot_set_a @ B )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_insert2
thf(fact_623_sumset__insert2,axiom,
! [B: set_a,A2: set_a,X: a] :
( ( nO_MAT3201932972334532047_set_a @ bot_bot_set_real @ B )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_insert2
thf(fact_624_sumset__insert2,axiom,
! [B: set_a,A2: set_a,X: a] :
( ( nO_MAT8518843428946182283_set_a @ bot_bot_set_set_a @ B )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_insert2
thf(fact_625_Min_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_626_Min_Obounded__iff,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A2 ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_int @ X @ X2 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_627_Min_Obounded__iff,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ X @ ( lattic3629708407755379051n_real @ A2 ) )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_real @ X @ X2 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_628_Min__singleton,axiom,
! [X: real] :
( ( lattic3629708407755379051n_real @ ( insert_real @ X @ bot_bot_set_real ) )
= X ) ).
% Min_singleton
thf(fact_629_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_630_div__mult__mult1__if,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ( C2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) )
= zero_zero_nat ) )
& ( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) )
= ( divide_divide_nat @ A @ B4 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_631_div__mult__mult1__if,axiom,
! [C2: int,A: int,B4: int] :
( ( ( C2 = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) )
= zero_zero_int ) )
& ( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) )
= ( divide_divide_int @ A @ B4 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_632_div__mult__mult2,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) )
= ( divide_divide_nat @ A @ B4 ) ) ) ).
% div_mult_mult2
thf(fact_633_div__mult__mult2,axiom,
! [C2: int,A: int,B4: int] :
( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ C2 ) )
= ( divide_divide_int @ A @ B4 ) ) ) ).
% div_mult_mult2
thf(fact_634_div__mult__mult1,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) )
= ( divide_divide_nat @ A @ B4 ) ) ) ).
% div_mult_mult1
thf(fact_635_div__mult__mult1,axiom,
! [C2: int,A: int,B4: int] :
( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) )
= ( divide_divide_int @ A @ B4 ) ) ) ).
% div_mult_mult1
thf(fact_636_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C2: real,A: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ C2 @ B4 ) )
= ( divide_divide_real @ A @ B4 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_637_nonzero__mult__div__cancel__right,axiom,
! [B4: real,A: real] :
( ( B4 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B4 ) @ B4 )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_638_nonzero__mult__div__cancel__right,axiom,
! [B4: nat,A: nat] :
( ( B4 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B4 ) @ B4 )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_639_nonzero__mult__div__cancel__right,axiom,
! [B4: int,A: int] :
( ( B4 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B4 ) @ B4 )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_640_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_641_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_642_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_643_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_644_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_645_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_646_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_647_Int__iff,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
= ( ( member_set_a @ C2 @ A2 )
& ( member_set_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_648_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_649_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_650_IntI,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_651_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_652_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_653_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_654_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_655_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_656_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_657_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_658_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_659_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_660_mult__cancel__right,axiom,
! [A: int,C2: int,B4: int] :
( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B4 @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A = B4 ) ) ) ).
% mult_cancel_right
thf(fact_661_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_662_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_663_mult__cancel__left,axiom,
! [C2: int,A: int,B4: int] :
( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B4 ) )
= ( ( C2 = zero_zero_int )
| ( A = B4 ) ) ) ).
% mult_cancel_left
thf(fact_664_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_665_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_666_mult__eq__0__iff,axiom,
! [A: int,B4: int] :
( ( ( times_times_int @ A @ B4 )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B4 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_667_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_668_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_669_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_670_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_671_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_672_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_673_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_674_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_675_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_676_divide__eq__0__iff,axiom,
! [A: real,B4: real] :
( ( ( divide_divide_real @ A @ B4 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B4 = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_677_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_678_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_679_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_680_divide__cancel__left,axiom,
! [C2: real,A: real,B4: real] :
( ( ( divide_divide_real @ C2 @ A )
= ( divide_divide_real @ C2 @ B4 ) )
= ( ( C2 = zero_zero_real )
| ( A = B4 ) ) ) ).
% divide_cancel_left
thf(fact_681_divide__cancel__right,axiom,
! [A: real,C2: real,B4: real] :
( ( ( divide_divide_real @ A @ C2 )
= ( divide_divide_real @ B4 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A = B4 ) ) ) ).
% divide_cancel_right
thf(fact_682_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_683_times__divide__eq__right,axiom,
! [A: real,B4: real,C2: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B4 @ C2 ) )
= ( divide_divide_real @ ( times_times_real @ A @ B4 ) @ C2 ) ) ).
% times_divide_eq_right
thf(fact_684_divide__divide__eq__right,axiom,
! [A: real,B4: real,C2: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B4 @ C2 ) )
= ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ B4 ) ) ).
% divide_divide_eq_right
thf(fact_685_divide__divide__eq__left,axiom,
! [A: real,B4: real,C2: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B4 ) @ C2 )
= ( divide_divide_real @ A @ ( times_times_real @ B4 @ C2 ) ) ) ).
% divide_divide_eq_left
thf(fact_686_times__divide__eq__left,axiom,
! [B4: real,C2: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B4 @ C2 ) @ A )
= ( divide_divide_real @ ( times_times_real @ B4 @ A ) @ C2 ) ) ).
% times_divide_eq_left
thf(fact_687_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_688_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_689_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_690_inf_Obounded__iff,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B4 @ C2 ) )
= ( ( ord_less_eq_int @ A @ B4 )
& ( ord_less_eq_int @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_691_le__inf__iff,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z ) )
= ( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_692_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_693_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_694_le__inf__iff,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) )
= ( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_695_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_696_inf__bot__left,axiom,
! [X: set_real] :
( ( inf_inf_set_real @ bot_bot_set_real @ X )
= bot_bot_set_real ) ).
% inf_bot_left
thf(fact_697_inf__bot__left,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X )
= bot_bot_set_set_a ) ).
% inf_bot_left
thf(fact_698_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_699_inf__bot__right,axiom,
! [X: set_real] :
( ( inf_inf_set_real @ X @ bot_bot_set_real )
= bot_bot_set_real ) ).
% inf_bot_right
thf(fact_700_inf__bot__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% inf_bot_right
thf(fact_701_finite__Int,axiom,
! [F: set_real,G: set_real] :
( ( ( finite_finite_real @ F )
| ( finite_finite_real @ G ) )
=> ( finite_finite_real @ ( inf_inf_set_real @ F @ G ) ) ) ).
% finite_Int
thf(fact_702_finite__Int,axiom,
! [F: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F @ G ) ) ) ).
% finite_Int
thf(fact_703_finite__Int,axiom,
! [F: set_int,G: set_int] :
( ( ( finite_finite_int @ F )
| ( finite_finite_int @ G ) )
=> ( finite_finite_int @ ( inf_inf_set_int @ F @ G ) ) ) ).
% finite_Int
thf(fact_704_finite__Int,axiom,
! [F: set_a,G: set_a] :
( ( ( finite_finite_a @ F )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F @ G ) ) ) ).
% finite_Int
thf(fact_705_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_706_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_707_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_708_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_709_Int__insert__left__if0,axiom,
! [A: set_a,C: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C )
= ( inf_inf_set_set_a @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_710_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_711_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_712_Int__insert__left__if1,axiom,
! [A: set_a,C: set_set_a,B: set_set_a] :
( ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_713_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_714_insert__inter__insert,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_715_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_716_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_717_Int__insert__right__if0,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_718_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_719_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_720_Int__insert__right__if1,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_721_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_722_Int__Un__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( sup_sup_set_a @ T2 @ ( inf_inf_set_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_723_Int__Un__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_724_Int__Un__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_725_Int__Un__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_726_Un__Int__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( inf_inf_set_a @ T2 @ ( sup_sup_set_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_727_Un__Int__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_728_Un__Int__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_729_Un__Int__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_730_mult__divide__mult__cancel__left__if,axiom,
! [C2: real,A: real,B4: real] :
( ( ( C2 = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) )
= zero_zero_real ) )
& ( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) )
= ( divide_divide_real @ A @ B4 ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_731_nonzero__mult__divide__mult__cancel__left,axiom,
! [C2: real,A: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) )
= ( divide_divide_real @ A @ B4 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_732_nonzero__mult__div__cancel__left,axiom,
! [A: real,B4: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B4 ) @ A )
= B4 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_733_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B4: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B4 ) @ A )
= B4 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_734_nonzero__mult__div__cancel__left,axiom,
! [A: int,B4: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B4 ) @ A )
= B4 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_735_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C2: real,A: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ B4 @ C2 ) )
= ( divide_divide_real @ A @ B4 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_736_nonzero__mult__divide__mult__cancel__right,axiom,
! [C2: real,A: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) )
= ( divide_divide_real @ A @ B4 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_737_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_738_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_739_insert__disjoint_I1_J,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B )
& ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_740_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_741_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_742_insert__disjoint_I2_J,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B ) )
= ( ~ ( member_set_a @ A @ B )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_743_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_744_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_745_disjoint__insert_I1_J,axiom,
! [B: set_set_a,A: set_a,A2: set_set_a] :
( ( ( inf_inf_set_set_a @ B @ ( insert_set_a @ A @ A2 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B )
& ( ( inf_inf_set_set_a @ B @ A2 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_746_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_747_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_748_disjoint__insert_I2_J,axiom,
! [A2: set_set_a,B4: set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ B4 @ B ) ) )
= ( ~ ( member_set_a @ B4 @ A2 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_749_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_750_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_751_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_752_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_753_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_754_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_755_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A7: set_a,B7: set_a] : ( inf_inf_set_a @ B7 @ A7 ) ) ) ).
% Int_commute
thf(fact_756_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_757_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_758_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_759_IntD2,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C2 @ B ) ) ).
% IntD2
thf(fact_760_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_761_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_762_IntD1,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_763_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_764_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_765_IntE,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ~ ( ( member_set_a @ C2 @ A2 )
=> ~ ( member_set_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_766_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_767_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_768_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_769_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y5: set_a] : ( inf_inf_set_a @ Y5 @ X2 ) ) ) ).
% inf_commute
thf(fact_770_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_771_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_772_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_773_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_774_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_775_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_776_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_777_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_778_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_779_div__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% div_mult2_eq
thf(fact_780_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_781_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_782_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_783_inf_OcoboundedI2,axiom,
! [B4: int,C2: int,A: int] :
( ( ord_less_eq_int @ B4 @ C2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_784_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_785_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_786_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_787_inf_OcoboundedI1,axiom,
! [A: int,C2: int,B4: int] :
( ( ord_less_eq_int @ A @ C2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_788_inf_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( inf_inf_real @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_789_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_790_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_791_inf_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( inf_inf_int @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_792_inf_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( inf_inf_real @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_793_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_794_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_795_inf_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( inf_inf_int @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_796_inf_Ocobounded2,axiom,
! [A: real,B4: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_797_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_798_inf_Ocobounded2,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_799_inf_Ocobounded2,axiom,
! [A: int,B4: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_800_inf_Ocobounded1,axiom,
! [A: real,B4: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_801_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_802_inf_Ocobounded1,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_803_inf_Ocobounded1,axiom,
! [A: int,B4: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_804_inf_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( A4
= ( inf_inf_real @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_805_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_806_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( A4
= ( inf_inf_nat @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_807_inf_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( A4
= ( inf_inf_int @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_808_inf__greatest,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Z )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_809_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_810_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_811_inf__greatest,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Z )
=> ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_812_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_813_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_814_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_815_inf_OboundedI,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( ord_less_eq_int @ A @ C2 )
=> ( ord_less_eq_int @ A @ ( inf_inf_int @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_816_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_817_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_818_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_819_inf_OboundedE,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_int @ A @ B4 )
=> ~ ( ord_less_eq_int @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_820_inf__absorb2,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( inf_inf_real @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_821_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_822_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_823_inf__absorb2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( inf_inf_int @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_824_inf__absorb1,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( inf_inf_real @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_825_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_826_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_827_inf__absorb1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( inf_inf_int @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_828_inf_Oabsorb2,axiom,
! [B4: real,A: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( inf_inf_real @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_829_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_830_inf_Oabsorb2,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( inf_inf_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_831_inf_Oabsorb2,axiom,
! [B4: int,A: int] :
( ( ord_less_eq_int @ B4 @ A )
=> ( ( inf_inf_int @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_832_inf_Oabsorb1,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( inf_inf_real @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_833_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_834_inf_Oabsorb1,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( inf_inf_nat @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_835_inf_Oabsorb1,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( inf_inf_int @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_836_le__iff__inf,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( inf_inf_real @ X2 @ Y5 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_837_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_838_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( inf_inf_nat @ X2 @ Y5 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_839_le__iff__inf,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y5: int] :
( ( inf_inf_int @ X2 @ Y5 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_840_inf__unique,axiom,
! [F3: real > real > real,X: real,Y: real] :
( ! [X3: real,Y2: real] : ( ord_less_eq_real @ ( F3 @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: real,Y2: real] : ( ord_less_eq_real @ ( F3 @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ord_less_eq_real @ X3 @ Z3 )
=> ( ord_less_eq_real @ X3 @ ( F3 @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_real @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_841_inf__unique,axiom,
! [F3: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F3 @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F3 @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a @ X3 @ Z3 )
=> ( ord_less_eq_set_a @ X3 @ ( F3 @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_842_inf__unique,axiom,
! [F3: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F3 @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F3 @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ ( F3 @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_843_inf__unique,axiom,
! [F3: int > int > int,X: int,Y: int] :
( ! [X3: int,Y2: int] : ( ord_less_eq_int @ ( F3 @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: int,Y2: int] : ( ord_less_eq_int @ ( F3 @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ( ord_less_eq_int @ X3 @ Z3 )
=> ( ord_less_eq_int @ X3 @ ( F3 @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_int @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_844_inf_OorderI,axiom,
! [A: real,B4: real] :
( ( A
= ( inf_inf_real @ A @ B4 ) )
=> ( ord_less_eq_real @ A @ B4 ) ) ).
% inf.orderI
thf(fact_845_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_846_inf_OorderI,axiom,
! [A: nat,B4: nat] :
( ( A
= ( inf_inf_nat @ A @ B4 ) )
=> ( ord_less_eq_nat @ A @ B4 ) ) ).
% inf.orderI
thf(fact_847_inf_OorderI,axiom,
! [A: int,B4: int] :
( ( A
= ( inf_inf_int @ A @ B4 ) )
=> ( ord_less_eq_int @ A @ B4 ) ) ).
% inf.orderI
thf(fact_848_inf_OorderE,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( A
= ( inf_inf_real @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_849_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_850_inf_OorderE,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( A
= ( inf_inf_nat @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_851_inf_OorderE,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( A
= ( inf_inf_int @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_852_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_853_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_854_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_855_le__infI2,axiom,
! [B4: int,X: int,A: int] :
( ( ord_less_eq_int @ B4 @ X )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_856_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_857_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_858_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_859_le__infI1,axiom,
! [A: int,X: int,B4: int] :
( ( ord_less_eq_int @ A @ X )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_860_inf__mono,axiom,
! [A: real,C2: real,B4: real,D: real] :
( ( ord_less_eq_real @ A @ C2 )
=> ( ( ord_less_eq_real @ B4 @ D )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ ( inf_inf_real @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_861_inf__mono,axiom,
! [A: set_a,C2: set_a,B4: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_862_inf__mono,axiom,
! [A: nat,C2: nat,B4: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_863_inf__mono,axiom,
! [A: int,C2: int,B4: int,D: int] :
( ( ord_less_eq_int @ A @ C2 )
=> ( ( ord_less_eq_int @ B4 @ D )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B4 ) @ ( inf_inf_int @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_864_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_865_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_866_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_867_le__infI,axiom,
! [X: int,A: int,B4: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ X @ B4 )
=> ( ord_less_eq_int @ X @ ( inf_inf_int @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_868_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_869_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_870_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_871_le__infE,axiom,
! [X: int,A: int,B4: int] :
( ( ord_less_eq_int @ X @ ( inf_inf_int @ A @ B4 ) )
=> ~ ( ( ord_less_eq_int @ X @ A )
=> ~ ( ord_less_eq_int @ X @ B4 ) ) ) ).
% le_infE
thf(fact_872_inf__le2,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_873_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_874_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_875_inf__le2,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_876_inf__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_877_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_878_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_879_inf__le1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_880_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_881_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_882_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_883_inf__sup__ord_I1_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_884_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_885_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_886_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_887_inf__sup__ord_I2_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_888_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X3: set_a,Y2: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y2 ) @ ( inf_inf_set_a @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_889_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X3: set_a,Y2: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y2 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ ( sup_sup_set_a @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_890_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_891_inf__sup__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_892_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_893_sup__inf__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_894_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_895_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_896_Int__emptyI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ~ ( member_set_a @ X3 @ B ) )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_897_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_898_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_899_disjoint__iff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ~ ( member_set_a @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_900_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_901_Int__empty__left,axiom,
! [B: set_real] :
( ( inf_inf_set_real @ bot_bot_set_real @ B )
= bot_bot_set_real ) ).
% Int_empty_left
thf(fact_902_Int__empty__left,axiom,
! [B: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B )
= bot_bot_set_set_a ) ).
% Int_empty_left
thf(fact_903_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_904_Int__empty__right,axiom,
! [A2: set_real] :
( ( inf_inf_set_real @ A2 @ bot_bot_set_real )
= bot_bot_set_real ) ).
% Int_empty_right
thf(fact_905_Int__empty__right,axiom,
! [A2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% Int_empty_right
thf(fact_906_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_907_disjoint__iff__not__equal,axiom,
! [A2: set_real,B: set_real] :
( ( ( inf_inf_set_real @ A2 @ B )
= bot_bot_set_real )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ! [Y5: real] :
( ( member_real @ Y5 @ B )
=> ( X2 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_908_disjoint__iff__not__equal,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ! [Y5: set_a] :
( ( member_set_a @ Y5 @ B )
=> ( X2 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_909_Int__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_910_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_911_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_912_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_913_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_914_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_915_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_916_Int__Collect__mono,axiom,
! [A2: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_917_Int__Collect__mono,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_918_Int__Collect__mono,axiom,
! [A2: set_int,B: set_int,P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ A2 @ B )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) @ ( inf_inf_set_int @ B @ ( collect_int @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_919_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_920_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_921_Int__insert__left,axiom,
! [A: set_a,C: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B @ C ) ) ) )
& ( ~ ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C )
= ( inf_inf_set_set_a @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_922_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_923_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_924_Int__insert__right,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_925_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_926_Un__Int__crazy,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ B @ C ) ) @ ( inf_inf_set_a @ C @ A2 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ B @ C ) ) @ ( sup_sup_set_a @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_927_Int__Un__distrib,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_928_Un__Int__distrib,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_929_Int__Un__distrib2,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C ) @ A2 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A2 ) @ ( inf_inf_set_a @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_930_Un__Int__distrib2,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C ) @ A2 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A2 ) @ ( sup_sup_set_a @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_931_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_932_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_933_distrib__inf__le,axiom,
! [X: real,Y: real,Z: real] : ( ord_less_eq_real @ ( sup_sup_real @ ( inf_inf_real @ X @ Y ) @ ( inf_inf_real @ X @ Z ) ) @ ( inf_inf_real @ X @ ( sup_sup_real @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_934_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_935_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_936_distrib__inf__le,axiom,
! [X: int,Y: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ ( inf_inf_int @ X @ Y ) @ ( inf_inf_int @ X @ Z ) ) @ ( inf_inf_int @ X @ ( sup_sup_int @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_937_distrib__sup__le,axiom,
! [X: real,Y: real,Z: real] : ( ord_less_eq_real @ ( sup_sup_real @ X @ ( inf_inf_real @ Y @ Z ) ) @ ( inf_inf_real @ ( sup_sup_real @ X @ Y ) @ ( sup_sup_real @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_938_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_939_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_940_distrib__sup__le,axiom,
! [X: int,Y: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ X @ ( inf_inf_int @ Y @ Z ) ) @ ( inf_inf_int @ ( sup_sup_int @ X @ Y ) @ ( sup_sup_int @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_941_Un__Int__assoc__eq,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) )
= ( ord_less_eq_set_a @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_942_mult__right__cancel,axiom,
! [C2: real,A: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B4 @ C2 ) )
= ( A = B4 ) ) ) ).
% mult_right_cancel
thf(fact_943_mult__right__cancel,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B4 @ C2 ) )
= ( A = B4 ) ) ) ).
% mult_right_cancel
thf(fact_944_mult__right__cancel,axiom,
! [C2: int,A: int,B4: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B4 @ C2 ) )
= ( A = B4 ) ) ) ).
% mult_right_cancel
thf(fact_945_mult__left__cancel,axiom,
! [C2: real,A: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B4 ) )
= ( A = B4 ) ) ) ).
% mult_left_cancel
thf(fact_946_mult__left__cancel,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B4 ) )
= ( A = B4 ) ) ) ).
% mult_left_cancel
thf(fact_947_mult__left__cancel,axiom,
! [C2: int,A: int,B4: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B4 ) )
= ( A = B4 ) ) ) ).
% mult_left_cancel
thf(fact_948_no__zero__divisors,axiom,
! [A: real,B4: real] :
( ( A != zero_zero_real )
=> ( ( B4 != zero_zero_real )
=> ( ( times_times_real @ A @ B4 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_949_no__zero__divisors,axiom,
! [A: nat,B4: nat] :
( ( A != zero_zero_nat )
=> ( ( B4 != zero_zero_nat )
=> ( ( times_times_nat @ A @ B4 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_950_no__zero__divisors,axiom,
! [A: int,B4: int] :
( ( A != zero_zero_int )
=> ( ( B4 != zero_zero_int )
=> ( ( times_times_int @ A @ B4 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_951_divisors__zero,axiom,
! [A: real,B4: real] :
( ( ( times_times_real @ A @ B4 )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B4 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_952_divisors__zero,axiom,
! [A: nat,B4: nat] :
( ( ( times_times_nat @ A @ B4 )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B4 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_953_divisors__zero,axiom,
! [A: int,B4: int] :
( ( ( times_times_int @ A @ B4 )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B4 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_954_mult__not__zero,axiom,
! [A: real,B4: real] :
( ( ( times_times_real @ A @ B4 )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B4 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_955_mult__not__zero,axiom,
! [A: nat,B4: nat] :
( ( ( times_times_nat @ A @ B4 )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B4 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_956_mult__not__zero,axiom,
! [A: int,B4: int] :
( ( ( times_times_int @ A @ B4 )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B4 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_957_divide__divide__eq__left_H,axiom,
! [A: real,B4: real,C2: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B4 ) @ C2 )
= ( divide_divide_real @ A @ ( times_times_real @ C2 @ B4 ) ) ) ).
% divide_divide_eq_left'
thf(fact_958_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_959_times__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_960_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_961_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_962_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_963_zero__le__mult__iff,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B4 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_964_zero__le__mult__iff,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B4 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B4 ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B4 @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_965_mult__nonneg__nonpos2,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B4 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B4 @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_966_mult__nonneg__nonpos2,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B4 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B4 @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_967_mult__nonneg__nonpos2,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B4 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B4 @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_968_mult__nonpos__nonneg,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B4 ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_969_mult__nonpos__nonneg,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_970_mult__nonpos__nonneg,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B4 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B4 ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_971_mult__nonneg__nonpos,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B4 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B4 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_972_mult__nonneg__nonpos,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B4 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_973_mult__nonneg__nonpos,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B4 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B4 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_974_mult__nonneg__nonneg,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B4 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_975_mult__nonneg__nonneg,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B4 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_976_mult__nonneg__nonneg,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B4 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_977_split__mult__neg__le,axiom,
! [A: real,B4: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B4 ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_978_split__mult__neg__le,axiom,
! [A: nat,B4: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B4 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B4 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_979_split__mult__neg__le,axiom,
! [A: int,B4: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B4 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B4 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B4 ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_980_mult__le__0__iff,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B4 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) ) ) ) ).
% mult_le_0_iff
thf(fact_981_mult__le__0__iff,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B4 ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B4 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B4 ) ) ) ) ).
% mult_le_0_iff
thf(fact_982_mult__right__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_983_mult__right__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_984_mult__right__mono,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_985_mult__right__mono__neg,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_986_mult__right__mono__neg,axiom,
! [B4: int,A: int,C2: int] :
( ( ord_less_eq_int @ B4 @ A )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_987_mult__left__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) ) ) ) ).
% mult_left_mono
thf(fact_988_mult__left__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% mult_left_mono
thf(fact_989_mult__left__mono,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) ) ) ) ).
% mult_left_mono
thf(fact_990_mult__nonpos__nonpos,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B4 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B4 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_991_mult__nonpos__nonpos,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B4 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B4 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_992_mult__left__mono__neg,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_993_mult__left__mono__neg,axiom,
! [B4: int,A: int,C2: int] :
( ( ord_less_eq_int @ B4 @ A )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_994_split__mult__pos__le,axiom,
! [A: real,B4: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B4 ) ) ) ).
% split_mult_pos_le
thf(fact_995_split__mult__pos__le,axiom,
! [A: int,B4: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B4 ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B4 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B4 ) ) ) ).
% split_mult_pos_le
thf(fact_996_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_997_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_998_mult__mono_H,axiom,
! [A: real,B4: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_999_mult__mono_H,axiom,
! [A: nat,B4: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1000_mult__mono_H,axiom,
! [A: int,B4: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1001_mult__mono,axiom,
! [A: real,B4: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1002_mult__mono,axiom,
! [A: nat,B4: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1003_mult__mono,axiom,
! [A: int,B4: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1004_divide__le__0__iff,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B4 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) ) ) ) ).
% divide_le_0_iff
thf(fact_1005_divide__right__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B4 @ C2 ) ) ) ) ).
% divide_right_mono
thf(fact_1006_zero__le__divide__iff,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B4 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_1007_divide__nonneg__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1008_divide__nonneg__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1009_divide__nonpos__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1010_divide__nonpos__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1011_divide__right__mono__neg,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B4 @ C2 ) @ ( divide_divide_real @ A @ C2 ) ) ) ) ).
% divide_right_mono_neg
thf(fact_1012_frac__eq__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X @ Z )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1013_divide__eq__eq,axiom,
! [B4: real,C2: real,A: real] :
( ( ( divide_divide_real @ B4 @ C2 )
= A )
= ( ( ( C2 != zero_zero_real )
=> ( B4
= ( times_times_real @ A @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_1014_eq__divide__eq,axiom,
! [A: real,B4: real,C2: real] :
( ( A
= ( divide_divide_real @ B4 @ C2 ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ A @ C2 )
= B4 ) )
& ( ( C2 = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_1015_divide__eq__imp,axiom,
! [C2: real,B4: real,A: real] :
( ( C2 != zero_zero_real )
=> ( ( B4
= ( times_times_real @ A @ C2 ) )
=> ( ( divide_divide_real @ B4 @ C2 )
= A ) ) ) ).
% divide_eq_imp
thf(fact_1016_eq__divide__imp,axiom,
! [C2: real,A: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ C2 )
= B4 )
=> ( A
= ( divide_divide_real @ B4 @ C2 ) ) ) ) ).
% eq_divide_imp
thf(fact_1017_nonzero__divide__eq__eq,axiom,
! [C2: real,B4: real,A: real] :
( ( C2 != zero_zero_real )
=> ( ( ( divide_divide_real @ B4 @ C2 )
= A )
= ( B4
= ( times_times_real @ A @ C2 ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1018_nonzero__eq__divide__eq,axiom,
! [C2: real,A: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B4 @ C2 ) )
= ( ( times_times_real @ A @ C2 )
= B4 ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1019_Min__le,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X ) ) ) ).
% Min_le
thf(fact_1020_Min__le,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ X @ A2 )
=> ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ A2 ) @ X ) ) ) ).
% Min_le
thf(fact_1021_Min__le,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ X @ A2 )
=> ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ A2 ) @ X ) ) ) ).
% Min_le
thf(fact_1022_Min__eqI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A2 )
=> ( ord_less_eq_nat @ X @ Y2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( lattic8721135487736765967in_nat @ A2 )
= X ) ) ) ) ).
% Min_eqI
thf(fact_1023_Min__eqI,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ A2 )
=> ( ord_less_eq_int @ X @ Y2 ) )
=> ( ( member_int @ X @ A2 )
=> ( ( lattic8718645017227715691in_int @ A2 )
= X ) ) ) ) ).
% Min_eqI
thf(fact_1024_Min__eqI,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ! [Y2: real] :
( ( member_real @ Y2 @ A2 )
=> ( ord_less_eq_real @ X @ Y2 ) )
=> ( ( member_real @ X @ A2 )
=> ( ( lattic3629708407755379051n_real @ A2 )
= X ) ) ) ) ).
% Min_eqI
thf(fact_1025_Min_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ A ) ) ) ).
% Min.coboundedI
thf(fact_1026_Min_OcoboundedI,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ A2 ) @ A ) ) ) ).
% Min.coboundedI
thf(fact_1027_Min_OcoboundedI,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ A2 ) @ A ) ) ) ).
% Min.coboundedI
thf(fact_1028_Min__in,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( member_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ A2 ) ) ) ).
% Min_in
thf(fact_1029_Min__in,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( member_int @ ( lattic8718645017227715691in_int @ A2 ) @ A2 ) ) ) ).
% Min_in
thf(fact_1030_Min__in,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( member_real @ ( lattic3629708407755379051n_real @ A2 ) @ A2 ) ) ) ).
% Min_in
thf(fact_1031_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F3: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ S2 )
=> ( ord_less_eq_real @ ( F3 @ Y3 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1032_finite__ranking__induct,axiom,
! [S: set_int,P: set_int > $o,F3: int > real] :
( ( finite_finite_int @ S )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X3: int,S2: set_int] :
( ( finite_finite_int @ S2 )
=> ( ! [Y3: int] :
( ( member_int @ Y3 @ S2 )
=> ( ord_less_eq_real @ ( F3 @ Y3 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_int @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1033_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F3: a > real] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ S2 )
=> ( ord_less_eq_real @ ( F3 @ Y3 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1034_finite__ranking__induct,axiom,
! [S: set_real,P: set_real > $o,F3: real > real] :
( ( finite_finite_real @ S )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,S2: set_real] :
( ( finite_finite_real @ S2 )
=> ( ! [Y3: real] :
( ( member_real @ Y3 @ S2 )
=> ( ord_less_eq_real @ ( F3 @ Y3 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_real @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1035_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F3: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ S2 )
=> ( ord_less_eq_nat @ ( F3 @ Y3 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1036_finite__ranking__induct,axiom,
! [S: set_int,P: set_int > $o,F3: int > nat] :
( ( finite_finite_int @ S )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X3: int,S2: set_int] :
( ( finite_finite_int @ S2 )
=> ( ! [Y3: int] :
( ( member_int @ Y3 @ S2 )
=> ( ord_less_eq_nat @ ( F3 @ Y3 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_int @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1037_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F3: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ S2 )
=> ( ord_less_eq_nat @ ( F3 @ Y3 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1038_finite__ranking__induct,axiom,
! [S: set_real,P: set_real > $o,F3: real > nat] :
( ( finite_finite_real @ S )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,S2: set_real] :
( ( finite_finite_real @ S2 )
=> ( ! [Y3: real] :
( ( member_real @ Y3 @ S2 )
=> ( ord_less_eq_nat @ ( F3 @ Y3 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_real @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1039_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F3: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ S2 )
=> ( ord_less_eq_int @ ( F3 @ Y3 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1040_finite__ranking__induct,axiom,
! [S: set_int,P: set_int > $o,F3: int > int] :
( ( finite_finite_int @ S )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X3: int,S2: set_int] :
( ( finite_finite_int @ S2 )
=> ( ! [Y3: int] :
( ( member_int @ Y3 @ S2 )
=> ( ord_less_eq_int @ ( F3 @ Y3 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_int @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1041_Min__eq__iff,axiom,
! [A2: set_nat,M: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A2 )
= M )
= ( ( member_nat @ M @ A2 )
& ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ M @ X2 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_1042_Min__eq__iff,axiom,
! [A2: set_int,M: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ( lattic8718645017227715691in_int @ A2 )
= M )
= ( ( member_int @ M @ A2 )
& ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_int @ M @ X2 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_1043_Min__eq__iff,axiom,
! [A2: set_real,M: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ( lattic3629708407755379051n_real @ A2 )
= M )
= ( ( member_real @ M @ A2 )
& ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_real @ M @ X2 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_1044_Min__le__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_1045_Min__le__iff,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ A2 ) @ X )
= ( ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( ord_less_eq_int @ X2 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_1046_Min__le__iff,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ A2 ) @ X )
= ( ? [X2: real] :
( ( member_real @ X2 @ A2 )
& ( ord_less_eq_real @ X2 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_1047_eq__Min__iff,axiom,
! [A2: set_nat,M: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( M
= ( lattic8721135487736765967in_nat @ A2 ) )
= ( ( member_nat @ M @ A2 )
& ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ M @ X2 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_1048_eq__Min__iff,axiom,
! [A2: set_int,M: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( M
= ( lattic8718645017227715691in_int @ A2 ) )
= ( ( member_int @ M @ A2 )
& ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_int @ M @ X2 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_1049_eq__Min__iff,axiom,
! [A2: set_real,M: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( M
= ( lattic3629708407755379051n_real @ A2 ) )
= ( ( member_real @ M @ A2 )
& ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_real @ M @ X2 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_1050_Min_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
=> ! [A9: nat] :
( ( member_nat @ A9 @ A2 )
=> ( ord_less_eq_nat @ X @ A9 ) ) ) ) ) ).
% Min.boundedE
thf(fact_1051_Min_OboundedE,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A2 ) )
=> ! [A9: int] :
( ( member_int @ A9 @ A2 )
=> ( ord_less_eq_int @ X @ A9 ) ) ) ) ) ).
% Min.boundedE
thf(fact_1052_Min_OboundedE,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ X @ ( lattic3629708407755379051n_real @ A2 ) )
=> ! [A9: real] :
( ( member_real @ A9 @ A2 )
=> ( ord_less_eq_real @ X @ A9 ) ) ) ) ) ).
% Min.boundedE
thf(fact_1053_Min_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ord_less_eq_nat @ X @ A3 ) )
=> ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.boundedI
thf(fact_1054_Min_OboundedI,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A2 )
=> ( ord_less_eq_int @ X @ A3 ) )
=> ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A2 ) ) ) ) ) ).
% Min.boundedI
thf(fact_1055_Min_OboundedI,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ord_less_eq_real @ X @ A3 ) )
=> ( ord_less_eq_real @ X @ ( lattic3629708407755379051n_real @ A2 ) ) ) ) ) ).
% Min.boundedI
thf(fact_1056_Min__insert2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ A @ B2 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ A @ A2 ) )
= A ) ) ) ).
% Min_insert2
thf(fact_1057_Min__insert2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ! [B2: int] :
( ( member_int @ B2 @ A2 )
=> ( ord_less_eq_int @ A @ B2 ) )
=> ( ( lattic8718645017227715691in_int @ ( insert_int @ A @ A2 ) )
= A ) ) ) ).
% Min_insert2
thf(fact_1058_Min__insert2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ A2 )
=> ( ord_less_eq_real @ A @ B2 ) )
=> ( ( lattic3629708407755379051n_real @ ( insert_real @ A @ A2 ) )
= A ) ) ) ).
% Min_insert2
thf(fact_1059_Min__antimono,axiom,
! [M2: set_nat,N2: set_nat] :
( ( ord_less_eq_set_nat @ M2 @ N2 )
=> ( ( M2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N2 ) @ ( lattic8721135487736765967in_nat @ M2 ) ) ) ) ) ).
% Min_antimono
thf(fact_1060_Min__antimono,axiom,
! [M2: set_int,N2: set_int] :
( ( ord_less_eq_set_int @ M2 @ N2 )
=> ( ( M2 != bot_bot_set_int )
=> ( ( finite_finite_int @ N2 )
=> ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ N2 ) @ ( lattic8718645017227715691in_int @ M2 ) ) ) ) ) ).
% Min_antimono
thf(fact_1061_Min__antimono,axiom,
! [M2: set_real,N2: set_real] :
( ( ord_less_eq_set_real @ M2 @ N2 )
=> ( ( M2 != bot_bot_set_real )
=> ( ( finite_finite_real @ N2 )
=> ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ N2 ) @ ( lattic3629708407755379051n_real @ M2 ) ) ) ) ) ).
% Min_antimono
thf(fact_1062_Min_Osubset__imp,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B ) @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_1063_Min_Osubset__imp,axiom,
! [A2: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A2 @ B )
=> ( ( A2 != bot_bot_set_int )
=> ( ( finite_finite_int @ B )
=> ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ B ) @ ( lattic8718645017227715691in_int @ A2 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_1064_Min_Osubset__imp,axiom,
! [A2: set_real,B: set_real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( A2 != bot_bot_set_real )
=> ( ( finite_finite_real @ B )
=> ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ B ) @ ( lattic3629708407755379051n_real @ A2 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_1065_gt0,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A2 @ a0 )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A2 ) ) ) ) ) ) ).
% gt0
thf(fact_1066_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_1067_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_1068_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_real] :
( ( inf_inf_set_real @ bot_bot_set_real @ X )
= bot_bot_set_real ) ).
% boolean_algebra.conj_zero_left
thf(fact_1069_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_1070_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_1071_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_real] :
( ( inf_inf_set_real @ X @ bot_bot_set_real )
= bot_bot_set_real ) ).
% boolean_algebra.conj_zero_right
thf(fact_1072_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_1073_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_1074_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_1075_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_1076_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_1077__092_060open_0620_A_060_AK_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ k ).
% \<open>0 < K\<close>
thf(fact_1078_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_1079_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_1080_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_1081_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_1082_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_1083_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_1084_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_1085_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_1086_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_1087_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_1088_DiffI,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( ~ ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_1089_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_1090_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_1091_Diff__iff,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= ( ( member_set_a @ C2 @ A2 )
& ~ ( member_set_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_1092_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_1093_Diff__idemp,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ B )
= ( minus_5736297505244876581_set_a @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_1094_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_1095_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_1096_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_1097_Diff__cancel,axiom,
! [A2: set_real] :
( ( minus_minus_set_real @ A2 @ A2 )
= bot_bot_set_real ) ).
% Diff_cancel
thf(fact_1098_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_1099_Diff__cancel,axiom,
! [A2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ A2 )
= bot_bot_set_set_a ) ).
% Diff_cancel
thf(fact_1100_empty__Diff,axiom,
! [A2: set_real] :
( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
= bot_bot_set_real ) ).
% empty_Diff
thf(fact_1101_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_1102_empty__Diff,axiom,
! [A2: set_set_a] :
( ( minus_5736297505244876581_set_a @ bot_bot_set_set_a @ A2 )
= bot_bot_set_set_a ) ).
% empty_Diff
thf(fact_1103_Diff__empty,axiom,
! [A2: set_real] :
( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
= A2 ) ).
% Diff_empty
thf(fact_1104_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_1105_Diff__empty,axiom,
! [A2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ bot_bot_set_set_a )
= A2 ) ).
% Diff_empty
thf(fact_1106_finite__Diff2,axiom,
! [B: set_real,A2: set_real] :
( ( finite_finite_real @ B )
=> ( ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B ) )
= ( finite_finite_real @ A2 ) ) ) ).
% finite_Diff2
thf(fact_1107_finite__Diff2,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_1108_finite__Diff2,axiom,
! [B: set_int,A2: set_int] :
( ( finite_finite_int @ B )
=> ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B ) )
= ( finite_finite_int @ A2 ) ) ) ).
% finite_Diff2
thf(fact_1109_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_1110_finite__Diff2,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= ( finite_finite_set_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_1111_finite__Diff,axiom,
! [A2: set_real,B: set_real] :
( ( finite_finite_real @ A2 )
=> ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_1112_finite__Diff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_1113_finite__Diff,axiom,
! [A2: set_int,B: set_int] :
( ( finite_finite_int @ A2 )
=> ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_1114_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_1115_finite__Diff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_1116_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_1117_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_1118_insert__Diff1,axiom,
! [X: set_a,B: set_set_a,A2: set_set_a] :
( ( member_set_a @ X @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B )
= ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_1119_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_1120_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_1121_Diff__insert0,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ B ) )
= ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_1122_Un__Diff__cancel2,axiom,
! [B: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B @ A2 ) @ A2 )
= ( sup_sup_set_a @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_1123_Un__Diff__cancel2,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( sup_sup_set_set_a @ ( minus_5736297505244876581_set_a @ B @ A2 ) @ A2 )
= ( sup_sup_set_set_a @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_1124_Un__Diff__cancel,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_1125_Un__Diff__cancel,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B @ A2 ) )
= ( sup_sup_set_set_a @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_1126_Diff__eq__empty__iff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1127_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_1128_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_1129_zdiv__int,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_1130_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1131_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1132_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1133_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1134_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_1135_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1136_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1137_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_1138_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_1139_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_1140_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_1141_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_1142_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_1143_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1144_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_1145_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1146_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1147__092_060open_062A_A_092_060in_062_APow_AA0_A_N_A_123_123_125_125_092_060close_062,axiom,
member_set_a @ a2 @ ( minus_5736297505244876581_set_a @ ( pow_a @ a0 ) @ ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ).
% \<open>A \<in> Pow A0 - {{}}\<close>
thf(fact_1148_zdiv__mono1,axiom,
! [A: int,A10: int,B4: int] :
( ( ord_less_eq_int @ A @ A10 )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A10 @ B4 ) ) ) ) ).
% zdiv_mono1
thf(fact_1149_zdiv__mono2,axiom,
! [A: int,B9: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B9 )
=> ( ( ord_less_eq_int @ B9 @ B4 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B9 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1150_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1151_zdiv__mono1__neg,axiom,
! [A: int,A10: int,B4: int] :
( ( ord_less_eq_int @ A @ A10 )
=> ( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A10 @ B4 ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1152_zdiv__mono2__neg,axiom,
! [A: int,B9: int,B4: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B9 )
=> ( ( ord_less_eq_int @ B9 @ B4 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B9 ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1153_zdiv__zmult2__eq,axiom,
! [C2: int,A: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B4 @ C2 ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B4 ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1154_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1155_div__neg__pos__less0,axiom,
! [A: int,B4: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ord_less_int @ ( divide_divide_int @ A @ B4 ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1156_div__nonneg__neg__le0,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1157_div__nonpos__pos__le0,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1158_neg__imp__zdiv__neg__iff,axiom,
! [B4: int,A: int] :
( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B4 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1159_pos__imp__zdiv__neg__iff,axiom,
! [B4: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B4 ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1160_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1161_neg__imp__zdiv__nonneg__iff,axiom,
! [B4: int,A: int] :
( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B4 ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1162_pos__imp__zdiv__nonneg__iff,axiom,
! [B4: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B4 ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1163_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B4 ) )
= ( ( ord_less_eq_int @ B4 @ A )
& ( ord_less_int @ zero_zero_int @ B4 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1164_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_1165_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_1166_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_1167_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_1168_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_1169_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1170_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1171_less__not__refl3,axiom,
! [S3: nat,T4: nat] :
( ( ord_less_nat @ S3 @ T4 )
=> ( S3 != T4 ) ) ).
% less_not_refl3
thf(fact_1172_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_1173_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1174_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1175_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1176_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_1177_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_1178_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1179_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_1180_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_1181_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_1182_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1183_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_1184_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_1185_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_1186_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_1187_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1188_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1189_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1190_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1191_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1192_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1193_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1194_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_1195_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_1196_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F3 @ I2 ) @ ( F3 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F3 @ I ) @ ( F3 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1197_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_1198_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_1199_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1200_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_1201_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1202_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1203_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_1204_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_1205_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1206_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1207_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1208_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1209_div__less__iff__less__mult,axiom,
! [Q2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1210_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1211__092_060open_062_092_060And_062thesis_O_A_I_092_060lbrakk_062A_A_092_060in_062_APow_AA0_A_N_A_123_123_125_125_059_AK_A_061_Areal_A_Icard_A_Isumset_AA_AB_J_J_A_P_Areal_A_Icard_AA_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ( ( member_set_a @ a2 @ ( minus_5736297505244876581_set_a @ ( pow_a @ a0 ) @ ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) )
=> ( k
!= ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a2 ) ) ) ) ) ).
% \<open>\<And>thesis. (\<lbrakk>A \<in> Pow A0 - {{}}; K = real (card (sumset A B)) / real (card A)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1212__092_060open_062_092_060exists_062A_O_AA_A_092_060in_062_APow_AA0_A_N_A_123_123_125_125_A_092_060and_062_AK_A_061_Areal_A_Icard_A_Isumset_AA_AB_J_J_A_P_Areal_A_Icard_AA_J_092_060close_062,axiom,
? [A6: set_a] :
( ( member_set_a @ A6 @ ( minus_5736297505244876581_set_a @ ( pow_a @ a0 ) @ ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) )
& ( k
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A6 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A6 ) ) ) ) ) ).
% \<open>\<exists>A. A \<in> Pow A0 - {{}} \<and> K = real (card (sumset A B)) / real (card A)\<close>
thf(fact_1213_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_1214_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_1215_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_1216_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_1217_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_1218_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1219_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_1220_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1221_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 )
=> ( ! [M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X ) @ C2 ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1222_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_1223_complete__real,axiom,
! [S: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y2: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S )
=> ( ord_less_eq_real @ X5 @ Y2 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y2 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_1224_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_1225_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y3: real] :
? [N3: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1226_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1227_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1228_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1229_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1230_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1231_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1232_int__diff__cases,axiom,
! [Z: int] :
~ ! [M5: nat,N3: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_1233_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_1234_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1235_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1236_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1237_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1238_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1239_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1240_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1241_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1242_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1243_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P2: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_1244_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P2: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_1245_plusinfinity,axiom,
! [D: int,P2: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [X_1: int] : ( P2 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_1246_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_1247_int__ops_I6_J,axiom,
! [A: nat,B4: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B4 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B4 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B4 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B4 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1248_Bolzano,axiom,
! [A: real,B4: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ! [A3: real,B2: real,C4: real] :
( ( P @ A3 @ B2 )
=> ( ( P @ B2 @ C4 )
=> ( ( ord_less_eq_real @ A3 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C4 )
=> ( P @ A3 @ C4 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B4 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A3: real,B2: real] :
( ( ( ord_less_eq_real @ A3 @ X3 )
& ( ord_less_eq_real @ X3 @ B2 )
& ( ord_less_real @ ( minus_minus_real @ B2 @ A3 ) @ D3 ) )
=> ( P @ A3 @ B2 ) ) ) ) )
=> ( P @ A @ B4 ) ) ) ) ).
% Bolzano
thf(fact_1249_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_1250_int__if,axiom,
! [P: $o,A: nat,B4: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B4 ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B4 ) )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% int_if
thf(fact_1251_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1252_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1253_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1254_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1255_int__ops_I7_J,axiom,
! [A: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B4 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ).
% int_ops(7)
thf(fact_1256_int__ops_I8_J,axiom,
! [A: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B4 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ).
% int_ops(8)
thf(fact_1257_A__def,axiom,
( a2
= ( fChoice_set_a
@ ^ [A7: set_a] :
( ( member_set_a @ A7 @ ( minus_5736297505244876581_set_a @ ( pow_a @ a0 ) @ ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) )
& ( k
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A7 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A7 ) ) ) ) ) ) ) ).
% A_def
thf(fact_1258_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1259_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1260_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1261_sumsetp__sumset__eq,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn895083305082786853setp_a @ g @ addition
@ ^ [X2: a] : ( member_a @ X2 @ A2 )
@ ^ [X2: a] : ( member_a @ X2 @ B ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumsetp_sumset_eq
thf(fact_1262_sumset__def,axiom,
( ( pluenn3038260743871226533mset_a @ g @ addition )
= ( ^ [A7: set_a,B7: set_a] :
( collect_a
@ ( pluenn895083305082786853setp_a @ g @ addition
@ ^ [X2: a] : ( member_a @ X2 @ A7 )
@ ^ [X2: a] : ( member_a @ X2 @ B7 ) ) ) ) ) ).
% sumset_def
thf(fact_1263_finite__interval__int1,axiom,
! [A: int,B4: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( ord_less_eq_int @ A @ I4 )
& ( ord_less_eq_int @ I4 @ B4 ) ) ) ) ).
% finite_interval_int1
thf(fact_1264_finite__interval__int4,axiom,
! [A: int,B4: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( ord_less_int @ A @ I4 )
& ( ord_less_int @ I4 @ B4 ) ) ) ) ).
% finite_interval_int4
thf(fact_1265_finite__interval__int3,axiom,
! [A: int,B4: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( ord_less_int @ A @ I4 )
& ( ord_less_eq_int @ I4 @ B4 ) ) ) ) ).
% finite_interval_int3
thf(fact_1266_finite__interval__int2,axiom,
! [A: int,B4: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( ord_less_eq_int @ A @ I4 )
& ( ord_less_int @ I4 @ B4 ) ) ) ) ).
% finite_interval_int2
thf(fact_1267_KS__def,axiom,
( ks
= ( image_set_a_real
@ ^ [A7: set_a] : ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A7 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A7 ) ) )
@ ( minus_5736297505244876581_set_a @ ( pow_a @ a0 ) @ ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ) ) ).
% KS_def
% Helper facts (4)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_fChoice_001t__Set__Oset_Itf__a_J_T,axiom,
! [P: set_a > $o] :
( ( P @ ( fChoice_set_a @ P ) )
= ( ? [X7: set_a] : ( P @ X7 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ c ) ) ) ) @ ( times_times_real @ k @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ c ) ) ) ) ).
%------------------------------------------------------------------------------