TPTP Problem File: SLH0326^1.p
View Solutions
- 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_00435_016189__12258218_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1371 ( 560 unt; 101 typ; 0 def)
% Number of atoms : 3771 (1244 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 11729 ( 456 ~; 83 |; 272 &;9175 @)
% ( 0 <=>;1743 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 259 ( 259 >; 0 *; 0 +; 0 <<)
% Number of symbols : 93 ( 92 usr; 17 con; 0-5 aty)
% Number of variables : 3148 ( 111 ^;2945 !; 92 ?;3148 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:21:25.551
%------------------------------------------------------------------------------
% 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 (92)
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__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_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
inf_inf_int: int > int > int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Real__Oreal,type,
inf_inf_real: real > real > real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Real__Oreal_J,type,
inf_inf_set_real: set_real > set_real > set_real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_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__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__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_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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__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_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 ).
thf(sy_v_thesis,type,
thesis: $o ).
% Relevant facts (1268)
thf(fact_0_assms_I6_J,axiom,
ord_less_eq_set_a @ b @ g ).
% assms(6)
thf(fact_1_assms_I3_J,axiom,
ord_less_eq_set_a @ a0 @ g ).
% assms(3)
thf(fact_2__092_060open_062K_A_092_060in_062_AKS_092_060close_062,axiom,
member_real @ k @ ks ).
% \<open>K \<in> KS\<close>
thf(fact_3_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_4_assms_I5_J,axiom,
finite_finite_a @ b ).
% assms(5)
thf(fact_5_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_6_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_7_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_8_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_9_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_10_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_11_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_12_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_13_assms_I2_J,axiom,
finite_finite_a @ a0 ).
% assms(2)
thf(fact_14_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_15_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_16_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_17_assms_I4_J,axiom,
a0 != bot_bot_set_a ).
% assms(4)
thf(fact_18_assms_I7_J,axiom,
b != bot_bot_set_a ).
% assms(7)
thf(fact_19_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_20_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_21_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( addition @ X @ Y )
= ( addition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_22_KS_I2_J,axiom,
ks != bot_bot_set_real ).
% KS(2)
thf(fact_23_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_24_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_25_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_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_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_28_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_29_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_30_KS_I1_J,axiom,
finite_finite_real @ ks ).
% KS(1)
thf(fact_31_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_32_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_33_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_34_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_35_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_36_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_37_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_38_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_39_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_40_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_41_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_42_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_43_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_44_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_45_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_46_K__def,axiom,
( k
= ( lattic3629708407755379051n_real @ ks ) ) ).
% K_def
thf(fact_47_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_48_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_49_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_50_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_51_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_52_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_53_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_54_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_55_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_56_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_57_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_58_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_59_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_60_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_61_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_62_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_63_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_64_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_67_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_68_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_70_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_71_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_72_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_73_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_74_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_75_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_76_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_77_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_78_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_79_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_80_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_81_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_82_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_83_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_84_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_85_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_86_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_87_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_88_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_89_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_90_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_91_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_92_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_93_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_94_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_95_sumset__mono,axiom,
! [A3: set_a,A2: set_a,B: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ A2 )
=> ( ( ord_less_eq_set_a @ B @ B2 )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A3 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ).
% sumset_mono
thf(fact_96_sumset__subset__carrier,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ g ) ).
% sumset_subset_carrier
thf(fact_97_finite__sumset,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ).
% finite_sumset
thf(fact_98_sumset__commute,axiom,
! [A2: set_a,B2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 )
= ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ A2 ) ) ).
% sumset_commute
thf(fact_99_sumset__assoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ C )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ C ) ) ) ).
% sumset_assoc
thf(fact_100_sumset_OsumsetI,axiom,
! [A: a,A2: set_a,B3: a,B2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( ( member_a @ B3 @ B2 )
=> ( ( member_a @ B3 @ g )
=> ( member_a @ ( addition @ A @ B3 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ) ) ).
% sumset.sumsetI
thf(fact_101_sumset_Osimps,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) )
= ( ? [A4: a,B4: a] :
( ( A
= ( addition @ A4 @ B4 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ g )
& ( member_a @ B4 @ B2 )
& ( member_a @ B4 @ g ) ) ) ) ).
% sumset.simps
thf(fact_102_sumset_Ocases,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) )
=> ~ ! [A5: a,B5: a] :
( ( A
= ( addition @ A5 @ B5 ) )
=> ( ( member_a @ A5 @ A2 )
=> ( ( member_a @ A5 @ g )
=> ( ( member_a @ B5 @ B2 )
=> ~ ( member_a @ B5 @ g ) ) ) ) ) ) ).
% sumset.cases
thf(fact_103_A_I1_J,axiom,
ord_less_eq_set_a @ a2 @ a0 ).
% A(1)
thf(fact_104__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_105_card__sumset__0__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ g )
=> ( ( ord_less_eq_set_a @ B2 @ g )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
| ( ( finite_card_a @ B2 )
= zero_zero_nat ) ) ) ) ) ).
% card_sumset_0_iff
thf(fact_106_A_I2_J,axiom,
a2 != bot_bot_set_a ).
% A(2)
thf(fact_107__092_060open_062finite_AA_092_060close_062,axiom,
finite_finite_a @ a2 ).
% \<open>finite A\<close>
thf(fact_108__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_109_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_110_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_111_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_112_card__le__sumset,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ g )
=> ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ) ) ) ) ).
% card_le_sumset
thf(fact_113_associative,axiom,
! [A: a,B3: a,C2: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B3 @ g )
=> ( ( member_a @ C2 @ g )
=> ( ( addition @ ( addition @ A @ B3 ) @ C2 )
= ( addition @ A @ ( addition @ B3 @ C2 ) ) ) ) ) ) ).
% associative
thf(fact_114_composition__closed,axiom,
! [A: a,B3: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B3 @ g )
=> ( member_a @ ( addition @ A @ B3 ) @ g ) ) ) ).
% composition_closed
thf(fact_115_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_116_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_117_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_118_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_119_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_120_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_121_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_122_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_123_additive__abelian__group_Osumset_Ocong,axiom,
pluenn3038260743871226533mset_a = pluenn3038260743871226533mset_a ).
% additive_abelian_group.sumset.cong
thf(fact_124_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_125_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_126_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_127_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_128_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_129_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_130_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_131_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_132_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_133_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_134_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_135_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_136_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_137_sumsetp_OsumsetI,axiom,
! [A2: a > $o,A: a,B2: a > $o,B3: a] :
( ( A2 @ A )
=> ( ( member_a @ A @ g )
=> ( ( B2 @ B3 )
=> ( ( member_a @ B3 @ g )
=> ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B2 @ ( addition @ A @ B3 ) ) ) ) ) ) ).
% sumsetp.sumsetI
thf(fact_138_sumsetp_Osimps,axiom,
! [A2: a > $o,B2: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B2 @ A )
= ( ? [A4: a,B4: a] :
( ( A
= ( addition @ A4 @ B4 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ g )
& ( B2 @ B4 )
& ( member_a @ B4 @ g ) ) ) ) ).
% sumsetp.simps
thf(fact_139_sumsetp_Ocases,axiom,
! [A2: a > $o,B2: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B2 @ A )
=> ~ ! [A5: a,B5: a] :
( ( A
= ( addition @ A5 @ B5 ) )
=> ( ( A2 @ A5 )
=> ( ( member_a @ A5 @ g )
=> ( ( B2 @ B5 )
=> ~ ( member_a @ B5 @ g ) ) ) ) ) ) ).
% sumsetp.cases
thf(fact_140_Min__gr__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_nat @ X @ X2 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_141_Min__gr__iff,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_int @ X @ ( lattic8718645017227715691in_int @ A2 ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_int @ X @ X2 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_142_Min__gr__iff,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ord_less_real @ X @ ( lattic3629708407755379051n_real @ A2 ) )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_real @ X @ X2 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_143_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_144_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_145_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_146_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_147_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_148_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_149_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_150_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_151_card_Oinfinite,axiom,
! [A2: set_real] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite_card_real @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_152_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_153_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_154_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_155_card_Oempty,axiom,
( ( finite_card_real @ bot_bot_set_real )
= zero_zero_nat ) ).
% card.empty
thf(fact_156_card_Oempty,axiom,
( ( finite_card_set_a @ bot_bot_set_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_157_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_158_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_159_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_160_that,axiom,
! [A2: set_a,K3: real] :
( ( ord_less_eq_set_a @ A2 @ a0 )
=> ( ( A2 != bot_bot_set_a )
=> ( ( ord_less_real @ zero_zero_real @ K3 )
=> ( ( ord_less_eq_real @ K3 @ k0 )
=> ( ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ g )
=> ( ( finite_finite_a @ C3 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ C3 ) ) ) ) @ ( times_times_real @ K3 @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ C3 ) ) ) ) ) ) )
=> thesis ) ) ) ) ) ).
% that
thf(fact_161_complete__real,axiom,
! [S2: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S2 )
=> ( ? [Z: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ? [Y2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Y2 ) )
& ! [Z: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ( ord_less_eq_real @ Y2 @ Z ) ) ) ) ) ).
% complete_real
thf(fact_162_mult_Oleft__commute,axiom,
! [B3: real,A: real,C2: real] :
( ( times_times_real @ B3 @ ( times_times_real @ A @ C2 ) )
= ( times_times_real @ A @ ( times_times_real @ B3 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_163_mult_Oleft__commute,axiom,
! [B3: nat,A: nat,C2: nat] :
( ( times_times_nat @ B3 @ ( times_times_nat @ A @ C2 ) )
= ( times_times_nat @ A @ ( times_times_nat @ B3 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_164_mult_Oleft__commute,axiom,
! [B3: int,A: int,C2: int] :
( ( times_times_int @ B3 @ ( times_times_int @ A @ C2 ) )
= ( times_times_int @ A @ ( times_times_int @ B3 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_165_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_166_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_167_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B4: int] : ( times_times_int @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_168_mult_Oassoc,axiom,
! [A: real,B3: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B3 ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B3 @ C2 ) ) ) ).
% mult.assoc
thf(fact_169_mult_Oassoc,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B3 ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B3 @ C2 ) ) ) ).
% mult.assoc
thf(fact_170_mult_Oassoc,axiom,
! [A: int,B3: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A @ B3 ) @ C2 )
= ( times_times_int @ A @ ( times_times_int @ B3 @ C2 ) ) ) ).
% mult.assoc
thf(fact_171_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B3: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B3 ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B3 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_172_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B3 ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B3 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_173_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B3: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A @ B3 ) @ C2 )
= ( times_times_int @ A @ ( times_times_int @ B3 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_174_additive__abelian__group_Osumsetp_Ocong,axiom,
pluenn895083305082786853setp_a = pluenn895083305082786853setp_a ).
% additive_abelian_group.sumsetp.cong
thf(fact_175_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_176_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_177_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_178_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y3: real] :
? [N2: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_179_finite__psubset__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ! [B6: set_a] :
( ( ord_less_set_a @ B6 @ A6 )
=> ( P @ B6 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_180_finite__psubset__induct,axiom,
! [A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ A2 )
=> ( ! [A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ! [B6: set_real] :
( ( ord_less_set_real @ B6 @ A6 )
=> ( P @ B6 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_181_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B6: set_nat] :
( ( ord_less_set_nat @ B6 @ A6 )
=> ( P @ B6 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_182_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C2 ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_183_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_184_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_185_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_186_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_187_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_188_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_189_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_190_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_191_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_192_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_193_finite_OemptyI,axiom,
finite_finite_real @ bot_bot_set_real ).
% finite.emptyI
thf(fact_194_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_195_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_196_infinite__imp__nonempty,axiom,
! [S2: set_a] :
( ~ ( finite_finite_a @ S2 )
=> ( S2 != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_197_infinite__imp__nonempty,axiom,
! [S2: set_real] :
( ~ ( finite_finite_real @ S2 )
=> ( S2 != bot_bot_set_real ) ) ).
% infinite_imp_nonempty
thf(fact_198_infinite__imp__nonempty,axiom,
! [S2: set_set_a] :
( ~ ( finite_finite_set_a @ S2 )
=> ( S2 != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_199_finite__subset,axiom,
! [A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( finite_finite_real @ B2 )
=> ( finite_finite_real @ A2 ) ) ) ).
% finite_subset
thf(fact_200_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_201_finite__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_202_infinite__super,axiom,
! [S2: set_real,T2: set_real] :
( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ~ ( finite_finite_real @ S2 )
=> ~ ( finite_finite_real @ T2 ) ) ) ).
% infinite_super
thf(fact_203_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_204_infinite__super,axiom,
! [S2: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S2 @ T2 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_205_rev__finite__subset,axiom,
! [B2: set_real,A2: set_real] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( finite_finite_real @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_206_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_207_rev__finite__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_208_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_209_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_210_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_211_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_212_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_213_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_214_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_215_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_216_ex__min__if__finite,axiom,
! [S2: set_set_a] :
( ( finite_finite_set_a @ S2 )
=> ( ( S2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ S2 )
& ~ ? [Xa: set_a] :
( ( member_set_a @ Xa @ S2 )
& ( ord_less_set_a @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_217_ex__min__if__finite,axiom,
! [S2: set_real] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ S2 )
& ~ ? [Xa: real] :
( ( member_real @ Xa @ S2 )
& ( ord_less_real @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_218_ex__min__if__finite,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S2 )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S2 )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_219_ex__min__if__finite,axiom,
! [S2: set_int] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ S2 )
& ~ ? [Xa: int] :
( ( member_int @ Xa @ S2 )
& ( ord_less_int @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_220_infinite__growing,axiom,
! [X5: set_real] :
( ( X5 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X5 )
=> ? [Xa: real] :
( ( member_real @ Xa @ X5 )
& ( ord_less_real @ X3 @ Xa ) ) )
=> ~ ( finite_finite_real @ X5 ) ) ) ).
% infinite_growing
thf(fact_221_infinite__growing,axiom,
! [X5: set_nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X5 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X5 ) ) ) ).
% infinite_growing
thf(fact_222_infinite__growing,axiom,
! [X5: set_int] :
( ( X5 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X5 )
=> ? [Xa: int] :
( ( member_int @ Xa @ X5 )
& ( ord_less_int @ X3 @ Xa ) ) )
=> ~ ( finite_finite_int @ X5 ) ) ) ).
% infinite_growing
thf(fact_223_card__subset__eq,axiom,
! [B2: set_real,A2: set_real] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ( finite_card_real @ A2 )
= ( finite_card_real @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_224_card__subset__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_225_card__subset__eq,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_226_infinite__arbitrarily__large,axiom,
! [A2: set_real,N: nat] :
( ~ ( finite_finite_real @ A2 )
=> ? [B7: set_real] :
( ( finite_finite_real @ B7 )
& ( ( finite_card_real @ B7 )
= N )
& ( ord_less_eq_set_real @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_227_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N )
& ( ord_less_eq_set_nat @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_228_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B7: set_a] :
( ( finite_finite_a @ B7 )
& ( ( finite_card_a @ B7 )
= N )
& ( ord_less_eq_set_a @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_229_psubset__card__mono,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_230_psubset__card__mono,axiom,
! [B2: set_real,A2: set_real] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_set_real @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_231_psubset__card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_232_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_233_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_234_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_235_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_236_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_237_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_238_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_239_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_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_card__psubset,axiom,
! [B2: set_real,A2: set_real] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B2 ) )
=> ( ord_less_set_real @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_248_card__psubset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_249_card__psubset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_250_card__ge__0__finite,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ( finite_finite_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_251_card__ge__0__finite,axiom,
! [A2: set_real] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A2 ) )
=> ( finite_finite_real @ A2 ) ) ).
% card_ge_0_finite
thf(fact_252_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_253_card__mono,axiom,
! [B2: set_real,A2: set_real] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B2 ) ) ) ) ).
% card_mono
thf(fact_254_card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_255_card__mono,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_256_card__seteq,axiom,
! [B2: set_real,A2: set_real] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_real @ B2 ) @ ( finite_card_real @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_257_card__seteq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_258_card__seteq,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_259_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 )
=> ? [B7: set_real] :
( ( ord_less_eq_set_real @ A2 @ B7 )
& ( ord_less_eq_set_real @ B7 @ C )
& ( ( finite_card_real @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_260_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C )
& ( ( finite_card_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_261_exists__subset__between,axiom,
! [A2: set_a,N: nat,C: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C ) )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( finite_finite_a @ C )
=> ? [B7: set_a] :
( ( ord_less_eq_set_a @ A2 @ B7 )
& ( ord_less_eq_set_a @ B7 @ C )
& ( ( finite_card_a @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_262_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_real] :
( ( ord_less_eq_nat @ N @ ( finite_card_real @ S2 ) )
=> ~ ! [T3: set_real] :
( ( ord_less_eq_set_real @ T3 @ S2 )
=> ( ( ( finite_card_real @ T3 )
= N )
=> ~ ( finite_finite_real @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_263_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S2 ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S2 )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_264_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S2 ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S2 )
=> ( ( ( finite_card_a @ T3 )
= N )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_265_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_real,C: nat] :
( ! [G: set_real] :
( ( ord_less_eq_set_real @ G @ F2 )
=> ( ( finite_finite_real @ G )
=> ( ord_less_eq_nat @ ( finite_card_real @ G ) @ C ) ) )
=> ( ( finite_finite_real @ F2 )
& ( ord_less_eq_nat @ ( finite_card_real @ F2 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_266_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_267_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C: nat] :
( ! [G: set_a] :
( ( ord_less_eq_set_a @ G @ F2 )
=> ( ( finite_finite_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ G ) @ C ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_268_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_269_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_270_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_271_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_272_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_273_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_274_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_275_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_276_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_277_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 ) )
=> ! [A7: nat] :
( ( member_nat @ A7 @ A2 )
=> ( ord_less_eq_nat @ X @ A7 ) ) ) ) ) ).
% Min.boundedE
thf(fact_278_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 ) )
=> ! [A7: int] :
( ( member_int @ A7 @ A2 )
=> ( ord_less_eq_int @ X @ A7 ) ) ) ) ) ).
% Min.boundedE
thf(fact_279_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 ) )
=> ! [A7: real] :
( ( member_real @ A7 @ A2 )
=> ( ord_less_eq_real @ X @ A7 ) ) ) ) ) ).
% Min.boundedE
thf(fact_280_Min_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ord_less_eq_nat @ X @ A5 ) )
=> ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.boundedI
thf(fact_281_Min_OboundedI,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ! [A5: int] :
( ( member_int @ A5 @ A2 )
=> ( ord_less_eq_int @ X @ A5 ) )
=> ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A2 ) ) ) ) ) ).
% Min.boundedI
thf(fact_282_Min_OboundedI,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ! [A5: real] :
( ( member_real @ A5 @ A2 )
=> ( ord_less_eq_real @ X @ A5 ) )
=> ( ord_less_eq_real @ X @ ( lattic3629708407755379051n_real @ A2 ) ) ) ) ) ).
% Min.boundedI
thf(fact_283_Min__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_nat @ X2 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_284_Min__less__iff,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_int @ ( lattic8718645017227715691in_int @ A2 ) @ X )
= ( ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( ord_less_int @ X2 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_285_Min__less__iff,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( ord_less_real @ ( lattic3629708407755379051n_real @ A2 ) @ X )
= ( ? [X2: real] :
( ( member_real @ X2 @ A2 )
& ( ord_less_real @ X2 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_286_card__gt__0__iff,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
= ( ( A2 != bot_bot_set_nat )
& ( finite_finite_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_287_card__gt__0__iff,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
= ( ( A2 != bot_bot_set_a )
& ( finite_finite_a @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_288_card__gt__0__iff,axiom,
! [A2: set_real] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A2 ) )
= ( ( A2 != bot_bot_set_real )
& ( finite_finite_real @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_289_card__gt__0__iff,axiom,
! [A2: set_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ A2 ) )
= ( ( A2 != bot_bot_set_set_a )
& ( finite_finite_set_a @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_290_Min__antimono,axiom,
! [M5: set_nat,N4: set_nat] :
( ( ord_less_eq_set_nat @ M5 @ N4 )
=> ( ( M5 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N4 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N4 ) @ ( lattic8721135487736765967in_nat @ M5 ) ) ) ) ) ).
% Min_antimono
thf(fact_291_Min__antimono,axiom,
! [M5: set_int,N4: set_int] :
( ( ord_less_eq_set_int @ M5 @ N4 )
=> ( ( M5 != bot_bot_set_int )
=> ( ( finite_finite_int @ N4 )
=> ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ N4 ) @ ( lattic8718645017227715691in_int @ M5 ) ) ) ) ) ).
% Min_antimono
thf(fact_292_Min__antimono,axiom,
! [M5: set_real,N4: set_real] :
( ( ord_less_eq_set_real @ M5 @ N4 )
=> ( ( M5 != bot_bot_set_real )
=> ( ( finite_finite_real @ N4 )
=> ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ N4 ) @ ( lattic3629708407755379051n_real @ M5 ) ) ) ) ) ).
% Min_antimono
thf(fact_293_Min_Osubset__imp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B2 ) @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_294_Min_Osubset__imp,axiom,
! [A2: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( finite_finite_int @ B2 )
=> ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ B2 ) @ ( lattic8718645017227715691in_int @ A2 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_295_Min_Osubset__imp,axiom,
! [A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ( finite_finite_real @ B2 )
=> ( ord_less_eq_real @ ( lattic3629708407755379051n_real @ B2 ) @ ( lattic3629708407755379051n_real @ A2 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_296_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_297_div__mult__mult1__if,axiom,
! [C2: nat,A: nat,B3: nat] :
( ( ( C2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B3 ) )
= zero_zero_nat ) )
& ( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B3 ) )
= ( divide_divide_nat @ A @ B3 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_298_div__mult__mult1__if,axiom,
! [C2: int,A: int,B3: int] :
( ( ( C2 = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) )
= zero_zero_int ) )
& ( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) )
= ( divide_divide_int @ A @ B3 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_299_div__mult__mult2,axiom,
! [C2: nat,A: nat,B3: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B3 @ C2 ) )
= ( divide_divide_nat @ A @ B3 ) ) ) ).
% div_mult_mult2
thf(fact_300_div__mult__mult2,axiom,
! [C2: int,A: int,B3: int] :
( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B3 @ C2 ) )
= ( divide_divide_int @ A @ B3 ) ) ) ).
% div_mult_mult2
thf(fact_301_div__mult__mult1,axiom,
! [C2: nat,A: nat,B3: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B3 ) )
= ( divide_divide_nat @ A @ B3 ) ) ) ).
% div_mult_mult1
thf(fact_302_div__mult__mult1,axiom,
! [C2: int,A: int,B3: int] :
( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) )
= ( divide_divide_int @ A @ B3 ) ) ) ).
% div_mult_mult1
thf(fact_303_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C2: real,A: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ C2 @ B3 ) )
= ( divide_divide_real @ A @ B3 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_304_nonzero__mult__div__cancel__right,axiom,
! [B3: real,A: real] :
( ( B3 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B3 ) @ B3 )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_305_nonzero__mult__div__cancel__right,axiom,
! [B3: nat,A: nat] :
( ( B3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B3 ) @ B3 )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_306_nonzero__mult__div__cancel__right,axiom,
! [B3: int,A: int] :
( ( B3 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B3 ) @ B3 )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_307_nonzero__mult__divide__mult__cancel__right,axiom,
! [C2: real,A: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B3 @ C2 ) )
= ( divide_divide_real @ A @ B3 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_308_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C2: real,A: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ B3 @ C2 ) )
= ( divide_divide_real @ A @ B3 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_309_nonzero__mult__div__cancel__left,axiom,
! [A: real,B3: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B3 ) @ A )
= B3 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_310_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B3: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B3 ) @ A )
= B3 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_311_nonzero__mult__div__cancel__left,axiom,
! [A: int,B3: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B3 ) @ A )
= B3 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_312_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_313_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_314_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_315_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_316_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_317_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_318_mult__eq__0__iff,axiom,
! [A: real,B3: real] :
( ( ( times_times_real @ A @ B3 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_319_mult__eq__0__iff,axiom,
! [A: nat,B3: nat] :
( ( ( times_times_nat @ A @ B3 )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_320_mult__eq__0__iff,axiom,
! [A: int,B3: int] :
( ( ( times_times_int @ A @ B3 )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B3 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_321_mult__cancel__left,axiom,
! [C2: real,A: real,B3: real] :
( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B3 ) )
= ( ( C2 = zero_zero_real )
| ( A = B3 ) ) ) ).
% mult_cancel_left
thf(fact_322_mult__cancel__left,axiom,
! [C2: nat,A: nat,B3: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B3 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B3 ) ) ) ).
% mult_cancel_left
thf(fact_323_mult__cancel__left,axiom,
! [C2: int,A: int,B3: int] :
( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B3 ) )
= ( ( C2 = zero_zero_int )
| ( A = B3 ) ) ) ).
% mult_cancel_left
thf(fact_324_mult__cancel__right,axiom,
! [A: real,C2: real,B3: real] :
( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B3 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A = B3 ) ) ) ).
% mult_cancel_right
thf(fact_325_mult__cancel__right,axiom,
! [A: nat,C2: nat,B3: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B3 @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B3 ) ) ) ).
% mult_cancel_right
thf(fact_326_mult__cancel__right,axiom,
! [A: int,C2: int,B3: int] :
( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B3 @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A = B3 ) ) ) ).
% mult_cancel_right
thf(fact_327_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_328_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_329_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_330_divide__eq__0__iff,axiom,
! [A: real,B3: real] :
( ( ( divide_divide_real @ A @ B3 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_331_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_332_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_333_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_334_divide__cancel__left,axiom,
! [C2: real,A: real,B3: real] :
( ( ( divide_divide_real @ C2 @ A )
= ( divide_divide_real @ C2 @ B3 ) )
= ( ( C2 = zero_zero_real )
| ( A = B3 ) ) ) ).
% divide_cancel_left
thf(fact_335_divide__cancel__right,axiom,
! [A: real,C2: real,B3: real] :
( ( ( divide_divide_real @ A @ C2 )
= ( divide_divide_real @ B3 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A = B3 ) ) ) ).
% divide_cancel_right
thf(fact_336_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_337_times__divide__eq__right,axiom,
! [A: real,B3: real,C2: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B3 @ C2 ) )
= ( divide_divide_real @ ( times_times_real @ A @ B3 ) @ C2 ) ) ).
% times_divide_eq_right
thf(fact_338_divide__divide__eq__right,axiom,
! [A: real,B3: real,C2: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B3 @ C2 ) )
= ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ B3 ) ) ).
% divide_divide_eq_right
thf(fact_339_divide__divide__eq__left,axiom,
! [A: real,B3: real,C2: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B3 ) @ C2 )
= ( divide_divide_real @ A @ ( times_times_real @ B3 @ C2 ) ) ) ).
% divide_divide_eq_left
thf(fact_340_times__divide__eq__left,axiom,
! [B3: real,C2: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B3 @ C2 ) @ A )
= ( divide_divide_real @ ( times_times_real @ B3 @ A ) @ C2 ) ) ).
% times_divide_eq_left
thf(fact_341_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_342_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_343_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_344_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_345_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_346_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_347_mult__divide__mult__cancel__left__if,axiom,
! [C2: real,A: real,B3: real] :
( ( ( C2 = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) )
= zero_zero_real ) )
& ( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) )
= ( divide_divide_real @ A @ B3 ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_348_nonzero__mult__divide__mult__cancel__left,axiom,
! [C2: real,A: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) )
= ( divide_divide_real @ A @ B3 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_349_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_350_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_351_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_352_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_353_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_354_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_355_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_356_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_357_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_358_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_359_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_360_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_361_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_362_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_363_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_364_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_365_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_366_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_367_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_368_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_369_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_370_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_371_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_372_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_373_linordered__field__no__lb,axiom,
! [X4: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X4 ) ).
% linordered_field_no_lb
thf(fact_374_linordered__field__no__ub,axiom,
! [X4: real] :
? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_375_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_376_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_377_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_378_mult__not__zero,axiom,
! [A: real,B3: real] :
( ( ( times_times_real @ A @ B3 )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B3 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_379_mult__not__zero,axiom,
! [A: nat,B3: nat] :
( ( ( times_times_nat @ A @ B3 )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B3 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_380_mult__not__zero,axiom,
! [A: int,B3: int] :
( ( ( times_times_int @ A @ B3 )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B3 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_381_divisors__zero,axiom,
! [A: real,B3: real] :
( ( ( times_times_real @ A @ B3 )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_382_divisors__zero,axiom,
! [A: nat,B3: nat] :
( ( ( times_times_nat @ A @ B3 )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_383_divisors__zero,axiom,
! [A: int,B3: int] :
( ( ( times_times_int @ A @ B3 )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B3 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_384_no__zero__divisors,axiom,
! [A: real,B3: real] :
( ( A != zero_zero_real )
=> ( ( B3 != zero_zero_real )
=> ( ( times_times_real @ A @ B3 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_385_no__zero__divisors,axiom,
! [A: nat,B3: nat] :
( ( A != zero_zero_nat )
=> ( ( B3 != zero_zero_nat )
=> ( ( times_times_nat @ A @ B3 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_386_no__zero__divisors,axiom,
! [A: int,B3: int] :
( ( A != zero_zero_int )
=> ( ( B3 != zero_zero_int )
=> ( ( times_times_int @ A @ B3 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_387_mult__left__cancel,axiom,
! [C2: real,A: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B3 ) )
= ( A = B3 ) ) ) ).
% mult_left_cancel
thf(fact_388_mult__left__cancel,axiom,
! [C2: nat,A: nat,B3: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B3 ) )
= ( A = B3 ) ) ) ).
% mult_left_cancel
thf(fact_389_mult__left__cancel,axiom,
! [C2: int,A: int,B3: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B3 ) )
= ( A = B3 ) ) ) ).
% mult_left_cancel
thf(fact_390_mult__right__cancel,axiom,
! [C2: real,A: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B3 @ C2 ) )
= ( A = B3 ) ) ) ).
% mult_right_cancel
thf(fact_391_mult__right__cancel,axiom,
! [C2: nat,A: nat,B3: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B3 @ C2 ) )
= ( A = B3 ) ) ) ).
% mult_right_cancel
thf(fact_392_mult__right__cancel,axiom,
! [C2: int,A: int,B3: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B3 @ C2 ) )
= ( A = B3 ) ) ) ).
% mult_right_cancel
thf(fact_393_divide__divide__eq__left_H,axiom,
! [A: real,B3: real,C2: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B3 ) @ C2 )
= ( divide_divide_real @ A @ ( times_times_real @ C2 @ B3 ) ) ) ).
% divide_divide_eq_left'
thf(fact_394_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z2: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z2 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z2 ) ) ) ).
% divide_divide_times_eq
thf(fact_395_times__divide__times__eq,axiom,
! [X: real,Y: real,Z2: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z2 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_396_mult__mono,axiom,
! [A: nat,B3: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_397_mult__mono,axiom,
! [A: real,B3: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_398_mult__mono,axiom,
! [A: int,B3: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_399_mult__mono_H,axiom,
! [A: nat,B3: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( 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 @ B3 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_400_mult__mono_H,axiom,
! [A: real,B3: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( 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 @ B3 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_401_mult__mono_H,axiom,
! [A: int,B3: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( 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 @ B3 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_402_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_403_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_404_split__mult__pos__le,axiom,
! [A: real,B3: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B3 ) ) ) ).
% split_mult_pos_le
thf(fact_405_split__mult__pos__le,axiom,
! [A: int,B3: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B3 ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B3 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B3 ) ) ) ).
% split_mult_pos_le
thf(fact_406_mult__left__mono__neg,axiom,
! [B3: real,A: real,C2: real] :
( ( ord_less_eq_real @ B3 @ A )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_407_mult__left__mono__neg,axiom,
! [B3: int,A: int,C2: int] :
( ( ord_less_eq_int @ B3 @ A )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_408_mult__nonpos__nonpos,axiom,
! [A: real,B3: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B3 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_409_mult__nonpos__nonpos,axiom,
! [A: int,B3: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B3 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B3 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_410_mult__left__mono,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_411_mult__left__mono,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_412_mult__left__mono,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_413_mult__right__mono__neg,axiom,
! [B3: real,A: real,C2: real] :
( ( ord_less_eq_real @ B3 @ A )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_414_mult__right__mono__neg,axiom,
! [B3: int,A: int,C2: int] :
( ( ord_less_eq_int @ B3 @ A )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B3 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_415_mult__right__mono,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B3 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_416_mult__right__mono,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_417_mult__right__mono,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B3 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_418_mult__le__0__iff,axiom,
! [A: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B3 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).
% mult_le_0_iff
thf(fact_419_mult__le__0__iff,axiom,
! [A: int,B3: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B3 ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B3 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ) ) ).
% mult_le_0_iff
thf(fact_420_split__mult__neg__le,axiom,
! [A: nat,B3: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B3 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B3 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_421_split__mult__neg__le,axiom,
! [A: real,B3: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B3 ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_422_split__mult__neg__le,axiom,
! [A: int,B3: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B3 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B3 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B3 ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_423_mult__nonneg__nonneg,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B3 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_424_mult__nonneg__nonneg,axiom,
! [A: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B3 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_425_mult__nonneg__nonneg,axiom,
! [A: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B3 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_426_mult__nonneg__nonpos,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_427_mult__nonneg__nonpos,axiom,
! [A: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B3 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_428_mult__nonneg__nonpos,axiom,
! [A: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B3 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B3 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_429_mult__nonpos__nonneg,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_430_mult__nonpos__nonneg,axiom,
! [A: real,B3: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B3 ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_431_mult__nonpos__nonneg,axiom,
! [A: int,B3: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B3 ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_432_mult__nonneg__nonpos2,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B3 @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_433_mult__nonneg__nonpos2,axiom,
! [A: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B3 @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_434_mult__nonneg__nonpos2,axiom,
! [A: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B3 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B3 @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_435_zero__le__mult__iff,axiom,
! [A: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B3 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_436_zero__le__mult__iff,axiom,
! [A: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B3 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B3 ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B3 @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_437_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_438_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_439_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_440_mult__neg__neg,axiom,
! [A: real,B3: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B3 ) ) ) ) ).
% mult_neg_neg
thf(fact_441_mult__neg__neg,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B3 ) ) ) ) ).
% mult_neg_neg
thf(fact_442_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_443_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_444_mult__less__0__iff,axiom,
! [A: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A @ B3 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B3 ) ) ) ) ).
% mult_less_0_iff
thf(fact_445_mult__less__0__iff,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ A @ B3 ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B3 @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B3 ) ) ) ) ).
% mult_less_0_iff
thf(fact_446_mult__neg__pos,axiom,
! [A: real,B3: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ ( times_times_real @ A @ B3 ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_447_mult__neg__pos,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ ( times_times_nat @ A @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_448_mult__neg__pos,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ ( times_times_int @ A @ B3 ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_449_mult__pos__neg,axiom,
! [A: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B3 ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_450_mult__pos__neg,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_451_mult__pos__neg,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B3 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_452_mult__pos__pos,axiom,
! [A: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B3 ) ) ) ) ).
% mult_pos_pos
thf(fact_453_mult__pos__pos,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B3 ) ) ) ) ).
% mult_pos_pos
thf(fact_454_mult__pos__pos,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B3 ) ) ) ) ).
% mult_pos_pos
thf(fact_455_mult__pos__neg2,axiom,
! [A: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B3 @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_456_mult__pos__neg2,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B3 @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_457_mult__pos__neg2,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B3 @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_458_zero__less__mult__iff,axiom,
! [A: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B3 @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_459_zero__less__mult__iff,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B3 ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B3 ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B3 @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_460_zero__less__mult__pos,axiom,
! [A: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_461_zero__less__mult__pos,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_462_zero__less__mult__pos,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B3 ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_463_zero__less__mult__pos2,axiom,
! [B3: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B3 @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_464_zero__less__mult__pos2,axiom,
! [B3: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B3 @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_465_zero__less__mult__pos2,axiom,
! [B3: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B3 @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_466_mult__less__cancel__left__neg,axiom,
! [C2: real,A: real,B3: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) )
= ( ord_less_real @ B3 @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_467_mult__less__cancel__left__neg,axiom,
! [C2: int,A: int,B3: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) )
= ( ord_less_int @ B3 @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_468_mult__less__cancel__left__pos,axiom,
! [C2: real,A: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) )
= ( ord_less_real @ A @ B3 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_469_mult__less__cancel__left__pos,axiom,
! [C2: int,A: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) )
= ( ord_less_int @ A @ B3 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_470_mult__strict__left__mono__neg,axiom,
! [B3: real,A: real,C2: real] :
( ( ord_less_real @ B3 @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_471_mult__strict__left__mono__neg,axiom,
! [B3: int,A: int,C2: int] :
( ( ord_less_int @ B3 @ A )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_472_mult__strict__left__mono,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_real @ A @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_473_mult__strict__left__mono,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B3 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_474_mult__strict__left__mono,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_int @ A @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_475_mult__less__cancel__left__disj,axiom,
! [C2: real,A: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A @ B3 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B3 @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_476_mult__less__cancel__left__disj,axiom,
! [C2: int,A: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A @ B3 ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B3 @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_477_mult__strict__right__mono__neg,axiom,
! [B3: real,A: real,C2: real] :
( ( ord_less_real @ B3 @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_478_mult__strict__right__mono__neg,axiom,
! [B3: int,A: int,C2: int] :
( ( ord_less_int @ B3 @ A )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B3 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_479_mult__strict__right__mono,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_real @ A @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_480_mult__strict__right__mono,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B3 @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_481_mult__strict__right__mono,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_int @ A @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B3 @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_482_mult__less__cancel__right__disj,axiom,
! [A: real,C2: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B3 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A @ B3 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B3 @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_483_mult__less__cancel__right__disj,axiom,
! [A: int,C2: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B3 @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A @ B3 ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B3 @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_484_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_real @ A @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_485_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_486_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_int @ A @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_487_divide__le__0__iff,axiom,
! [A: real,B3: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B3 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).
% divide_le_0_iff
thf(fact_488_divide__right__mono,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B3 @ C2 ) ) ) ) ).
% divide_right_mono
thf(fact_489_zero__le__divide__iff,axiom,
! [A: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B3 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_490_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_491_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_492_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_493_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_494_divide__right__mono__neg,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C2 ) @ ( divide_divide_real @ A @ C2 ) ) ) ) ).
% divide_right_mono_neg
thf(fact_495_divide__neg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_496_divide__neg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_497_divide__pos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_498_divide__pos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_499_divide__less__0__iff,axiom,
! [A: real,B3: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B3 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B3 ) ) ) ) ).
% divide_less_0_iff
thf(fact_500_divide__less__cancel,axiom,
! [A: real,C2: real,B3: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B3 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B3 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B3 @ A ) )
& ( C2 != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_501_zero__less__divide__iff,axiom,
! [A: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B3 @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_502_divide__strict__right__mono,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_real @ A @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B3 @ C2 ) ) ) ) ).
% divide_strict_right_mono
thf(fact_503_divide__strict__right__mono__neg,axiom,
! [B3: real,A: real,C2: real] :
( ( ord_less_real @ B3 @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B3 @ C2 ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_504_frac__eq__eq,axiom,
! [Y: real,Z2: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W @ Z2 ) )
= ( ( times_times_real @ X @ Z2 )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_505_divide__eq__eq,axiom,
! [B3: real,C2: real,A: real] :
( ( ( divide_divide_real @ B3 @ C2 )
= A )
= ( ( ( C2 != zero_zero_real )
=> ( B3
= ( times_times_real @ A @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_506_eq__divide__eq,axiom,
! [A: real,B3: real,C2: real] :
( ( A
= ( divide_divide_real @ B3 @ C2 ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ A @ C2 )
= B3 ) )
& ( ( C2 = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_507_divide__eq__imp,axiom,
! [C2: real,B3: real,A: real] :
( ( C2 != zero_zero_real )
=> ( ( B3
= ( times_times_real @ A @ C2 ) )
=> ( ( divide_divide_real @ B3 @ C2 )
= A ) ) ) ).
% divide_eq_imp
thf(fact_508_eq__divide__imp,axiom,
! [C2: real,A: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ C2 )
= B3 )
=> ( A
= ( divide_divide_real @ B3 @ C2 ) ) ) ) ).
% eq_divide_imp
thf(fact_509_nonzero__divide__eq__eq,axiom,
! [C2: real,B3: real,A: real] :
( ( C2 != zero_zero_real )
=> ( ( ( divide_divide_real @ B3 @ C2 )
= A )
= ( B3
= ( times_times_real @ A @ C2 ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_510_nonzero__eq__divide__eq,axiom,
! [C2: real,A: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B3 @ C2 ) )
= ( ( times_times_real @ A @ C2 )
= B3 ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_511_mult__le__cancel__left,axiom,
! [C2: real,A: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B3 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_512_mult__le__cancel__left,axiom,
! [C2: int,A: int,B3: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B3 ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B3 @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_513_mult__le__cancel__right,axiom,
! [A: real,C2: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B3 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B3 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_514_mult__le__cancel__right,axiom,
! [A: int,C2: int,B3: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B3 @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B3 ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B3 @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_515_mult__left__less__imp__less,axiom,
! [C2: nat,A: nat,B3: nat] :
( ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B3 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A @ B3 ) ) ) ).
% mult_left_less_imp_less
thf(fact_516_mult__left__less__imp__less,axiom,
! [C2: real,A: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B3 ) ) ) ).
% mult_left_less_imp_less
thf(fact_517_mult__left__less__imp__less,axiom,
! [C2: int,A: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B3 ) ) ) ).
% mult_left_less_imp_less
thf(fact_518_mult__strict__mono,axiom,
! [A: nat,B3: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_519_mult__strict__mono,axiom,
! [A: real,B3: real,C2: real,D: real] :
( ( ord_less_real @ A @ B3 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_520_mult__strict__mono,axiom,
! [A: int,B3: int,C2: int,D: int] :
( ( ord_less_int @ A @ B3 )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_521_mult__less__cancel__left,axiom,
! [C2: real,A: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B3 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B3 @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_522_mult__less__cancel__left,axiom,
! [C2: int,A: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B3 ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B3 @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_523_mult__right__less__imp__less,axiom,
! [A: nat,C2: nat,B3: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B3 @ C2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A @ B3 ) ) ) ).
% mult_right_less_imp_less
thf(fact_524_mult__right__less__imp__less,axiom,
! [A: real,C2: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B3 @ C2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B3 ) ) ) ).
% mult_right_less_imp_less
thf(fact_525_mult__right__less__imp__less,axiom,
! [A: int,C2: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B3 @ C2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B3 ) ) ) ).
% mult_right_less_imp_less
thf(fact_526_mult__strict__mono_H,axiom,
! [A: nat,B3: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_527_mult__strict__mono_H,axiom,
! [A: real,B3: real,C2: real,D: real] :
( ( ord_less_real @ A @ B3 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_528_mult__strict__mono_H,axiom,
! [A: int,B3: int,C2: int,D: int] :
( ( ord_less_int @ A @ B3 )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_529_mult__less__cancel__right,axiom,
! [A: real,C2: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B3 @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B3 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B3 @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_530_mult__less__cancel__right,axiom,
! [A: int,C2: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B3 @ C2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B3 ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B3 @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_531_mult__le__cancel__left__neg,axiom,
! [C2: real,A: real,B3: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) )
= ( ord_less_eq_real @ B3 @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_532_mult__le__cancel__left__neg,axiom,
! [C2: int,A: int,B3: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) )
= ( ord_less_eq_int @ B3 @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_533_mult__le__cancel__left__pos,axiom,
! [C2: real,A: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) )
= ( ord_less_eq_real @ A @ B3 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_534_mult__le__cancel__left__pos,axiom,
! [C2: int,A: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) )
= ( ord_less_eq_int @ A @ B3 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_535_mult__left__le__imp__le,axiom,
! [C2: nat,A: nat,B3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A @ B3 ) ) ) ).
% mult_left_le_imp_le
thf(fact_536_mult__left__le__imp__le,axiom,
! [C2: real,A: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B3 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B3 ) ) ) ).
% mult_left_le_imp_le
thf(fact_537_mult__left__le__imp__le,axiom,
! [C2: int,A: int,B3: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B3 ) )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B3 ) ) ) ).
% mult_left_le_imp_le
thf(fact_538_mult__right__le__imp__le,axiom,
! [A: nat,C2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B3 @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A @ B3 ) ) ) ).
% mult_right_le_imp_le
thf(fact_539_mult__right__le__imp__le,axiom,
! [A: real,C2: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B3 @ C2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B3 ) ) ) ).
% mult_right_le_imp_le
thf(fact_540_mult__right__le__imp__le,axiom,
! [A: int,C2: int,B3: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B3 @ C2 ) )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B3 ) ) ) ).
% mult_right_le_imp_le
thf(fact_541_mult__le__less__imp__less,axiom,
! [A: nat,B3: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_542_mult__le__less__imp__less,axiom,
! [A: real,B3: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_543_mult__le__less__imp__less,axiom,
! [A: int,B3: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_544_mult__less__le__imp__less,axiom,
! [A: nat,B3: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_545_mult__less__le__imp__less,axiom,
! [A: real,B3: real,C2: real,D: real] :
( ( ord_less_real @ A @ B3 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_546_mult__less__le__imp__less,axiom,
! [A: int,B3: int,C2: int,D: int] :
( ( ord_less_int @ A @ B3 )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_547_frac__le,axiom,
! [Y: real,X: real,W: real,Z2: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% frac_le
thf(fact_548_frac__less,axiom,
! [X: real,Y: real,W: real,Z2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% frac_less
thf(fact_549_frac__less2,axiom,
! [X: real,Y: real,W: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_real @ W @ Z2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% frac_less2
thf(fact_550_divide__le__cancel,axiom,
! [A: real,C2: real,B3: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B3 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B3 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_551_divide__nonneg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_552_divide__nonneg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_pos
thf(fact_553_divide__nonpos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_neg
thf(fact_554_divide__nonpos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_555_divide__less__eq,axiom,
! [B3: real,C2: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B3 @ C2 ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ B3 @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B3 ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_556_less__divide__eq,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_real @ A @ ( divide_divide_real @ B3 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B3 ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B3 @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_557_neg__divide__less__eq,axiom,
! [C2: real,B3: real,A: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B3 @ C2 ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B3 ) ) ) ).
% neg_divide_less_eq
thf(fact_558_neg__less__divide__eq,axiom,
! [C2: real,A: real,B3: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B3 @ C2 ) )
= ( ord_less_real @ B3 @ ( times_times_real @ A @ C2 ) ) ) ) ).
% neg_less_divide_eq
thf(fact_559_pos__divide__less__eq,axiom,
! [C2: real,B3: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( divide_divide_real @ B3 @ C2 ) @ A )
= ( ord_less_real @ B3 @ ( times_times_real @ A @ C2 ) ) ) ) ).
% pos_divide_less_eq
thf(fact_560_pos__less__divide__eq,axiom,
! [C2: real,A: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B3 @ C2 ) )
= ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B3 ) ) ) ).
% pos_less_divide_eq
thf(fact_561_mult__imp__div__pos__less,axiom,
! [Y: real,X: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z2 @ Y ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z2 ) ) ) ).
% mult_imp_div_pos_less
thf(fact_562_mult__imp__less__div__pos,axiom,
! [Y: real,Z2: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( times_times_real @ Z2 @ Y ) @ X )
=> ( ord_less_real @ Z2 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_563_divide__strict__left__mono,axiom,
! [B3: real,A: real,C2: real] :
( ( ord_less_real @ B3 @ A )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B3 ) )
=> ( ord_less_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B3 ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_564_divide__strict__left__mono__neg,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_real @ A @ B3 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B3 ) )
=> ( ord_less_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B3 ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_565_divide__le__eq,axiom,
! [B3: real,C2: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C2 ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ B3 @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B3 ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_566_le__divide__eq,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B3 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B3 ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_567_divide__left__mono,axiom,
! [B3: real,A: real,C2: real] :
( ( ord_less_eq_real @ B3 @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B3 ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B3 ) ) ) ) ) ).
% divide_left_mono
thf(fact_568_neg__divide__le__eq,axiom,
! [C2: real,B3: real,A: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C2 ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B3 ) ) ) ).
% neg_divide_le_eq
thf(fact_569_neg__le__divide__eq,axiom,
! [C2: real,A: real,B3: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B3 @ C2 ) )
= ( ord_less_eq_real @ B3 @ ( times_times_real @ A @ C2 ) ) ) ) ).
% neg_le_divide_eq
thf(fact_570_pos__divide__le__eq,axiom,
! [C2: real,B3: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C2 ) @ A )
= ( ord_less_eq_real @ B3 @ ( times_times_real @ A @ C2 ) ) ) ) ).
% pos_divide_le_eq
thf(fact_571_pos__le__divide__eq,axiom,
! [C2: real,A: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B3 @ C2 ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B3 ) ) ) ).
% pos_le_divide_eq
thf(fact_572_mult__imp__div__pos__le,axiom,
! [Y: real,X: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ ( times_times_real @ Z2 @ Y ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z2 ) ) ) ).
% mult_imp_div_pos_le
thf(fact_573_mult__imp__le__div__pos,axiom,
! [Y: real,Z2: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ Y ) @ X )
=> ( ord_less_eq_real @ Z2 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_574_divide__left__mono__neg,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B3 ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B3 ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_575_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_576_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_577_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_578_psubsetI,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_579_empty__subsetI,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% empty_subsetI
thf(fact_580_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_581_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_582_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_583_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_584_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_585_infinite__sumset__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) )
= ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
& ( ( inf_inf_set_a @ B2 @ g )
!= bot_bot_set_a ) )
| ( ( ( inf_inf_set_a @ A2 @ g )
!= bot_bot_set_a )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ B2 @ g ) ) ) ) ) ).
% infinite_sumset_iff
thf(fact_586_infinite__sumset__aux,axiom,
! [A2: set_a,B2: set_a] :
( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) )
= ( ( inf_inf_set_a @ B2 @ g )
!= bot_bot_set_a ) ) ) ).
% infinite_sumset_aux
thf(fact_587_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_588_empty__iff,axiom,
! [C2: real] :
~ ( member_real @ C2 @ bot_bot_set_real ) ).
% empty_iff
thf(fact_589_empty__iff,axiom,
! [C2: set_a] :
~ ( member_set_a @ C2 @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_590_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_591_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_592_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_593_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_594_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_595_Collect__empty__eq,axiom,
! [P: real > $o] :
( ( ( collect_real @ P )
= bot_bot_set_real )
= ( ! [X2: real] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_596_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_597_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_598_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_599_empty__Collect__eq,axiom,
! [P: real > $o] :
( ( bot_bot_set_real
= ( collect_real @ P ) )
= ( ! [X2: real] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_600_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_601_subsetI,axiom,
! [A2: set_real,B2: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ X3 @ B2 ) )
=> ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% subsetI
thf(fact_602_subsetI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_603_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_604_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_605_insertCI,axiom,
! [A: real,B2: set_real,B3: real] :
( ( ~ ( member_real @ A @ B2 )
=> ( A = B3 ) )
=> ( member_real @ A @ ( insert_real @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_606_insertCI,axiom,
! [A: a,B2: set_a,B3: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B3 ) )
=> ( member_a @ A @ ( insert_a @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_607_insertCI,axiom,
! [A: set_a,B2: set_set_a,B3: set_a] :
( ( ~ ( member_set_a @ A @ B2 )
=> ( A = B3 ) )
=> ( member_set_a @ A @ ( insert_set_a @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_608_insert__iff,axiom,
! [A: real,B3: real,A2: set_real] :
( ( member_real @ A @ ( insert_real @ B3 @ A2 ) )
= ( ( A = B3 )
| ( member_real @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_609_insert__iff,axiom,
! [A: a,B3: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B3 @ A2 ) )
= ( ( A = B3 )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_610_insert__iff,axiom,
! [A: set_a,B3: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B3 @ A2 ) )
= ( ( A = B3 )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_611_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_612_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_613_IntI,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ A2 )
=> ( ( member_real @ C2 @ B2 )
=> ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_614_IntI,axiom,
! [C2: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( ( member_set_a @ C2 @ B2 )
=> ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_615_IntI,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ( member_a @ C2 @ B2 )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_616_Int__iff,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B2 ) )
= ( ( member_real @ C2 @ A2 )
& ( member_real @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_617_Int__iff,axiom,
! [C2: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B2 ) )
= ( ( member_set_a @ C2 @ A2 )
& ( member_set_a @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_618_Int__iff,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( member_a @ C2 @ A2 )
& ( member_a @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_619_sumset__empty_H_I2_J,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% sumset_empty'(2)
thf(fact_620_sumset__empty_H_I1_J,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ A2 )
= bot_bot_set_a ) ) ).
% sumset_empty'(1)
thf(fact_621_finite__sumset_H,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B2 @ g ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ).
% finite_sumset'
thf(fact_622_sumset__subset__insert_I2_J,axiom,
! [A2: set_a,B2: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B2 ) ) ).
% sumset_subset_insert(2)
thf(fact_623_sumset__subset__insert_I1_J,axiom,
! [A2: set_a,B2: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B2 ) ) ) ).
% sumset_subset_insert(1)
thf(fact_624_card__sumset__0__iff_H,axiom,
! [A2: set_a,B2: set_a] :
( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) )
= zero_zero_nat )
| ( ( finite_card_a @ ( inf_inf_set_a @ B2 @ g ) )
= zero_zero_nat ) ) ) ).
% card_sumset_0_iff'
thf(fact_625_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_626_singletonI,axiom,
! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).
% singletonI
thf(fact_627_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_628_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_629_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_630_finite__insert,axiom,
! [A: real,A2: set_real] :
( ( finite_finite_real @ ( insert_real @ A @ A2 ) )
= ( finite_finite_real @ A2 ) ) ).
% finite_insert
thf(fact_631_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_632_insert__subset,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ ( insert_real @ X @ A2 ) @ B2 )
= ( ( member_real @ X @ B2 )
& ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_633_insert__subset,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( ( member_set_a @ X @ B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_634_insert__subset,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( ( member_a @ X @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_635_finite__Int,axiom,
! [F2: set_real,G2: set_real] :
( ( ( finite_finite_real @ F2 )
| ( finite_finite_real @ G2 ) )
=> ( finite_finite_real @ ( inf_inf_set_real @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_636_finite__Int,axiom,
! [F2: set_nat,G2: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G2 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_637_finite__Int,axiom,
! [F2: set_a,G2: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G2 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_638_Int__subset__iff,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ C @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_639_Int__insert__left__if0,axiom,
! [A: real,C: set_real,B2: set_real] :
( ~ ( member_real @ A @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A @ B2 ) @ C )
= ( inf_inf_set_real @ B2 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_640_Int__insert__left__if0,axiom,
! [A: set_a,C: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B2 ) @ C )
= ( inf_inf_set_set_a @ B2 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_641_Int__insert__left__if0,axiom,
! [A: a,C: set_a,B2: set_a] :
( ~ ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C )
= ( inf_inf_set_a @ B2 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_642_Int__insert__left__if1,axiom,
! [A: real,C: set_real,B2: set_real] :
( ( member_real @ A @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A @ B2 ) @ C )
= ( insert_real @ A @ ( inf_inf_set_real @ B2 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_643_Int__insert__left__if1,axiom,
! [A: set_a,C: set_set_a,B2: set_set_a] :
( ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B2 ) @ C )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_644_Int__insert__left__if1,axiom,
! [A: a,C: set_a,B2: set_a] :
( ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C )
= ( insert_a @ A @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_645_insert__inter__insert,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ ( insert_set_a @ A @ B2 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_646_insert__inter__insert,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_647_Int__insert__right__if0,axiom,
! [A: real,A2: set_real,B2: set_real] :
( ~ ( member_real @ A @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B2 ) )
= ( inf_inf_set_real @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_648_Int__insert__right__if0,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B2 ) )
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_649_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_650_Int__insert__right__if1,axiom,
! [A: real,A2: set_real,B2: set_real] :
( ( member_real @ A @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B2 ) )
= ( insert_real @ A @ ( inf_inf_set_real @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_651_Int__insert__right__if1,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B2 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_652_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_653_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_654_singleton__insert__inj__eq,axiom,
! [B3: real,A: real,A2: set_real] :
( ( ( insert_real @ B3 @ bot_bot_set_real )
= ( insert_real @ A @ A2 ) )
= ( ( A = B3 )
& ( ord_less_eq_set_real @ A2 @ ( insert_real @ B3 @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_655_singleton__insert__inj__eq,axiom,
! [B3: set_a,A: set_a,A2: set_set_a] :
( ( ( insert_set_a @ B3 @ bot_bot_set_set_a )
= ( insert_set_a @ A @ A2 ) )
= ( ( A = B3 )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B3 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_656_singleton__insert__inj__eq,axiom,
! [B3: a,A: a,A2: set_a] :
( ( ( insert_a @ B3 @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B3 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_657_singleton__insert__inj__eq_H,axiom,
! [A: real,A2: set_real,B3: real] :
( ( ( insert_real @ A @ A2 )
= ( insert_real @ B3 @ bot_bot_set_real ) )
= ( ( A = B3 )
& ( ord_less_eq_set_real @ A2 @ ( insert_real @ B3 @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_658_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A2: set_set_a,B3: set_a] :
( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B3 @ bot_bot_set_set_a ) )
= ( ( A = B3 )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B3 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_659_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B3: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B3 @ bot_bot_set_a ) )
= ( ( A = B3 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_660_disjoint__insert_I2_J,axiom,
! [A2: set_a,B3: a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B3 @ B2 ) ) )
= ( ~ ( member_a @ B3 @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_661_disjoint__insert_I2_J,axiom,
! [A2: set_real,B3: real,B2: set_real] :
( ( bot_bot_set_real
= ( inf_inf_set_real @ A2 @ ( insert_real @ B3 @ B2 ) ) )
= ( ~ ( member_real @ B3 @ A2 )
& ( bot_bot_set_real
= ( inf_inf_set_real @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_662_disjoint__insert_I2_J,axiom,
! [A2: set_set_a,B3: set_a,B2: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ B3 @ B2 ) ) )
= ( ~ ( member_set_a @ B3 @ A2 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_663_disjoint__insert_I1_J,axiom,
! [B2: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B2 @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ B2 @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_664_disjoint__insert_I1_J,axiom,
! [B2: set_real,A: real,A2: set_real] :
( ( ( inf_inf_set_real @ B2 @ ( insert_real @ A @ A2 ) )
= bot_bot_set_real )
= ( ~ ( member_real @ A @ B2 )
& ( ( inf_inf_set_real @ B2 @ A2 )
= bot_bot_set_real ) ) ) ).
% disjoint_insert(1)
thf(fact_665_disjoint__insert_I1_J,axiom,
! [B2: set_set_a,A: set_a,A2: set_set_a] :
( ( ( inf_inf_set_set_a @ B2 @ ( insert_set_a @ A @ A2 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B2 )
& ( ( inf_inf_set_set_a @ B2 @ A2 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_666_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member_a @ A @ B2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_667_insert__disjoint_I2_J,axiom,
! [A: real,A2: set_real,B2: set_real] :
( ( bot_bot_set_real
= ( inf_inf_set_real @ ( insert_real @ A @ A2 ) @ B2 ) )
= ( ~ ( member_real @ A @ B2 )
& ( bot_bot_set_real
= ( inf_inf_set_real @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_668_insert__disjoint_I2_J,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member_set_a @ A @ B2 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_669_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B2 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_670_insert__disjoint_I1_J,axiom,
! [A: real,A2: set_real,B2: set_real] :
( ( ( inf_inf_set_real @ ( insert_real @ A @ A2 ) @ B2 )
= bot_bot_set_real )
= ( ~ ( member_real @ A @ B2 )
& ( ( inf_inf_set_real @ A2 @ B2 )
= bot_bot_set_real ) ) ) ).
% insert_disjoint(1)
thf(fact_671_insert__disjoint_I1_J,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B2 )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B2 )
& ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_672_Min__singleton,axiom,
! [X: real] :
( ( lattic3629708407755379051n_real @ ( insert_real @ X @ bot_bot_set_real ) )
= X ) ).
% Min_singleton
thf(fact_673_sumset__Int__carrier__eq_I2_J,axiom,
! [A2: set_a,B2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( inf_inf_set_a @ A2 @ g ) @ B2 )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ).
% sumset_Int_carrier_eq(2)
thf(fact_674_sumset__Int__carrier__eq_I1_J,axiom,
! [A2: set_a,B2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( inf_inf_set_a @ B2 @ g ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ).
% sumset_Int_carrier_eq(1)
thf(fact_675_sumset__Int__carrier,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ g )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ).
% sumset_Int_carrier
thf(fact_676_sumset__is__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B2 @ g )
= bot_bot_set_a ) ) ) ).
% sumset_is_empty_iff
thf(fact_677_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_678_disjoint__iff__not__equal,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ B2 )
=> ( X2 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_679_disjoint__iff__not__equal,axiom,
! [A2: set_real,B2: set_real] :
( ( ( inf_inf_set_real @ A2 @ B2 )
= bot_bot_set_real )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ! [Y4: real] :
( ( member_real @ Y4 @ B2 )
=> ( X2 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_680_disjoint__iff__not__equal,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ B2 )
=> ( X2 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_681_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_682_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_683_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_684_Int__empty__left,axiom,
! [B2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B2 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_685_Int__empty__left,axiom,
! [B2: set_real] :
( ( inf_inf_set_real @ bot_bot_set_real @ B2 )
= bot_bot_set_real ) ).
% Int_empty_left
thf(fact_686_Int__empty__left,axiom,
! [B2: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B2 )
= bot_bot_set_set_a ) ).
% Int_empty_left
thf(fact_687_disjoint__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_688_disjoint__iff,axiom,
! [A2: set_real,B2: set_real] :
( ( ( inf_inf_set_real @ A2 @ B2 )
= bot_bot_set_real )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ~ ( member_real @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_689_disjoint__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ~ ( member_set_a @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_690_Int__emptyI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_691_Int__emptyI,axiom,
! [A2: set_real,B2: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ~ ( member_real @ X3 @ B2 ) )
=> ( ( inf_inf_set_real @ A2 @ B2 )
= bot_bot_set_real ) ) ).
% Int_emptyI
thf(fact_692_Int__emptyI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ~ ( member_set_a @ X3 @ B2 ) )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_693_singleton__inject,axiom,
! [A: a,B3: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B3 @ bot_bot_set_a ) )
=> ( A = B3 ) ) ).
% singleton_inject
thf(fact_694_singleton__inject,axiom,
! [A: real,B3: real] :
( ( ( insert_real @ A @ bot_bot_set_real )
= ( insert_real @ B3 @ bot_bot_set_real ) )
=> ( A = B3 ) ) ).
% singleton_inject
thf(fact_695_singleton__inject,axiom,
! [A: set_a,B3: set_a] :
( ( ( insert_set_a @ A @ bot_bot_set_set_a )
= ( insert_set_a @ B3 @ bot_bot_set_set_a ) )
=> ( A = B3 ) ) ).
% singleton_inject
thf(fact_696_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_697_insert__not__empty,axiom,
! [A: real,A2: set_real] :
( ( insert_real @ A @ A2 )
!= bot_bot_set_real ) ).
% insert_not_empty
thf(fact_698_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_699_doubleton__eq__iff,axiom,
! [A: a,B3: a,C2: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B3 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C2 )
& ( B3 = D ) )
| ( ( A = D )
& ( B3 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_700_doubleton__eq__iff,axiom,
! [A: real,B3: real,C2: real,D: real] :
( ( ( insert_real @ A @ ( insert_real @ B3 @ bot_bot_set_real ) )
= ( insert_real @ C2 @ ( insert_real @ D @ bot_bot_set_real ) ) )
= ( ( ( A = C2 )
& ( B3 = D ) )
| ( ( A = D )
& ( B3 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_701_doubleton__eq__iff,axiom,
! [A: set_a,B3: set_a,C2: set_a,D: set_a] :
( ( ( insert_set_a @ A @ ( insert_set_a @ B3 @ bot_bot_set_set_a ) )
= ( insert_set_a @ C2 @ ( insert_set_a @ D @ bot_bot_set_set_a ) ) )
= ( ( ( A = C2 )
& ( B3 = D ) )
| ( ( A = D )
& ( B3 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_702_singleton__iff,axiom,
! [B3: a,A: a] :
( ( member_a @ B3 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B3 = A ) ) ).
% singleton_iff
thf(fact_703_singleton__iff,axiom,
! [B3: real,A: real] :
( ( member_real @ B3 @ ( insert_real @ A @ bot_bot_set_real ) )
= ( B3 = A ) ) ).
% singleton_iff
thf(fact_704_singleton__iff,axiom,
! [B3: set_a,A: set_a] :
( ( member_set_a @ B3 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B3 = A ) ) ).
% singleton_iff
thf(fact_705_singletonD,axiom,
! [B3: a,A: a] :
( ( member_a @ B3 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B3 = A ) ) ).
% singletonD
thf(fact_706_singletonD,axiom,
! [B3: real,A: real] :
( ( member_real @ B3 @ ( insert_real @ A @ bot_bot_set_real ) )
=> ( B3 = A ) ) ).
% singletonD
thf(fact_707_singletonD,axiom,
! [B3: set_a,A: set_a] :
( ( member_set_a @ B3 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B3 = A ) ) ).
% singletonD
thf(fact_708_Int__Collect__mono,axiom,
! [A2: set_real,B2: set_real,P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ! [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 @ B2 @ ( collect_real @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_709_Int__Collect__mono,axiom,
! [A2: set_set_a,B2: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ! [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 @ B2 @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_710_Int__Collect__mono,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [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 @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_711_Int__Collect__mono,axiom,
! [A2: set_a,B2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_712_Int__greatest,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_713_Int__absorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_714_Int__absorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_715_Int__lower2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_716_Int__lower1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_717_Int__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D3: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D3 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C @ D3 ) ) ) ) ).
% Int_mono
thf(fact_718_subset__insertI2,axiom,
! [A2: set_set_a,B2: set_set_a,B3: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B3 @ B2 ) ) ) ).
% subset_insertI2
thf(fact_719_subset__insertI2,axiom,
! [A2: set_a,B2: set_a,B3: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B3 @ B2 ) ) ) ).
% subset_insertI2
thf(fact_720_subset__insertI,axiom,
! [B2: set_set_a,A: set_a] : ( ord_le3724670747650509150_set_a @ B2 @ ( insert_set_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_721_subset__insertI,axiom,
! [B2: set_a,A: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_722_subset__insert,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) )
= ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_723_subset__insert,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_724_subset__insert,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_725_insert__mono,axiom,
! [C: set_set_a,D3: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ C @ D3 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A @ C ) @ ( insert_set_a @ A @ D3 ) ) ) ).
% insert_mono
thf(fact_726_insert__mono,axiom,
! [C: set_a,D3: set_a,A: a] :
( ( ord_less_eq_set_a @ C @ D3 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C ) @ ( insert_a @ A @ D3 ) ) ) ).
% insert_mono
thf(fact_727_IntE,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B2 ) )
=> ~ ( ( member_real @ C2 @ A2 )
=> ~ ( member_real @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_728_IntE,axiom,
! [C2: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ~ ( ( member_set_a @ C2 @ A2 )
=> ~ ( member_set_a @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_729_IntE,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ~ ( member_a @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_730_IntD1,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B2 ) )
=> ( member_real @ C2 @ A2 ) ) ).
% IntD1
thf(fact_731_IntD1,axiom,
! [C2: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_732_IntD1,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_733_IntD2,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B2 ) )
=> ( member_real @ C2 @ B2 ) ) ).
% IntD2
thf(fact_734_IntD2,axiom,
! [C2: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C2 @ B2 ) ) ).
% IntD2
thf(fact_735_IntD2,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C2 @ B2 ) ) ).
% IntD2
thf(fact_736_insertE,axiom,
! [A: real,B3: real,A2: set_real] :
( ( member_real @ A @ ( insert_real @ B3 @ A2 ) )
=> ( ( A != B3 )
=> ( member_real @ A @ A2 ) ) ) ).
% insertE
thf(fact_737_insertE,axiom,
! [A: a,B3: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B3 @ A2 ) )
=> ( ( A != B3 )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_738_insertE,axiom,
! [A: set_a,B3: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B3 @ A2 ) )
=> ( ( A != B3 )
=> ( member_set_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_739_insertI1,axiom,
! [A: real,B2: set_real] : ( member_real @ A @ ( insert_real @ A @ B2 ) ) ).
% insertI1
thf(fact_740_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a @ A @ ( insert_a @ A @ B2 ) ) ).
% insertI1
thf(fact_741_insertI1,axiom,
! [A: set_a,B2: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B2 ) ) ).
% insertI1
thf(fact_742_insertI2,axiom,
! [A: real,B2: set_real,B3: real] :
( ( member_real @ A @ B2 )
=> ( member_real @ A @ ( insert_real @ B3 @ B2 ) ) ) ).
% insertI2
thf(fact_743_insertI2,axiom,
! [A: a,B2: set_a,B3: a] :
( ( member_a @ A @ B2 )
=> ( member_a @ A @ ( insert_a @ B3 @ B2 ) ) ) ).
% insertI2
thf(fact_744_insertI2,axiom,
! [A: set_a,B2: set_set_a,B3: set_a] :
( ( member_set_a @ A @ B2 )
=> ( member_set_a @ A @ ( insert_set_a @ B3 @ B2 ) ) ) ).
% insertI2
thf(fact_745_Int__assoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% Int_assoc
thf(fact_746_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_747_Set_Oset__insert,axiom,
! [X: real,A2: set_real] :
( ( member_real @ X @ A2 )
=> ~ ! [B7: set_real] :
( ( A2
= ( insert_real @ X @ B7 ) )
=> ( member_real @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_748_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B7: set_a] :
( ( A2
= ( insert_a @ X @ B7 ) )
=> ( member_a @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_749_Set_Oset__insert,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ~ ! [B7: set_set_a] :
( ( A2
= ( insert_set_a @ X @ B7 ) )
=> ( member_set_a @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_750_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A8: set_a,B8: set_a] : ( inf_inf_set_a @ B8 @ A8 ) ) ) ).
% Int_commute
thf(fact_751_insert__ident,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ~ ( member_real @ X @ B2 )
=> ( ( ( insert_real @ X @ A2 )
= ( insert_real @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_752_insert__ident,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_753_insert__ident,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ~ ( member_set_a @ X @ B2 )
=> ( ( ( insert_set_a @ X @ A2 )
= ( insert_set_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_754_insert__absorb,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ( ( insert_real @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_755_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_756_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_757_insert__eq__iff,axiom,
! [A: real,A2: set_real,B3: real,B2: set_real] :
( ~ ( member_real @ A @ A2 )
=> ( ~ ( member_real @ B3 @ B2 )
=> ( ( ( insert_real @ A @ A2 )
= ( insert_real @ B3 @ B2 ) )
= ( ( ( A = B3 )
=> ( A2 = B2 ) )
& ( ( A != B3 )
=> ? [C4: set_real] :
( ( A2
= ( insert_real @ B3 @ C4 ) )
& ~ ( member_real @ B3 @ C4 )
& ( B2
= ( insert_real @ A @ C4 ) )
& ~ ( member_real @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_758_insert__eq__iff,axiom,
! [A: a,A2: set_a,B3: a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B3 @ B2 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B3 @ B2 ) )
= ( ( ( A = B3 )
=> ( A2 = B2 ) )
& ( ( A != B3 )
=> ? [C4: set_a] :
( ( A2
= ( insert_a @ B3 @ C4 ) )
& ~ ( member_a @ B3 @ C4 )
& ( B2
= ( insert_a @ A @ C4 ) )
& ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_759_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B3: set_a,B2: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B3 @ B2 )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B3 @ B2 ) )
= ( ( ( A = B3 )
=> ( A2 = B2 ) )
& ( ( A != B3 )
=> ? [C4: set_set_a] :
( ( A2
= ( insert_set_a @ B3 @ C4 ) )
& ~ ( member_set_a @ B3 @ C4 )
& ( B2
= ( insert_set_a @ A @ C4 ) )
& ~ ( member_set_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_760_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_761_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_762_Int__insert__left,axiom,
! [A: real,C: set_real,B2: set_real] :
( ( ( member_real @ A @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A @ B2 ) @ C )
= ( insert_real @ A @ ( inf_inf_set_real @ B2 @ C ) ) ) )
& ( ~ ( member_real @ A @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A @ B2 ) @ C )
= ( inf_inf_set_real @ B2 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_763_Int__insert__left,axiom,
! [A: set_a,C: set_set_a,B2: set_set_a] :
( ( ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B2 ) @ C )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C ) ) ) )
& ( ~ ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B2 ) @ C )
= ( inf_inf_set_set_a @ B2 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_764_Int__insert__left,axiom,
! [A: a,C: set_a,B2: set_a] :
( ( ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C )
= ( insert_a @ A @ ( inf_inf_set_a @ B2 @ C ) ) ) )
& ( ~ ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C )
= ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_765_Int__left__absorb,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_766_Int__insert__right,axiom,
! [A: real,A2: set_real,B2: set_real] :
( ( ( member_real @ A @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B2 ) )
= ( insert_real @ A @ ( inf_inf_set_real @ A2 @ B2 ) ) ) )
& ( ~ ( member_real @ A @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B2 ) )
= ( inf_inf_set_real @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_767_Int__insert__right,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B2 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B2 ) )
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_768_Int__insert__right,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_769_Int__left__commute,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
= ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_left_commute
thf(fact_770_mk__disjoint__insert,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ? [B7: set_real] :
( ( A2
= ( insert_real @ A @ B7 ) )
& ~ ( member_real @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_771_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B7: set_a] :
( ( A2
= ( insert_a @ A @ B7 ) )
& ~ ( member_a @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_772_mk__disjoint__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ? [B7: set_set_a] :
( ( A2
= ( insert_set_a @ A @ B7 ) )
& ~ ( member_set_a @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_773_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_774_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_775_finite_OinsertI,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( finite_finite_real @ ( insert_real @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_776_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_777_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_778_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_779_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_780_subset__singleton__iff,axiom,
! [X5: set_real,A: real] :
( ( ord_less_eq_set_real @ X5 @ ( insert_real @ A @ bot_bot_set_real ) )
= ( ( X5 = bot_bot_set_real )
| ( X5
= ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% subset_singleton_iff
thf(fact_781_subset__singleton__iff,axiom,
! [X5: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( ( X5 = bot_bot_set_set_a )
| ( X5
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_782_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_783_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A5: nat] :
( A
= ( insert_nat @ A5 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_784_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A6: set_a] :
( ? [A5: a] :
( A
= ( insert_a @ A5 @ A6 ) )
=> ~ ( finite_finite_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_785_finite_Ocases,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ~ ! [A6: set_real] :
( ? [A5: real] :
( A
= ( insert_real @ A5 @ A6 ) )
=> ~ ( finite_finite_real @ A6 ) ) ) ) ).
% finite.cases
thf(fact_786_finite_Ocases,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ~ ! [A6: set_set_a] :
( ? [A5: set_a] :
( A
= ( insert_set_a @ A5 @ A6 ) )
=> ~ ( finite_finite_set_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_787_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A8: set_nat,B4: nat] :
( ( A4
= ( insert_nat @ B4 @ A8 ) )
& ( finite_finite_nat @ A8 ) ) ) ) ) ).
% finite.simps
thf(fact_788_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A8: set_a,B4: a] :
( ( A4
= ( insert_a @ B4 @ A8 ) )
& ( finite_finite_a @ A8 ) ) ) ) ) ).
% finite.simps
thf(fact_789_finite_Osimps,axiom,
( finite_finite_real
= ( ^ [A4: set_real] :
( ( A4 = bot_bot_set_real )
| ? [A8: set_real,B4: real] :
( ( A4
= ( insert_real @ B4 @ A8 ) )
& ( finite_finite_real @ A8 ) ) ) ) ) ).
% finite.simps
thf(fact_790_finite_Osimps,axiom,
( finite_finite_set_a
= ( ^ [A4: set_set_a] :
( ( A4 = bot_bot_set_set_a )
| ? [A8: set_set_a,B4: set_a] :
( ( A4
= ( insert_set_a @ B4 @ A8 ) )
& ( finite_finite_set_a @ A8 ) ) ) ) ) ).
% finite.simps
thf(fact_791_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_792_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_793_finite__induct,axiom,
! [F2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ~ ( member_real @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_794_finite__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_795_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_796_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_797_finite__ne__induct,axiom,
! [F2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( F2 != bot_bot_set_real )
=> ( ! [X3: real] : ( P @ ( insert_real @ X3 @ bot_bot_set_real ) )
=> ( ! [X3: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( F3 != bot_bot_set_real )
=> ( ~ ( member_real @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_798_finite__ne__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( F2 != bot_bot_set_set_a )
=> ( ! [X3: set_a] : ( P @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
=> ( ! [X3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( F3 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_799_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,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_800_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,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_801_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,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ~ ( member_real @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_802_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,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_803_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_804_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_805_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_806_finite__ranking__induct,axiom,
! [S2: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X3 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_807_finite__ranking__induct,axiom,
! [S2: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S3: set_a] :
( ( finite_finite_a @ S3 )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a @ X3 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_808_finite__ranking__induct,axiom,
! [S2: set_real,P: set_real > $o,F: real > nat] :
( ( finite_finite_real @ S2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,S3: set_real] :
( ( finite_finite_real @ S3 )
=> ( ! [Y3: real] :
( ( member_real @ Y3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_real @ X3 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_809_finite__ranking__induct,axiom,
! [S2: set_nat,P: set_nat > $o,F: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ S3 )
=> ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X3 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_810_finite__ranking__induct,axiom,
! [S2: set_a,P: set_a > $o,F: a > real] :
( ( finite_finite_a @ S2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S3: set_a] :
( ( finite_finite_a @ S3 )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ S3 )
=> ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a @ X3 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_811_finite__ranking__induct,axiom,
! [S2: set_real,P: set_real > $o,F: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,S3: set_real] :
( ( finite_finite_real @ S3 )
=> ( ! [Y3: real] :
( ( member_real @ Y3 @ S3 )
=> ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_real @ X3 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_812_finite__ranking__induct,axiom,
! [S2: set_nat,P: set_nat > $o,F: nat > int] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ S3 )
=> ( ord_less_eq_int @ ( F @ Y3 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X3 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_813_finite__ranking__induct,axiom,
! [S2: set_a,P: set_a > $o,F: a > int] :
( ( finite_finite_a @ S2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S3: set_a] :
( ( finite_finite_a @ S3 )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ S3 )
=> ( ord_less_eq_int @ ( F @ Y3 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a @ X3 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_814_finite__ranking__induct,axiom,
! [S2: set_real,P: set_real > $o,F: real > int] :
( ( finite_finite_real @ S2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,S3: set_real] :
( ( finite_finite_real @ S3 )
=> ( ! [Y3: real] :
( ( member_real @ Y3 @ S3 )
=> ( ord_less_eq_int @ ( F @ Y3 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_real @ X3 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_815_finite__ranking__induct,axiom,
! [S2: set_set_a,P: set_set_a > $o,F: set_a > nat] :
( ( finite_finite_set_a @ S2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,S3: set_set_a] :
( ( finite_finite_set_a @ S3 )
=> ( ! [Y3: set_a] :
( ( member_set_a @ Y3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_set_a @ X3 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_816_finite__linorder__max__induct,axiom,
! [A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [B5: real,A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A6 )
=> ( ord_less_real @ X4 @ B5 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_real @ B5 @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_817_finite__linorder__max__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B5: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( ord_less_nat @ X4 @ B5 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_nat @ B5 @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_818_finite__linorder__max__induct,axiom,
! [A2: set_int,P: set_int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [B5: int,A6: set_int] :
( ( finite_finite_int @ A6 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A6 )
=> ( ord_less_int @ X4 @ B5 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_int @ B5 @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_819_finite__linorder__min__induct,axiom,
! [A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [B5: real,A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A6 )
=> ( ord_less_real @ B5 @ X4 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_real @ B5 @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_820_finite__linorder__min__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B5: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( ord_less_nat @ B5 @ X4 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_nat @ B5 @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_821_finite__linorder__min__induct,axiom,
! [A2: set_int,P: set_int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [B5: int,A6: set_int] :
( ( finite_finite_int @ A6 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A6 )
=> ( ord_less_int @ B5 @ X4 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_int @ B5 @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_822_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A5 @ A2 )
=> ( ~ ( member_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_823_finite__subset__induct,axiom,
! [F2: set_real,A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( ord_less_eq_set_real @ F2 @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A5: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( member_real @ A5 @ A2 )
=> ( ~ ( member_real @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_824_finite__subset__induct,axiom,
! [F2: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A5: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A5 @ A2 )
=> ( ~ ( member_set_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_825_finite__subset__induct,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A5 @ A2 )
=> ( ~ ( member_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_826_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A5 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_827_finite__subset__induct_H,axiom,
! [F2: set_real,A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( ord_less_eq_set_real @ F2 @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A5: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( member_real @ A5 @ A2 )
=> ( ( ord_less_eq_set_real @ F3 @ A2 )
=> ( ~ ( member_real @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_828_finite__subset__induct_H,axiom,
! [F2: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A5: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A5 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A2 )
=> ( ~ ( member_set_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_829_finite__subset__induct_H,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_830_Min__insert2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ A2 )
=> ( ord_less_eq_nat @ A @ B5 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ A @ A2 ) )
= A ) ) ) ).
% Min_insert2
thf(fact_831_Min__insert2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ! [B5: int] :
( ( member_int @ B5 @ A2 )
=> ( ord_less_eq_int @ A @ B5 ) )
=> ( ( lattic8718645017227715691in_int @ ( insert_int @ A @ A2 ) )
= A ) ) ) ).
% Min_insert2
thf(fact_832_Min__insert2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ! [B5: real] :
( ( member_real @ B5 @ A2 )
=> ( ord_less_eq_real @ A @ B5 ) )
=> ( ( lattic3629708407755379051n_real @ ( insert_real @ A @ A2 ) )
= A ) ) ) ).
% Min_insert2
thf(fact_833_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_834_emptyE,axiom,
! [A: real] :
~ ( member_real @ A @ bot_bot_set_real ) ).
% emptyE
thf(fact_835_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_836_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_837_equals0D,axiom,
! [A2: set_real,A: real] :
( ( A2 = bot_bot_set_real )
=> ~ ( member_real @ A @ A2 ) ) ).
% equals0D
thf(fact_838_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_839_equals0I,axiom,
! [A2: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_840_equals0I,axiom,
! [A2: set_real] :
( ! [Y2: real] :
~ ( member_real @ Y2 @ A2 )
=> ( A2 = bot_bot_set_real ) ) ).
% equals0I
thf(fact_841_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_842_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_843_ex__in__conv,axiom,
! [A2: set_real] :
( ( ? [X2: real] : ( member_real @ X2 @ A2 ) )
= ( A2 != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_844_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_845_in__mono,axiom,
! [A2: set_real,B2: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B2 ) ) ) ).
% in_mono
thf(fact_846_in__mono,axiom,
! [A2: set_set_a,B2: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_847_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_848_subsetD,axiom,
! [A2: set_real,B2: set_real,C2: real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_849_subsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_850_subsetD,axiom,
! [A2: set_a,B2: set_a,C2: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_851_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_852_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A8: set_real,B8: set_real] :
! [X2: real] :
( ( member_real @ X2 @ A8 )
=> ( member_real @ X2 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_853_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A8: set_set_a,B8: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A8 )
=> ( member_set_a @ X2 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_854_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B8: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A8 )
=> ( member_a @ X2 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_855_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_856_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_857_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A8: set_real,B8: set_real] :
! [T4: real] :
( ( member_real @ T4 @ A8 )
=> ( member_real @ T4 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_858_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A8: set_set_a,B8: set_set_a] :
! [T4: set_a] :
( ( member_set_a @ T4 @ A8 )
=> ( member_set_a @ T4 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_859_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B8: set_a] :
! [T4: a] :
( ( member_a @ T4 @ A8 )
=> ( member_a @ T4 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_860_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_861_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_862_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_863_subset__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_864_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A8: set_a,B8: set_a] :
( ( ord_less_eq_set_a @ A8 @ B8 )
& ( ord_less_eq_set_a @ B8 @ A8 ) ) ) ) ).
% set_eq_subset
thf(fact_865_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_866_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_867_psubsetD,axiom,
! [A2: set_real,B2: set_real,C2: real] :
( ( ord_less_set_real @ A2 @ B2 )
=> ( ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_868_psubsetD,axiom,
! [A2: set_a,B2: set_a,C2: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_869_psubsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_870_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_871_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_872_not__psubset__empty,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_873_not__psubset__empty,axiom,
! [A2: set_real] :
~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).
% not_psubset_empty
thf(fact_874_not__psubset__empty,axiom,
! [A2: set_set_a] :
~ ( ord_less_set_set_a @ A2 @ bot_bot_set_set_a ) ).
% not_psubset_empty
thf(fact_875_psubsetE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_876_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A8: set_a,B8: set_a] :
( ( ord_less_eq_set_a @ A8 @ B8 )
& ( A8 != B8 ) ) ) ) ).
% psubset_eq
thf(fact_877_psubset__imp__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_878_psubset__subset__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_879_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A8: set_a,B8: set_a] :
( ( ord_less_eq_set_a @ A8 @ B8 )
& ~ ( ord_less_eq_set_a @ B8 @ A8 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_880_subset__psubset__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_881_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B8: set_a] :
( ( ord_less_set_a @ A8 @ B8 )
| ( A8 = B8 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_882_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_883_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_884_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_885_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_886_sumsetdiff__sing,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ B2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% sumsetdiff_sing
thf(fact_887_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_888_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_889_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_890_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_891_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_892_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_893_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_894_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_895_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_896_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_897_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_898_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_899_inf_Obounded__iff,axiom,
! [A: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B3 @ C2 ) )
= ( ( ord_less_eq_set_a @ A @ B3 )
& ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_900_inf_Obounded__iff,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B3 @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B3 )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_901_inf_Obounded__iff,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B3 @ C2 ) )
= ( ( ord_less_eq_real @ A @ B3 )
& ( ord_less_eq_real @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_902_inf_Obounded__iff,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B3 @ C2 ) )
= ( ( ord_less_eq_int @ A @ B3 )
& ( ord_less_eq_int @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_903_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_904_inf_Oright__idem,axiom,
! [A: set_a,B3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B3 ) @ B3 )
= ( inf_inf_set_a @ A @ B3 ) ) ).
% inf.right_idem
thf(fact_905_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_906_inf_Oleft__idem,axiom,
! [A: set_a,B3: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B3 ) )
= ( inf_inf_set_a @ A @ B3 ) ) ).
% inf.left_idem
thf(fact_907_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_908_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_909_DiffI,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ A2 )
=> ( ~ ( member_real @ C2 @ B2 )
=> ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_910_DiffI,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ~ ( member_a @ C2 @ B2 )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_911_DiffI,axiom,
! [C2: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( ~ ( member_set_a @ C2 @ B2 )
=> ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_912_Diff__iff,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B2 ) )
= ( ( member_real @ C2 @ A2 )
& ~ ( member_real @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_913_Diff__iff,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C2 @ A2 )
& ~ ( member_a @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_914_Diff__iff,axiom,
! [C2: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( ( member_set_a @ C2 @ A2 )
& ~ ( member_set_a @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_915_Diff__idemp,axiom,
! [A2: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_916_Diff__idemp,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) @ B2 )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_917_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_918_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_919_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_920_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_921_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_922_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_923_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_924_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_925_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_926_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_927_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_928_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_929_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_930_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_931_le__inf__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_932_le__inf__iff,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z2 ) )
= ( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_933_le__inf__iff,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z2 ) )
= ( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_934_Diff__cancel,axiom,
! [A2: set_real] :
( ( minus_minus_set_real @ A2 @ A2 )
= bot_bot_set_real ) ).
% Diff_cancel
thf(fact_935_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_936_Diff__cancel,axiom,
! [A2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ A2 )
= bot_bot_set_set_a ) ).
% Diff_cancel
thf(fact_937_empty__Diff,axiom,
! [A2: set_real] :
( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
= bot_bot_set_real ) ).
% empty_Diff
thf(fact_938_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_939_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_940_Diff__empty,axiom,
! [A2: set_real] :
( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
= A2 ) ).
% Diff_empty
thf(fact_941_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_942_Diff__empty,axiom,
! [A2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ bot_bot_set_set_a )
= A2 ) ).
% Diff_empty
thf(fact_943_finite__Diff2,axiom,
! [B2: set_real,A2: set_real] :
( ( finite_finite_real @ B2 )
=> ( ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B2 ) )
= ( finite_finite_real @ A2 ) ) ) ).
% finite_Diff2
thf(fact_944_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_945_finite__Diff2,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_946_finite__Diff2,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( finite_finite_set_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_947_finite__Diff,axiom,
! [A2: set_real,B2: set_real] :
( ( finite_finite_real @ A2 )
=> ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_948_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_949_finite__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_950_finite__Diff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_951_insert__Diff1,axiom,
! [X: real,B2: set_real,A2: set_real] :
( ( member_real @ X @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
= ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_952_insert__Diff1,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_953_insert__Diff1,axiom,
! [X: set_a,B2: set_set_a,A2: set_set_a] :
( ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_954_Diff__insert0,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ( minus_minus_set_real @ A2 @ ( insert_real @ X @ B2 ) )
= ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_955_Diff__insert0,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_956_Diff__insert0,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_957_diff__ge__0__iff__ge,axiom,
! [A: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B3 ) )
= ( ord_less_eq_real @ B3 @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_958_diff__ge__0__iff__ge,axiom,
! [A: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B3 ) )
= ( ord_less_eq_int @ B3 @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_959_diff__gt__0__iff__gt,axiom,
! [A: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B3 ) )
= ( ord_less_real @ B3 @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_960_diff__gt__0__iff__gt,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B3 ) )
= ( ord_less_int @ B3 @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_961_Diff__eq__empty__iff,axiom,
! [A2: set_real,B2: set_real] :
( ( ( minus_minus_set_real @ A2 @ B2 )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_962_Diff__eq__empty__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_963_Diff__eq__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_964_insert__Diff__single,axiom,
! [A: real,A2: set_real] :
( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( insert_real @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_965_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_966_insert__Diff__single,axiom,
! [A: set_a,A2: set_set_a] :
( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= ( insert_set_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_967_finite__Diff__insert,axiom,
! [A2: set_real,A: real,B2: set_real] :
( ( finite_finite_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) ) )
= ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_968_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_969_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_970_finite__Diff__insert,axiom,
! [A2: set_set_a,A: set_a,B2: set_set_a] :
( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B2 ) ) )
= ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_971_Diff__disjoint,axiom,
! [A2: set_real,B2: set_real] :
( ( inf_inf_set_real @ A2 @ ( minus_minus_set_real @ B2 @ A2 ) )
= bot_bot_set_real ) ).
% Diff_disjoint
thf(fact_972_Diff__disjoint,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B2 @ A2 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_973_Diff__disjoint,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B2 @ A2 ) )
= bot_bot_set_set_a ) ).
% Diff_disjoint
thf(fact_974_zdiv__mono1,axiom,
! [A: int,A9: int,B3: int] :
( ( ord_less_eq_int @ A @ A9 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B3 ) @ ( divide_divide_int @ A9 @ B3 ) ) ) ) ).
% zdiv_mono1
thf(fact_975_zdiv__mono2,axiom,
! [A: int,B9: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B9 )
=> ( ( ord_less_eq_int @ B9 @ B3 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B3 ) @ ( divide_divide_int @ A @ B9 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_976_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_977_zdiv__mono1__neg,axiom,
! [A: int,A9: int,B3: int] :
( ( ord_less_eq_int @ A @ A9 )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A9 @ B3 ) @ ( divide_divide_int @ A @ B3 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_978_zdiv__mono2__neg,axiom,
! [A: int,B9: int,B3: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B9 )
=> ( ( ord_less_eq_int @ B9 @ B3 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B9 ) @ ( divide_divide_int @ A @ B3 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_979_zdiv__zmult2__eq,axiom,
! [C2: int,A: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B3 @ C2 ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B3 ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_980_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_981_div__nonneg__neg__le0,axiom,
! [A: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B3 ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_982_div__nonpos__pos__le0,axiom,
! [A: int,B3: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B3 ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_983_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_984_neg__imp__zdiv__nonneg__iff,axiom,
! [B3: int,A: int] :
( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B3 ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_985_pos__imp__zdiv__nonneg__iff,axiom,
! [B3: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B3 ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_986_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B3 ) )
= ( ( ord_less_eq_int @ B3 @ A )
& ( ord_less_int @ zero_zero_int @ B3 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_987_pos__imp__zdiv__neg__iff,axiom,
! [B3: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B3 ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_988_neg__imp__zdiv__neg__iff,axiom,
! [B3: int,A: int] :
( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B3 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_989_div__neg__pos__less0,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ ( divide_divide_int @ A @ B3 ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_990_DiffE,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ~ ( ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_991_DiffE,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_992_DiffE,axiom,
! [C2: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ~ ( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_993_DiffD1,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ( member_real @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_994_DiffD1,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_995_DiffD1,axiom,
! [C2: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_996_DiffD2,axiom,
! [C2: real,A2: set_real,B2: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ~ ( member_real @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_997_DiffD2,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_998_DiffD2,axiom,
! [C2: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ~ ( member_set_a @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_999_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1000_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1001_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1002_diff__eq__diff__eq,axiom,
! [A: real,B3: real,C2: real,D: real] :
( ( ( minus_minus_real @ A @ B3 )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( A = B3 )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_1003_diff__eq__diff__eq,axiom,
! [A: int,B3: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B3 )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( A = B3 )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_1004_diff__right__commute,axiom,
! [A: nat,C2: nat,B3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B3 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B3 ) @ C2 ) ) ).
% diff_right_commute
thf(fact_1005_diff__right__commute,axiom,
! [A: real,C2: real,B3: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B3 )
= ( minus_minus_real @ ( minus_minus_real @ A @ B3 ) @ C2 ) ) ).
% diff_right_commute
thf(fact_1006_diff__right__commute,axiom,
! [A: int,C2: int,B3: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B3 )
= ( minus_minus_int @ ( minus_minus_int @ A @ B3 ) @ C2 ) ) ).
% diff_right_commute
thf(fact_1007_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1008_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_1009_bot__set__def,axiom,
( bot_bot_set_real
= ( collect_real @ bot_bot_real_o ) ) ).
% bot_set_def
thf(fact_1010_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_1011_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1012_diff__eq__diff__less__eq,axiom,
! [A: real,B3: real,C2: real,D: real] :
( ( ( minus_minus_real @ A @ B3 )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( ord_less_eq_real @ A @ B3 )
= ( ord_less_eq_real @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1013_diff__eq__diff__less__eq,axiom,
! [A: int,B3: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B3 )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_eq_int @ A @ B3 )
= ( ord_less_eq_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1014_diff__right__mono,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B3 @ C2 ) ) ) ).
% diff_right_mono
thf(fact_1015_diff__right__mono,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B3 @ C2 ) ) ) ).
% diff_right_mono
thf(fact_1016_diff__left__mono,axiom,
! [B3: real,A: real,C2: real] :
( ( ord_less_eq_real @ B3 @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B3 ) ) ) ).
% diff_left_mono
thf(fact_1017_diff__left__mono,axiom,
! [B3: int,A: int,C2: int] :
( ( ord_less_eq_int @ B3 @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B3 ) ) ) ).
% diff_left_mono
thf(fact_1018_diff__mono,axiom,
! [A: real,B3: real,D: real,C2: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ord_less_eq_real @ D @ C2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B3 @ D ) ) ) ) ).
% diff_mono
thf(fact_1019_diff__mono,axiom,
! [A: int,B3: int,D: int,C2: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_eq_int @ D @ C2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B3 @ D ) ) ) ) ).
% diff_mono
thf(fact_1020_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
= ( ^ [A4: real,B4: real] :
( ( minus_minus_real @ A4 @ B4 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1021_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A4: int,B4: int] :
( ( minus_minus_int @ A4 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1022_diff__strict__mono,axiom,
! [A: real,B3: real,D: real,C2: real] :
( ( ord_less_real @ A @ B3 )
=> ( ( ord_less_real @ D @ C2 )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B3 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1023_diff__strict__mono,axiom,
! [A: int,B3: int,D: int,C2: int] :
( ( ord_less_int @ A @ B3 )
=> ( ( ord_less_int @ D @ C2 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B3 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1024_diff__eq__diff__less,axiom,
! [A: real,B3: real,C2: real,D: real] :
( ( ( minus_minus_real @ A @ B3 )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( ord_less_real @ A @ B3 )
= ( ord_less_real @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1025_diff__eq__diff__less,axiom,
! [A: int,B3: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B3 )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_int @ A @ B3 )
= ( ord_less_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1026_diff__strict__left__mono,axiom,
! [B3: real,A: real,C2: real] :
( ( ord_less_real @ B3 @ A )
=> ( ord_less_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B3 ) ) ) ).
% diff_strict_left_mono
thf(fact_1027_diff__strict__left__mono,axiom,
! [B3: int,A: int,C2: int] :
( ( ord_less_int @ B3 @ A )
=> ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B3 ) ) ) ).
% diff_strict_left_mono
thf(fact_1028_diff__strict__right__mono,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_real @ A @ B3 )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B3 @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_1029_diff__strict__right__mono,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_int @ A @ B3 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B3 @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_1030_left__diff__distrib,axiom,
! [A: real,B3: real,C2: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B3 ) @ C2 )
= ( minus_minus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B3 @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_1031_left__diff__distrib,axiom,
! [A: int,B3: int,C2: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B3 ) @ C2 )
= ( minus_minus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B3 @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_1032_right__diff__distrib,axiom,
! [A: real,B3: real,C2: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B3 @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A @ B3 ) @ ( times_times_real @ A @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_1033_right__diff__distrib,axiom,
! [A: int,B3: int,C2: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B3 @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ A @ B3 ) @ ( times_times_int @ A @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_1034_left__diff__distrib_H,axiom,
! [B3: real,C2: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B3 @ C2 ) @ A )
= ( minus_minus_real @ ( times_times_real @ B3 @ A ) @ ( times_times_real @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1035_left__diff__distrib_H,axiom,
! [B3: nat,C2: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B3 @ C2 ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B3 @ A ) @ ( times_times_nat @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1036_left__diff__distrib_H,axiom,
! [B3: int,C2: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B3 @ C2 ) @ A )
= ( minus_minus_int @ ( times_times_int @ B3 @ A ) @ ( times_times_int @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1037_right__diff__distrib_H,axiom,
! [A: real,B3: real,C2: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B3 @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A @ B3 ) @ ( times_times_real @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_1038_right__diff__distrib_H,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B3 @ C2 ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B3 ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_1039_right__diff__distrib_H,axiom,
! [A: int,B3: int,C2: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B3 @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ A @ B3 ) @ ( times_times_int @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_1040_diff__divide__distrib,axiom,
! [A: real,B3: real,C2: real] :
( ( divide_divide_real @ ( minus_minus_real @ A @ B3 ) @ C2 )
= ( minus_minus_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B3 @ C2 ) ) ) ).
% diff_divide_distrib
thf(fact_1041_Diff__infinite__finite,axiom,
! [T2: set_real,S2: set_real] :
( ( finite_finite_real @ T2 )
=> ( ~ ( finite_finite_real @ S2 )
=> ~ ( finite_finite_real @ ( minus_minus_set_real @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1042_Diff__infinite__finite,axiom,
! [T2: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1043_Diff__infinite__finite,axiom,
! [T2: set_a,S2: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1044_Diff__infinite__finite,axiom,
! [T2: set_set_a,S2: set_set_a] :
( ( finite_finite_set_a @ T2 )
=> ( ~ ( finite_finite_set_a @ S2 )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1045_Diff__mono,axiom,
! [A2: set_set_a,C: set_set_a,D3: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ D3 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) @ ( minus_5736297505244876581_set_a @ C @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_1046_Diff__mono,axiom,
! [A2: set_a,C: set_a,D3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ D3 @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_1047_Diff__subset,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_1048_Diff__subset,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_1049_double__diff,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ( minus_5736297505244876581_set_a @ B2 @ ( minus_5736297505244876581_set_a @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1050_double__diff,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1051_insert__Diff__if,axiom,
! [X: real,B2: set_real,A2: set_real] :
( ( ( member_real @ X @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
= ( minus_minus_set_real @ A2 @ B2 ) ) )
& ( ~ ( member_real @ X @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
= ( insert_real @ X @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1052_insert__Diff__if,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1053_insert__Diff__if,axiom,
! [X: set_a,B2: set_set_a,A2: set_set_a] :
( ( ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( insert_set_a @ X @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1054_Diff__Int__distrib2,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_1055_Diff__Int__distrib2,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) @ C )
= ( minus_5736297505244876581_set_a @ ( inf_inf_set_set_a @ A2 @ C ) @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_1056_Diff__Int__distrib,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C @ A2 ) @ ( inf_inf_set_a @ C @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_1057_Diff__Int__distrib,axiom,
! [C: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( minus_5736297505244876581_set_a @ ( inf_inf_set_set_a @ C @ A2 ) @ ( inf_inf_set_set_a @ C @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_1058_Diff__Diff__Int,axiom,
! [A2: set_a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_1059_Diff__Diff__Int,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_1060_Diff__Int2,axiom,
! [A2: set_a,C: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B2 @ C ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ B2 ) ) ).
% Diff_Int2
thf(fact_1061_Diff__Int2,axiom,
! [A2: set_set_a,C: set_set_a,B2: set_set_a] :
( ( minus_5736297505244876581_set_a @ ( inf_inf_set_set_a @ A2 @ C ) @ ( inf_inf_set_set_a @ B2 @ C ) )
= ( minus_5736297505244876581_set_a @ ( inf_inf_set_set_a @ A2 @ C ) @ B2 ) ) ).
% Diff_Int2
thf(fact_1062_Int__Diff,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C )
= ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C ) ) ) ).
% Int_Diff
thf(fact_1063_Int__Diff,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( minus_5736297505244876581_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ C )
= ( inf_inf_set_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B2 @ C ) ) ) ).
% Int_Diff
thf(fact_1064_psubset__imp__ex__mem,axiom,
! [A2: set_real,B2: set_real] :
( ( ord_less_set_real @ A2 @ B2 )
=> ? [B5: real] : ( member_real @ B5 @ ( minus_minus_set_real @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1065_psubset__imp__ex__mem,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ? [B5: a] : ( member_a @ B5 @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1066_psubset__imp__ex__mem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ? [B5: set_a] : ( member_set_a @ B5 @ ( minus_5736297505244876581_set_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1067_diff__shunt__var,axiom,
! [X: set_real,Y: set_real] :
( ( ( minus_minus_set_real @ X @ Y )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1068_diff__shunt__var,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ X @ Y )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1069_diff__shunt__var,axiom,
! [X: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1070_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_1071_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1072_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_1073_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_1074_Diff__insert__absorb,axiom,
! [X: real,A2: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1075_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1076_Diff__insert__absorb,axiom,
! [X: set_a,A2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1077_Diff__insert2,axiom,
! [A2: set_real,A: real,B2: set_real] :
( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
= ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1078_Diff__insert2,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1079_Diff__insert2,axiom,
! [A2: set_set_a,A: set_a,B2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B2 ) )
= ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1080_insert__Diff,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1081_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1082_insert__Diff,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1083_Diff__insert,axiom,
! [A2: set_real,A: real,B2: set_real] :
( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
= ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B2 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).
% Diff_insert
thf(fact_1084_Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_1085_Diff__insert,axiom,
! [A2: set_set_a,A: set_a,B2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B2 ) )
= ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ).
% Diff_insert
thf(fact_1086_subset__Diff__insert,axiom,
! [A2: set_real,B2: set_real,X: real,C: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ ( insert_real @ X @ C ) ) )
= ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ C ) )
& ~ ( member_real @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1087_subset__Diff__insert,axiom,
! [A2: set_set_a,B2: set_set_a,X: set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B2 @ ( insert_set_a @ X @ C ) ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B2 @ C ) )
& ~ ( member_set_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1088_subset__Diff__insert,axiom,
! [A2: set_a,B2: set_a,X: a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ ( insert_a @ X @ C ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1089_Int__Diff__disjoint,axiom,
! [A2: set_real,B2: set_real] :
( ( inf_inf_set_real @ ( inf_inf_set_real @ A2 @ B2 ) @ ( minus_minus_set_real @ A2 @ B2 ) )
= bot_bot_set_real ) ).
% Int_Diff_disjoint
thf(fact_1090_Int__Diff__disjoint,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ A2 @ B2 ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_1091_Int__Diff__disjoint,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= bot_bot_set_set_a ) ).
% Int_Diff_disjoint
thf(fact_1092_Diff__triv,axiom,
! [A2: set_real,B2: set_real] :
( ( ( inf_inf_set_real @ A2 @ B2 )
= bot_bot_set_real )
=> ( ( minus_minus_set_real @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_1093_Diff__triv,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_1094_Diff__triv,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
=> ( ( minus_5736297505244876581_set_a @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_1095_add__divide__eq__if__simps_I4_J,axiom,
! [Z2: real,A: real,B3: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B3 @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B3 @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z2 ) @ B3 ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1096_diff__frac__eq,axiom,
! [Y: real,Z2: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1097_diff__divide__eq__iff,axiom,
! [Z2: real,X: real,Y: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% diff_divide_eq_iff
thf(fact_1098_divide__diff__eq__iff,axiom,
! [Z2: real,X: real,Y: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z2 ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% divide_diff_eq_iff
thf(fact_1099_infinite__remove,axiom,
! [S2: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1100_infinite__remove,axiom,
! [S2: set_real,A: real] :
( ~ ( finite_finite_real @ S2 )
=> ~ ( finite_finite_real @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% infinite_remove
thf(fact_1101_infinite__remove,axiom,
! [S2: set_a,A: a] :
( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_1102_infinite__remove,axiom,
! [S2: set_set_a,A: set_a] :
( ~ ( finite_finite_set_a @ S2 )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% infinite_remove
thf(fact_1103_infinite__coinduct,axiom,
! [X5: set_nat > $o,A2: set_nat] :
( ( X5 @ A2 )
=> ( ! [A6: set_nat] :
( ( X5 @ A6 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A6 )
& ( ( X5 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1104_infinite__coinduct,axiom,
! [X5: set_real > $o,A2: set_real] :
( ( X5 @ A2 )
=> ( ! [A6: set_real] :
( ( X5 @ A6 )
=> ? [X4: real] :
( ( member_real @ X4 @ A6 )
& ( ( X5 @ ( minus_minus_set_real @ A6 @ ( insert_real @ X4 @ bot_bot_set_real ) ) )
| ~ ( finite_finite_real @ ( minus_minus_set_real @ A6 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ) )
=> ~ ( finite_finite_real @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1105_infinite__coinduct,axiom,
! [X5: set_a > $o,A2: set_a] :
( ( X5 @ A2 )
=> ( ! [A6: set_a] :
( ( X5 @ A6 )
=> ? [X4: a] :
( ( member_a @ X4 @ A6 )
& ( ( X5 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1106_infinite__coinduct,axiom,
! [X5: set_set_a > $o,A2: set_set_a] :
( ( X5 @ A2 )
=> ( ! [A6: set_set_a] :
( ( X5 @ A6 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A6 )
& ( ( X5 @ ( minus_5736297505244876581_set_a @ A6 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) )
| ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A6 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) ) ) )
=> ~ ( finite_finite_set_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1107_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1108_finite__empty__induct,axiom,
! [A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: real,A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ( member_real @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_real @ A6 @ ( insert_real @ A5 @ bot_bot_set_real ) ) ) ) ) )
=> ( P @ bot_bot_set_real ) ) ) ) ).
% finite_empty_induct
thf(fact_1109_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: a,A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( member_a @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ A5 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1110_finite__empty__induct,axiom,
! [A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: set_a,A6: set_set_a] :
( ( finite_finite_set_a @ A6 )
=> ( ( member_set_a @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_5736297505244876581_set_a @ A6 @ ( insert_set_a @ A5 @ bot_bot_set_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1111_Diff__single__insert,axiom,
! [A2: set_real,X: real,B2: set_real] :
( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 )
=> ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1112_Diff__single__insert,axiom,
! [A2: set_set_a,X: set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1113_Diff__single__insert,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1114_subset__insert__iff,axiom,
! [A2: set_real,X: real,B2: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) )
= ( ( ( member_real @ X @ A2 )
=> ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) )
& ( ~ ( member_real @ X @ A2 )
=> ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1115_subset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( ( ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B2 ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1116_subset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1117_card__less__sym__Diff,axiom,
! [A2: set_real,B2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_real @ B2 )
=> ( ( ord_less_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B2 ) )
=> ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ B2 ) ) @ ( finite_card_real @ ( minus_minus_set_real @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1118_card__less__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1119_card__less__sym__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1120_card__less__sym__Diff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) )
=> ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1121_card__le__sym__Diff,axiom,
! [A2: set_real,B2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ B2 ) ) @ ( finite_card_real @ ( minus_minus_set_real @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1122_card__le__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1123_card__le__sym__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1124_card__le__sym__Diff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1125_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_1126_inf_Oleft__commute,axiom,
! [B3: set_a,A: set_a,C2: set_a] :
( ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A @ C2 ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_1127_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_a,K: set_a,B3: set_a,A: set_a] :
( ( B2
= ( inf_inf_set_a @ K @ B3 ) )
=> ( ( inf_inf_set_a @ A @ B2 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_1128_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_a,K: set_a,A: set_a,B3: set_a] :
( ( A2
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_1129_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X2 ) ) ) ).
% inf_commute
thf(fact_1130_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_1131_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_1132_inf_Oassoc,axiom,
! [A: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B3 ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ).
% inf.assoc
thf(fact_1133_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_1134_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z2 )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_1135_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_1136_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_1137_frac__le__eq,axiom,
! [Y: real,Z2: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
= ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% frac_le_eq
thf(fact_1138_frac__less__eq,axiom,
! [Y: real,Z2: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_1139_finite__remove__induct,axiom,
! [B2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1140_finite__remove__induct,axiom,
! [B2: set_real,P: set_real > $o] :
( ( finite_finite_real @ B2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ( A6 != bot_bot_set_real )
=> ( ( ord_less_eq_set_real @ A6 @ B2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A6 )
=> ( P @ ( minus_minus_set_real @ A6 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1141_finite__remove__induct,axiom,
! [B2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ B2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A6: set_set_a] :
( ( finite_finite_set_a @ A6 )
=> ( ( A6 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A6 @ B2 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A6 )
=> ( P @ ( minus_5736297505244876581_set_a @ A6 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1142_finite__remove__induct,axiom,
! [B2: set_a,P: set_a > $o] :
( ( finite_finite_a @ B2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1143_remove__induct,axiom,
! [P: set_nat > $o,B2: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1144_remove__induct,axiom,
! [P: set_real > $o,B2: set_real] :
( ( P @ bot_bot_set_real )
=> ( ( ~ ( finite_finite_real @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ( A6 != bot_bot_set_real )
=> ( ( ord_less_eq_set_real @ A6 @ B2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A6 )
=> ( P @ ( minus_minus_set_real @ A6 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1145_remove__induct,axiom,
! [P: set_set_a > $o,B2: set_set_a] :
( ( P @ bot_bot_set_set_a )
=> ( ( ~ ( finite_finite_set_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_set_a] :
( ( finite_finite_set_a @ A6 )
=> ( ( A6 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A6 @ B2 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A6 )
=> ( P @ ( minus_5736297505244876581_set_a @ A6 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1146_remove__induct,axiom,
! [P: set_a > $o,B2: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1147_card__Diff1__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_1148_card__Diff1__le,axiom,
! [A2: set_real,X: real] : ( ord_less_eq_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) ) ).
% card_Diff1_le
thf(fact_1149_card__Diff1__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ).
% card_Diff1_le
thf(fact_1150_card__Diff1__le,axiom,
! [A2: set_set_a,X: set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A2 ) ) ).
% card_Diff1_le
thf(fact_1151_finite__induct__select,axiom,
! [S2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T3: set_nat] :
( ( ord_less_set_nat @ T3 @ S2 )
=> ( ( P @ T3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ S2 @ T3 ) )
& ( P @ ( insert_nat @ X4 @ T3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_1152_finite__induct__select,axiom,
! [S2: set_real,P: set_real > $o] :
( ( finite_finite_real @ S2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [T3: set_real] :
( ( ord_less_set_real @ T3 @ S2 )
=> ( ( P @ T3 )
=> ? [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ S2 @ T3 ) )
& ( P @ ( insert_real @ X4 @ T3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_1153_finite__induct__select,axiom,
! [S2: set_a,P: set_a > $o] :
( ( finite_finite_a @ S2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [T3: set_a] :
( ( ord_less_set_a @ T3 @ S2 )
=> ( ( P @ T3 )
=> ? [X4: a] :
( ( member_a @ X4 @ ( minus_minus_set_a @ S2 @ T3 ) )
& ( P @ ( insert_a @ X4 @ T3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_1154_finite__induct__select,axiom,
! [S2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ S2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [T3: set_set_a] :
( ( ord_less_set_set_a @ T3 @ S2 )
=> ( ( P @ T3 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ ( minus_5736297505244876581_set_a @ S2 @ T3 ) )
& ( P @ ( insert_set_a @ X4 @ T3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_1155_psubset__insert__iff,axiom,
! [A2: set_real,X: real,B2: set_real] :
( ( ord_less_set_real @ A2 @ ( insert_real @ X @ B2 ) )
= ( ( ( member_real @ X @ B2 )
=> ( ord_less_set_real @ A2 @ B2 ) )
& ( ~ ( member_real @ X @ B2 )
=> ( ( ( member_real @ X @ A2 )
=> ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) )
& ( ~ ( member_real @ X @ A2 )
=> ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1156_psubset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( ( ( member_set_a @ X @ B2 )
=> ( ord_less_set_set_a @ A2 @ B2 ) )
& ( ~ ( member_set_a @ X @ B2 )
=> ( ( ( member_set_a @ X @ A2 )
=> ( ord_less_set_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B2 ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1157_psubset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( ( member_a @ X @ A2 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1158_card__Diff1__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) )
= ( ( finite_finite_nat @ A2 )
& ( member_nat @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1159_card__Diff1__less__iff,axiom,
! [A2: set_real,X: real] :
( ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) )
= ( ( finite_finite_real @ A2 )
& ( member_real @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1160_card__Diff1__less__iff,axiom,
! [A2: set_a,X: a] :
( ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) )
= ( ( finite_finite_a @ A2 )
& ( member_a @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1161_card__Diff1__less__iff,axiom,
! [A2: set_set_a,X: set_a] :
( ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A2 ) )
= ( ( finite_finite_set_a @ A2 )
& ( member_set_a @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1162_card__Diff2__less,axiom,
! [A2: set_nat,X: nat,Y: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1163_card__Diff2__less,axiom,
! [A2: set_real,X: real,Y: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ X @ A2 )
=> ( ( member_real @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ ( insert_real @ Y @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1164_card__Diff2__less,axiom,
! [A2: set_a,X: a,Y: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( insert_a @ Y @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1165_card__Diff2__less,axiom,
! [A2: set_set_a,X: set_a,Y: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( member_set_a @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ ( insert_set_a @ Y @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1166_card__Diff1__less,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1167_card__Diff1__less,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1168_card__Diff1__less,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1169_card__Diff1__less,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1170_inf_OcoboundedI2,axiom,
! [B3: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_1171_inf_OcoboundedI2,axiom,
! [B3: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_1172_inf_OcoboundedI2,axiom,
! [B3: real,C2: real,A: real] :
( ( ord_less_eq_real @ B3 @ C2 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_1173_inf_OcoboundedI2,axiom,
! [B3: int,C2: int,A: int] :
( ( ord_less_eq_int @ B3 @ C2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_1174_inf_OcoboundedI1,axiom,
! [A: set_a,C2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_1175_inf_OcoboundedI1,axiom,
! [A: nat,C2: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_1176_inf_OcoboundedI1,axiom,
! [A: real,C2: real,B3: real] :
( ( ord_less_eq_real @ A @ C2 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_1177_inf_OcoboundedI1,axiom,
! [A: int,C2: int,B3: int] :
( ( ord_less_eq_int @ A @ C2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_1178_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_1179_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_1180_inf_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A4: real] :
( ( inf_inf_real @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_1181_inf_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A4: int] :
( ( inf_inf_int @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_1182_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1183_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1184_inf_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] :
( ( inf_inf_real @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1185_inf_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] :
( ( inf_inf_int @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1186_inf_Ocobounded2,axiom,
! [A: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_1187_inf_Ocobounded2,axiom,
! [A: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_1188_inf_Ocobounded2,axiom,
! [A: real,B3: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_1189_inf_Ocobounded2,axiom,
! [A: int,B3: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_1190_inf_Ocobounded1,axiom,
! [A: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ A ) ).
% inf.cobounded1
thf(fact_1191_inf_Ocobounded1,axiom,
! [A: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ A ) ).
% inf.cobounded1
thf(fact_1192_inf_Ocobounded1,axiom,
! [A: real,B3: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B3 ) @ A ) ).
% inf.cobounded1
thf(fact_1193_inf_Ocobounded1,axiom,
! [A: int,B3: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B3 ) @ A ) ).
% inf.cobounded1
thf(fact_1194_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1195_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( A4
= ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1196_inf_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] :
( A4
= ( inf_inf_real @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1197_inf_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] :
( A4
= ( inf_inf_int @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1198_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1199_inf__greatest,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1200_inf__greatest,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Z2 )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1201_inf__greatest,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Z2 )
=> ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1202_inf_OboundedI,axiom,
! [A: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1203_inf_OboundedI,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1204_inf_OboundedI,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ord_less_eq_real @ A @ C2 )
=> ( ord_less_eq_real @ A @ ( inf_inf_real @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1205_inf_OboundedI,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_eq_int @ A @ C2 )
=> ( ord_less_eq_int @ A @ ( inf_inf_int @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1206_inf_OboundedE,axiom,
! [A: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B3 @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B3 )
=> ~ ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1207_inf_OboundedE,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B3 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A @ B3 )
=> ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1208_inf_OboundedE,axiom,
! [A: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B3 @ C2 ) )
=> ~ ( ( ord_less_eq_real @ A @ B3 )
=> ~ ( ord_less_eq_real @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1209_inf_OboundedE,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B3 @ C2 ) )
=> ~ ( ( ord_less_eq_int @ A @ B3 )
=> ~ ( ord_less_eq_int @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1210_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_1211_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1212_inf__absorb2,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( inf_inf_real @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1213_inf__absorb2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( inf_inf_int @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1214_inf__absorb1,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( inf_inf_real @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1215_inf__absorb1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( inf_inf_int @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1216_sumset__subset__Un_I1_J,axiom,
! [A2: set_a,B2: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sumset_subset_Un(1)
thf(fact_1217_sumset__subset__Un_I2_J,axiom,
! [A2: set_a,B2: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ C ) @ B2 ) ) ).
% sumset_subset_Un(2)
thf(fact_1218_sumset__subset__Un1,axiom,
! [A2: set_a,A3: set_a,B2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ A3 ) @ B2 )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A3 @ B2 ) ) ) ).
% sumset_subset_Un1
thf(fact_1219_sumset__subset__Un2,axiom,
! [A2: set_a,B2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B2 @ B ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ).
% sumset_subset_Un2
thf(fact_1220_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1221_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1222_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_1223_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_1224_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_1225_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_1226__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_1227_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_1228_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_1229_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1230_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_1231_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_1232_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_1233_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_1234_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_1235_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_1236_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1237_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1238_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_1239_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_1240_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_1241_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_1242_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_1243_diff__less__mono,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B3 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1244_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_1245__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_1246__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_1247_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_1248_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_1249_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_1250_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1251_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_1252_Bolzano,axiom,
! [A: real,B3: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ! [A5: real,B5: real,C5: real] :
( ( P @ A5 @ B5 )
=> ( ( P @ B5 @ C5 )
=> ( ( ord_less_eq_real @ A5 @ B5 )
=> ( ( ord_less_eq_real @ B5 @ C5 )
=> ( P @ A5 @ C5 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B3 )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [A5: real,B5: real] :
( ( ( ord_less_eq_real @ A5 @ X3 )
& ( ord_less_eq_real @ X3 @ B5 )
& ( ord_less_real @ ( minus_minus_real @ B5 @ A5 ) @ D4 ) )
=> ( P @ A5 @ B5 ) ) ) ) )
=> ( P @ A @ B3 ) ) ) ) ).
% Bolzano
thf(fact_1253_A__def,axiom,
( a2
= ( fChoice_set_a
@ ^ [A8: set_a] :
( ( member_set_a @ A8 @ ( 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 @ A8 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A8 ) ) ) ) ) ) ) ).
% A_def
thf(fact_1254_int__ops_I8_J,axiom,
! [A: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B3 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).
% int_ops(8)
thf(fact_1255_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1256_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_1257_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1258_sumsetp__sumset__eq,axiom,
! [A2: set_a,B2: set_a] :
( ( pluenn895083305082786853setp_a @ g @ addition
@ ^ [X2: a] : ( member_a @ X2 @ A2 )
@ ^ [X2: a] : ( member_a @ X2 @ B2 ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 ) ) ) ) ).
% sumsetp_sumset_eq
thf(fact_1259_sumset__def,axiom,
( ( pluenn3038260743871226533mset_a @ g @ addition )
= ( ^ [A8: set_a,B8: set_a] :
( collect_a
@ ( pluenn895083305082786853setp_a @ g @ addition
@ ^ [X2: a] : ( member_a @ X2 @ A8 )
@ ^ [X2: a] : ( member_a @ X2 @ B8 ) ) ) ) ) ).
% sumset_def
thf(fact_1260_KS__def,axiom,
( ks
= ( image_set_a_real
@ ^ [A8: set_a] : ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A8 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A8 ) ) )
@ ( minus_5736297505244876581_set_a @ ( pow_a @ a0 ) @ ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ) ) ).
% KS_def
thf(fact_1261_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_1262_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1263_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1264_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1265_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1266_int__ops_I7_J,axiom,
! [A: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B3 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).
% int_ops(7)
thf(fact_1267_sumset__eq,axiom,
! [A2: set_a,B2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B2 )
= ( collect_a
@ ^ [C6: a] :
? [X2: a] :
( ( member_a @ X2 @ ( inf_inf_set_a @ A2 @ g ) )
& ? [Y4: a] :
( ( member_a @ Y4 @ ( inf_inf_set_a @ B2 @ g ) )
& ( C6
= ( addition @ X2 @ Y4 ) ) ) ) ) ) ).
% sumset_eq
% Helper facts (1)
thf(help_fChoice_1_1_fChoice_001t__Set__Oset_Itf__a_J_T,axiom,
! [P: set_a > $o] :
( ( P @ ( fChoice_set_a @ P ) )
= ( ? [X6: set_a] : ( P @ X6 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_real @ zero_zero_real @ k ).
%------------------------------------------------------------------------------