TPTP Problem File: SLH0233^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Pluennecke_Ruzsa_Inequality/0003_Pluennecke_Ruzsa_Inequality/prob_00499_019945__12316334_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1358 ( 709 unt; 88 typ; 0 def)
% Number of atoms : 3090 (1368 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 10296 ( 335 ~; 68 |; 206 &;8586 @)
% ( 0 <=>;1101 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 188 ( 188 >; 0 *; 0 +; 0 <<)
% Number of symbols : 81 ( 80 usr; 19 con; 0-5 aty)
% Number of variables : 3039 ( 107 ^;2878 !; 54 ?;3039 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:24:27.864
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (80)
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__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__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_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Real__Oreal,type,
inf_inf_real: real > real > real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Real__Oreal_J,type,
inf_inf_set_real: set_real > set_real > set_real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_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__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Real__Oreal,type,
sup_sup_real: real > real > real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Real__Oreal_J,type,
sup_sup_set_real: set_real > set_real > set_real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__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__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__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__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__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__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_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_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_H____,type,
a2: set_a ).
thf(sy_v_A____,type,
a3: set_a ).
thf(sy_v_B,type,
b: set_a ).
thf(sy_v_C____,type,
c: set_a ).
thf(sy_v_Ca____,type,
ca: 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_x____,type,
x: a ).
% Relevant facts (1269)
thf(fact_0_insert_Ohyps_I2_J,axiom,
~ ( member_a @ x @ ca ) ).
% insert.hyps(2)
thf(fact_1_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( addition @ X @ Y )
= ( addition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_2__092_060open_062x_A_092_060in_062_AG_092_060close_062,axiom,
member_a @ x @ g ).
% \<open>x \<in> G\<close>
thf(fact_3_sumset_Ocases,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( addition @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ g )
=> ( ( member_a @ B2 @ B )
=> ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).
% sumset.cases
thf(fact_4_sumset_Osimps,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= ( ? [A4: a,B3: a] :
( ( A
= ( addition @ A4 @ B3 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ g )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumset.simps
thf(fact_5_sumset_OsumsetI,axiom,
! [A: a,A2: set_a,B4: a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ B )
=> ( ( member_a @ B4 @ g )
=> ( member_a @ ( addition @ A @ B4 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ) ) ).
% sumset.sumsetI
thf(fact_6_sumset__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ C )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ C ) ) ) ).
% sumset_assoc
thf(fact_7_sumset__commute,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A2 ) ) ).
% sumset_commute
thf(fact_8_insert_Ohyps_I1_J,axiom,
finite_finite_a @ ca ).
% insert.hyps(1)
thf(fact_9__092_060open_062K_A_092_060le_062_AK0_092_060close_062,axiom,
ord_less_eq_real @ k @ k0 ).
% \<open>K \<le> K0\<close>
thf(fact_10_associative,axiom,
! [A: a,B4: a,C2: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ g )
=> ( ( member_a @ C2 @ g )
=> ( ( addition @ ( addition @ A @ B4 ) @ C2 )
= ( addition @ A @ ( addition @ B4 @ C2 ) ) ) ) ) ) ).
% associative
thf(fact_11_composition__closed,axiom,
! [A: a,B4: a] :
( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ g )
=> ( member_a @ ( addition @ A @ B4 ) @ g ) ) ) ).
% composition_closed
thf(fact_12_insert_Oprems,axiom,
ord_less_eq_set_a @ ( insert_a @ x @ ca ) @ g ).
% insert.prems
thf(fact_13_additive__abelian__group_Osumset_Ocong,axiom,
pluenn3038260743871226533mset_a = pluenn3038260743871226533mset_a ).
% additive_abelian_group.sumset.cong
thf(fact_14_sumsetp_Ocases,axiom,
! [A2: a > $o,B: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ A )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( addition @ A3 @ B2 ) )
=> ( ( A2 @ A3 )
=> ( ( member_a @ A3 @ g )
=> ( ( B @ B2 )
=> ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).
% sumsetp.cases
thf(fact_15_sumsetp_Osimps,axiom,
! [A2: a > $o,B: a > $o,A: a] :
( ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ A )
= ( ? [A4: a,B3: a] :
( ( A
= ( addition @ A4 @ B3 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ g )
& ( B @ B3 )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumsetp.simps
thf(fact_16_sumsetp_OsumsetI,axiom,
! [A2: a > $o,A: a,B: a > $o,B4: a] :
( ( A2 @ A )
=> ( ( member_a @ A @ g )
=> ( ( B @ B4 )
=> ( ( member_a @ B4 @ g )
=> ( pluenn895083305082786853setp_a @ g @ addition @ A2 @ B @ ( addition @ A @ B4 ) ) ) ) ) ) ).
% sumsetp.sumsetI
thf(fact_17_le__add__diff__inverse,axiom,
! [B4: real,A: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( plus_plus_real @ B4 @ ( minus_minus_real @ A @ B4 ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_18_le__add__diff__inverse,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( plus_plus_nat @ B4 @ ( minus_minus_nat @ A @ B4 ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_19_le__add__diff__inverse2,axiom,
! [B4: real,A: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B4 ) @ B4 )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_20_le__add__diff__inverse2,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B4 ) @ B4 )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_21_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_22_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_add
thf(fact_23_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_24_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_25_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_26_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_27_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_28_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_29_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_30_add__diff__cancel,axiom,
! [A: real,B4: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B4 ) @ B4 )
= A ) ).
% add_diff_cancel
thf(fact_31_diff__add__cancel,axiom,
! [A: real,B4: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B4 ) @ B4 )
= A ) ).
% diff_add_cancel
thf(fact_32_add__diff__cancel__left,axiom,
! [C2: real,A: real,B4: real] :
( ( minus_minus_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B4 ) )
= ( minus_minus_real @ A @ B4 ) ) ).
% add_diff_cancel_left
thf(fact_33_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B4 ) )
= ( minus_minus_nat @ A @ B4 ) ) ).
% add_diff_cancel_left
thf(fact_34_that_I1_J,axiom,
finite_finite_a @ c ).
% that(1)
thf(fact_35_that_I2_J,axiom,
ord_less_eq_set_a @ c @ g ).
% that(2)
thf(fact_36__092_060open_062finite_AA_092_060close_062,axiom,
finite_finite_a @ a3 ).
% \<open>finite A\<close>
thf(fact_37__092_060open_062finite_AA_H_092_060close_062,axiom,
finite_finite_a @ a2 ).
% \<open>finite A'\<close>
thf(fact_38_assms_I2_J,axiom,
finite_finite_a @ a0 ).
% assms(2)
thf(fact_39__092_060open_062A_A_092_060subseteq_062_AG_092_060close_062,axiom,
ord_less_eq_set_a @ a3 @ g ).
% \<open>A \<subseteq> G\<close>
thf(fact_40__092_060open_062C_A_092_060subseteq_062_AG_092_060close_062,axiom,
ord_less_eq_set_a @ ca @ g ).
% \<open>C \<subseteq> G\<close>
thf(fact_41_add__right__cancel,axiom,
! [B4: real,A: real,C2: real] :
( ( ( plus_plus_real @ B4 @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B4 = C2 ) ) ).
% add_right_cancel
thf(fact_42_add__right__cancel,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B4 @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B4 = C2 ) ) ).
% add_right_cancel
thf(fact_43_add__left__cancel,axiom,
! [A: real,B4: real,C2: real] :
( ( ( plus_plus_real @ A @ B4 )
= ( plus_plus_real @ A @ C2 ) )
= ( B4 = C2 ) ) ).
% add_left_cancel
thf(fact_44_add__left__cancel,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ A @ C2 ) )
= ( B4 = C2 ) ) ).
% add_left_cancel
thf(fact_45_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_46_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_47_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_48_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_49__092_060open_062A_H_A_092_060subseteq_062_AA_092_060close_062,axiom,
ord_less_eq_set_a @ a2 @ a3 ).
% \<open>A' \<subseteq> A\<close>
thf(fact_50_sumset__subset__carrier,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ g ) ).
% sumset_subset_carrier
thf(fact_51_sumset__mono,axiom,
! [A5: set_a,A2: set_a,B5: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ B5 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% sumset_mono
thf(fact_52_finite__sumset,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% finite_sumset
thf(fact_53_sumset__subset__insert_I1_J,axiom,
! [A2: set_a,B: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ B ) ) ) ).
% sumset_subset_insert(1)
thf(fact_54_sumset__subset__insert_I2_J,axiom,
! [A2: set_a,B: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A2 ) @ B ) ) ).
% sumset_subset_insert(2)
thf(fact_55_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_57_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_58_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_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_61_assms_I3_J,axiom,
ord_less_eq_set_a @ a0 @ g ).
% assms(3)
thf(fact_62_add__le__cancel__right,axiom,
! [A: real,C2: real,B4: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B4 @ C2 ) )
= ( ord_less_eq_real @ A @ B4 ) ) ).
% add_le_cancel_right
thf(fact_63_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B4 @ C2 ) )
= ( ord_less_eq_nat @ A @ B4 ) ) ).
% add_le_cancel_right
thf(fact_64_add__le__cancel__left,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B4 ) )
= ( ord_less_eq_real @ A @ B4 ) ) ).
% add_le_cancel_left
thf(fact_65_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B4 ) )
= ( ord_less_eq_nat @ A @ B4 ) ) ).
% add_le_cancel_left
thf(fact_66_add__diff__cancel__right_H,axiom,
! [A: real,B4: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B4 ) @ B4 )
= A ) ).
% add_diff_cancel_right'
thf(fact_67_add__diff__cancel__right_H,axiom,
! [A: nat,B4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B4 ) @ B4 )
= A ) ).
% add_diff_cancel_right'
thf(fact_68_add__diff__cancel__right,axiom,
! [A: real,C2: real,B4: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B4 @ C2 ) )
= ( minus_minus_real @ A @ B4 ) ) ).
% add_diff_cancel_right
thf(fact_69_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B4 @ C2 ) )
= ( minus_minus_nat @ A @ B4 ) ) ).
% add_diff_cancel_right
thf(fact_70_add__diff__cancel__left_H,axiom,
! [A: real,B4: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B4 ) @ A )
= B4 ) ).
% add_diff_cancel_left'
thf(fact_71_add__diff__cancel__left_H,axiom,
! [A: nat,B4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B4 ) @ A )
= B4 ) ).
% add_diff_cancel_left'
thf(fact_72_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_73_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_74_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_75_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_76_card__le__sumset,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ g )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ) ) ) ) ).
% card_le_sumset
thf(fact_77__092_060open_062card_A_Isumset_AA_AC_J_A_L_Acard_A_IA_A_N_AA_H_J_A_061_Acard_A_Isumset_AA_AC_J_A_L_A_Icard_AA_A_N_Acard_AA_H_J_092_060close_062,axiom,
( ( plus_plus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ca ) ) @ ( finite_card_a @ ( minus_minus_set_a @ a3 @ a2 ) ) )
= ( plus_plus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ca ) ) @ ( minus_minus_nat @ ( finite_card_a @ a3 ) @ ( finite_card_a @ a2 ) ) ) ) ).
% \<open>card (sumset A C) + card (A - A') = card (sumset A C) + (card A - card A')\<close>
thf(fact_78__C_K_C,axiom,
( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( insert_a @ x @ ca ) ) )
= ( plus_plus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ca ) ) @ ( minus_minus_nat @ ( finite_card_a @ a3 ) @ ( finite_card_a @ a2 ) ) ) ) ).
% "*"
thf(fact_79__092_060open_062real_A_Icard_AA_A_N_Acard_AA_H_J_A_061_Areal_A_Icard_AA_J_A_N_Areal_A_Icard_AA_H_J_092_060close_062,axiom,
( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( finite_card_a @ a3 ) @ ( finite_card_a @ a2 ) ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a3 ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a2 ) ) ) ) ).
% \<open>real (card A - card A') = real (card A) - real (card A')\<close>
thf(fact_80_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_81_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_82_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_83_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_84_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_85_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_86_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_87_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_88_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_89_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_90_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_91_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_92_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_93_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_94_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_95_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_96_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_97_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_98_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_99_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_100_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_101_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_102_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B4 ) )
= ( ord_less_eq_nat @ B4 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_103_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_104_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_105_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_106_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_107_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_108_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_109_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_110_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_111_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_112_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_113_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_114_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_115_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_116_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_117_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_118_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_119_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_120_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_121_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B4: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B4 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_122_additive__abelian__group_Osumsetp_Ocong,axiom,
pluenn895083305082786853setp_a = pluenn895083305082786853setp_a ).
% additive_abelian_group.sumsetp.cong
thf(fact_123_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_124_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_125_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_126_mult_Oleft__commute,axiom,
! [B4: real,A: real,C2: real] :
( ( times_times_real @ B4 @ ( times_times_real @ A @ C2 ) )
= ( times_times_real @ A @ ( times_times_real @ B4 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_127_mult_Oleft__commute,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( times_times_nat @ B4 @ ( times_times_nat @ A @ C2 ) )
= ( times_times_nat @ A @ ( times_times_nat @ B4 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_128_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B3: real] : ( times_times_real @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_129_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_130_mult_Oassoc,axiom,
! [A: real,B4: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B4 ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B4 @ C2 ) ) ) ).
% mult.assoc
thf(fact_131_mult_Oassoc,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B4 ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B4 @ C2 ) ) ) ).
% mult.assoc
thf(fact_132_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B4: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B4 ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B4 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_133_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B4 ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B4 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_134_add__right__imp__eq,axiom,
! [B4: real,A: real,C2: real] :
( ( ( plus_plus_real @ B4 @ A )
= ( plus_plus_real @ C2 @ A ) )
=> ( B4 = C2 ) ) ).
% add_right_imp_eq
thf(fact_135_add__right__imp__eq,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B4 @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B4 = C2 ) ) ).
% add_right_imp_eq
thf(fact_136_add__left__imp__eq,axiom,
! [A: real,B4: real,C2: real] :
( ( ( plus_plus_real @ A @ B4 )
= ( plus_plus_real @ A @ C2 ) )
=> ( B4 = C2 ) ) ).
% add_left_imp_eq
thf(fact_137_add__left__imp__eq,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B4 = C2 ) ) ).
% add_left_imp_eq
thf(fact_138_add_Oleft__commute,axiom,
! [B4: real,A: real,C2: real] :
( ( plus_plus_real @ B4 @ ( plus_plus_real @ A @ C2 ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B4 @ C2 ) ) ) ).
% add.left_commute
thf(fact_139_add_Oleft__commute,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B4 @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B4 @ C2 ) ) ) ).
% add.left_commute
thf(fact_140_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B3: real] : ( plus_plus_real @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_141_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_142_add_Oright__cancel,axiom,
! [B4: real,A: real,C2: real] :
( ( ( plus_plus_real @ B4 @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B4 = C2 ) ) ).
% add.right_cancel
thf(fact_143_add_Oleft__cancel,axiom,
! [A: real,B4: real,C2: real] :
( ( ( plus_plus_real @ A @ B4 )
= ( plus_plus_real @ A @ C2 ) )
= ( B4 = C2 ) ) ).
% add.left_cancel
thf(fact_144_add_Oassoc,axiom,
! [A: real,B4: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B4 ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B4 @ C2 ) ) ) ).
% add.assoc
thf(fact_145_add_Oassoc,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B4 ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B4 @ C2 ) ) ) ).
% add.assoc
thf(fact_146_group__cancel_Oadd2,axiom,
! [B: real,K: real,B4: real,A: real] :
( ( B
= ( plus_plus_real @ K @ B4 ) )
=> ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B4 ) ) ) ) ).
% group_cancel.add2
thf(fact_147_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B4: nat,A: nat] :
( ( B
= ( plus_plus_nat @ K @ B4 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B4 ) ) ) ) ).
% group_cancel.add2
thf(fact_148_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B4: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B4 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B4 ) ) ) ) ).
% group_cancel.add1
thf(fact_149_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B4: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B4 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B4 ) ) ) ) ).
% group_cancel.add1
thf(fact_150_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_151_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_152_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B4: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B4 ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B4 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_153_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B4 ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B4 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_154_diff__right__commute,axiom,
! [A: real,C2: real,B4: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B4 )
= ( minus_minus_real @ ( minus_minus_real @ A @ B4 ) @ C2 ) ) ).
% diff_right_commute
thf(fact_155_diff__right__commute,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B4 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B4 ) @ C2 ) ) ).
% diff_right_commute
thf(fact_156_diff__eq__diff__eq,axiom,
! [A: real,B4: real,C2: real,D: real] :
( ( ( minus_minus_real @ A @ B4 )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( A = B4 )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_157_add__le__imp__le__right,axiom,
! [A: real,C2: real,B4: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B4 @ C2 ) )
=> ( ord_less_eq_real @ A @ B4 ) ) ).
% add_le_imp_le_right
thf(fact_158_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B4 @ C2 ) )
=> ( ord_less_eq_nat @ A @ B4 ) ) ).
% add_le_imp_le_right
thf(fact_159_add__le__imp__le__left,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B4 ) )
=> ( ord_less_eq_real @ A @ B4 ) ) ).
% add_le_imp_le_left
thf(fact_160_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B4 ) )
=> ( ord_less_eq_nat @ A @ B4 ) ) ).
% add_le_imp_le_left
thf(fact_161_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_162_add__right__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B4 @ C2 ) ) ) ).
% add_right_mono
thf(fact_163_add__right__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B4 @ C2 ) ) ) ).
% add_right_mono
thf(fact_164_less__eqE,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ~ ! [C4: nat] :
( B4
!= ( plus_plus_nat @ A @ C4 ) ) ) ).
% less_eqE
thf(fact_165_add__left__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B4 ) ) ) ).
% add_left_mono
thf(fact_166_add__left__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B4 ) ) ) ).
% add_left_mono
thf(fact_167_add__mono,axiom,
! [A: real,B4: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B4 @ D ) ) ) ) ).
% add_mono
thf(fact_168_add__mono,axiom,
! [A: nat,B4: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B4 @ D ) ) ) ) ).
% add_mono
thf(fact_169_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_170_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_171_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_172_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_173_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_174_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_175_diff__eq__diff__less__eq,axiom,
! [A: real,B4: real,C2: real,D: real] :
( ( ( minus_minus_real @ A @ B4 )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( ord_less_eq_real @ A @ B4 )
= ( ord_less_eq_real @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_176_diff__right__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B4 @ C2 ) ) ) ).
% diff_right_mono
thf(fact_177_diff__left__mono,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B4 ) ) ) ).
% diff_left_mono
thf(fact_178_diff__mono,axiom,
! [A: real,B4: real,D: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ D @ C2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B4 @ D ) ) ) ) ).
% diff_mono
thf(fact_179_combine__common__factor,axiom,
! [A: real,E: real,B4: real,C2: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B4 @ E ) @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B4 ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_180_combine__common__factor,axiom,
! [A: nat,E: nat,B4: nat,C2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B4 @ E ) @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B4 ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_181_distrib__right,axiom,
! [A: real,B4: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B4 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ).
% distrib_right
thf(fact_182_distrib__right,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B4 ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) ) ) ).
% distrib_right
thf(fact_183_distrib__left,axiom,
! [A: real,B4: real,C2: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B4 @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ B4 ) @ ( times_times_real @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_184_distrib__left,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B4 @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B4 ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_185_comm__semiring__class_Odistrib,axiom,
! [A: real,B4: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B4 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_186_comm__semiring__class_Odistrib,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B4 ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_187_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B4: real,C2: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B4 @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ B4 ) @ ( times_times_real @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_188_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B4: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B4 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_189_right__diff__distrib_H,axiom,
! [A: real,B4: real,C2: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B4 @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A @ B4 ) @ ( times_times_real @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_190_right__diff__distrib_H,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B4 @ C2 ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B4 ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_191_left__diff__distrib_H,axiom,
! [B4: real,C2: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B4 @ C2 ) @ A )
= ( minus_minus_real @ ( times_times_real @ B4 @ A ) @ ( times_times_real @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_192_left__diff__distrib_H,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B4 @ C2 ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B4 @ A ) @ ( times_times_nat @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_193_right__diff__distrib,axiom,
! [A: real,B4: real,C2: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B4 @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A @ B4 ) @ ( times_times_real @ A @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_194_left__diff__distrib,axiom,
! [A: real,B4: real,C2: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B4 ) @ C2 )
= ( minus_minus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_195_diff__diff__eq,axiom,
! [A: real,B4: real,C2: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B4 ) @ C2 )
= ( minus_minus_real @ A @ ( plus_plus_real @ B4 @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_196_diff__diff__eq,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B4 ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B4 @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_197_add__implies__diff,axiom,
! [C2: real,B4: real,A: real] :
( ( ( plus_plus_real @ C2 @ B4 )
= A )
=> ( C2
= ( minus_minus_real @ A @ B4 ) ) ) ).
% add_implies_diff
thf(fact_198_add__implies__diff,axiom,
! [C2: nat,B4: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B4 )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B4 ) ) ) ).
% add_implies_diff
thf(fact_199_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B4: real,C2: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B4 @ C2 ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B4 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_200_diff__add__eq,axiom,
! [A: real,B4: real,C2: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B4 ) @ C2 )
= ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B4 ) ) ).
% diff_add_eq
thf(fact_201_diff__diff__eq2,axiom,
! [A: real,B4: real,C2: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B4 @ C2 ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B4 ) ) ).
% diff_diff_eq2
thf(fact_202_add__diff__eq,axiom,
! [A: real,B4: real,C2: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B4 @ C2 ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B4 ) @ C2 ) ) ).
% add_diff_eq
thf(fact_203_eq__diff__eq,axiom,
! [A: real,C2: real,B4: real] :
( ( A
= ( minus_minus_real @ C2 @ B4 ) )
= ( ( plus_plus_real @ A @ B4 )
= C2 ) ) ).
% eq_diff_eq
thf(fact_204_diff__eq__eq,axiom,
! [A: real,B4: real,C2: real] :
( ( ( minus_minus_real @ A @ B4 )
= C2 )
= ( A
= ( plus_plus_real @ C2 @ B4 ) ) ) ).
% diff_eq_eq
thf(fact_205_group__cancel_Osub1,axiom,
! [A2: real,K: real,A: real,B4: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A2 @ B4 )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B4 ) ) ) ) ).
% group_cancel.sub1
thf(fact_206_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_207_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_208_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_209_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_210_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_211_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_212_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ( minus_minus_nat @ B4 @ A )
= C2 )
= ( B4
= ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_213_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B4 @ A ) )
= B4 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_214_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B4 @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B4 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_215_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B4 @ C2 ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B4 @ A ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_216_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B4 @ A ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B4 @ C2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_217_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B4 ) @ A )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B4 @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_218_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B4 @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B4 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_219_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B4 @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B4 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_220_le__add__diff,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B4 @ C2 ) @ A ) ) ) ).
% le_add_diff
thf(fact_221_add__le__add__imp__diff__le,axiom,
! [I: real,K: real,N: real,J: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_222_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_223_diff__add,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B4 @ A ) @ A )
= B4 ) ) ).
% diff_add
thf(fact_224_add__le__imp__le__diff,axiom,
! [I: real,K: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_225_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_226_le__diff__eq,axiom,
! [A: real,C2: real,B4: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C2 @ B4 ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B4 ) @ C2 ) ) ).
% le_diff_eq
thf(fact_227_diff__le__eq,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B4 ) @ C2 )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C2 @ B4 ) ) ) ).
% diff_le_eq
thf(fact_228_square__diff__square__factored,axiom,
! [X: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_229_eq__add__iff2,axiom,
! [A: real,E: real,C2: real,B4: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B4 @ E ) @ D ) )
= ( C2
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B4 @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_230_eq__add__iff1,axiom,
! [A: real,E: real,C2: real,B4: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B4 @ E ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B4 ) @ E ) @ C2 )
= D ) ) ).
% eq_add_iff1
thf(fact_231_ordered__ring__class_Ole__add__iff2,axiom,
! [A: real,E: real,C2: real,B4: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B4 @ E ) @ D ) )
= ( ord_less_eq_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B4 @ A ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_232_ordered__ring__class_Ole__add__iff1,axiom,
! [A: real,E: real,C2: real,B4: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B4 @ E ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B4 ) @ E ) @ C2 ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_233__092_060open_062card_A_Isumset_AA_A_Iinsert_Ax_AC_J_J_A_061_Acard_A_Isumset_AA_AC_J_A_L_Acard_A_Isumset_A_IA_A_N_AA_H_J_A_123x_125_J_092_060close_062,axiom,
( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( insert_a @ x @ ca ) ) )
= ( plus_plus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ca ) ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( minus_minus_set_a @ a3 @ a2 ) @ ( insert_a @ x @ bot_bot_set_a ) ) ) ) ) ).
% \<open>card (sumset A (insert x C)) = card (sumset A C) + card (sumset (A - A') {x})\<close>
thf(fact_234_calculation,axiom,
( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( semiri5074537144036343181t_real @ Y2 ) ) )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ( insert_a @ x @ ca ) ) ) ) ) @ ( plus_plus_real @ ( times_times_real @ k @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ca ) ) ) ) @ ( minus_minus_real @ ( times_times_real @ k @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a3 ) ) ) @ ( times_times_real @ k @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a2 ) ) ) ) ) ) ) ).
% calculation
thf(fact_235__092_060open_062card_A_Isumset_AA_AC_J_A_L_Acard_A_I_IA_A_N_AA_H_J_A_092_060inter_062_AG_J_A_061_Acard_A_Isumset_AA_AC_J_A_L_Acard_A_IA_A_N_AA_H_J_092_060close_062,axiom,
( ( plus_plus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ca ) ) @ ( finite_card_a @ ( inf_inf_set_a @ ( minus_minus_set_a @ a3 @ a2 ) @ g ) ) )
= ( plus_plus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ca ) ) @ ( finite_card_a @ ( minus_minus_set_a @ a3 @ a2 ) ) ) ) ).
% \<open>card (sumset A C) + card ((A - A') \<inter> G) = card (sumset A C) + card (A - A')\<close>
thf(fact_236__092_060open_062real_A_Icard_A_Isumset_AA_A_Isumset_AB_AC_J_J_A_L_A_Icard_A_Isumset_AA_AB_J_A_N_Acard_A_Isumset_AA_H_AB_J_J_J_A_092_060le_062_AK_A_K_Areal_A_Icard_A_Isumset_AA_AC_J_J_A_L_A_IK_A_K_Areal_A_Icard_AA_J_A_N_AK_A_K_Areal_A_Icard_AA_H_J_J_092_060close_062,axiom,
ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ca ) ) ) @ ( minus_minus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ b ) ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ b ) ) ) ) ) @ ( plus_plus_real @ ( times_times_real @ k @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ca ) ) ) ) @ ( minus_minus_real @ ( times_times_real @ k @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a3 ) ) ) @ ( times_times_real @ k @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a2 ) ) ) ) ) ).
% \<open>real (card (sumset A (sumset B C)) + (card (sumset A B) - card (sumset A' B))) \<le> K * real (card (sumset A C)) + (K * real (card A) - K * real (card A'))\<close>
thf(fact_237_assms_I6_J,axiom,
ord_less_eq_set_a @ b @ g ).
% assms(6)
thf(fact_238_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_239_card__sumset__0__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ g )
=> ( ( ord_less_eq_set_a @ B @ g )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
| ( ( finite_card_a @ B )
= zero_zero_nat ) ) ) ) ) ).
% card_sumset_0_iff
thf(fact_240_insert__subset,axiom,
! [X: real,A2: set_real,B: set_real] :
( ( ord_less_eq_set_real @ ( insert_real @ X @ A2 ) @ B )
= ( ( member_real @ X @ B )
& ( ord_less_eq_set_real @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_241_insert__subset,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A2 ) @ B )
= ( ( member_set_a @ X @ B )
& ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_242_insert__subset,axiom,
! [X: a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_243_finite__Diff__insert,axiom,
! [A2: set_real,A: real,B: set_real] :
( ( finite_finite_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B ) ) )
= ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_244_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_245_finite__Diff__insert,axiom,
! [A2: set_set_a,A: set_a,B: set_set_a] :
( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B ) ) )
= ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_246_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_247_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_248_finite__insert,axiom,
! [A: real,A2: set_real] :
( ( finite_finite_real @ ( insert_real @ A @ A2 ) )
= ( finite_finite_real @ A2 ) ) ).
% finite_insert
thf(fact_249_sumsetdiff__sing,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( minus_minus_set_a @ A2 @ B ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% sumsetdiff_sing
thf(fact_250_KS_I1_J,axiom,
finite_finite_real @ ks ).
% KS(1)
thf(fact_251_assms_I7_J,axiom,
b != bot_bot_set_a ).
% assms(7)
thf(fact_252_assms_I5_J,axiom,
finite_finite_a @ b ).
% assms(5)
thf(fact_253_assms_I4_J,axiom,
a0 != bot_bot_set_a ).
% assms(4)
thf(fact_254_A_I2_J,axiom,
a3 != bot_bot_set_a ).
% A(2)
thf(fact_255_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_256_empty__Collect__eq,axiom,
! [P: real > $o] :
( ( bot_bot_set_real
= ( collect_real @ P ) )
= ( ! [X2: real] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_257_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_258_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_259_Collect__empty__eq,axiom,
! [P: real > $o] :
( ( ( collect_real @ P )
= bot_bot_set_real )
= ( ! [X2: real] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_260_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_261_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_262_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_263_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_264_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_265_empty__iff,axiom,
! [C2: real] :
~ ( member_real @ C2 @ bot_bot_set_real ) ).
% empty_iff
thf(fact_266_empty__iff,axiom,
! [C2: set_a] :
~ ( member_set_a @ C2 @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_267_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_268_subsetI,axiom,
! [A2: set_real,B: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ X3 @ B ) )
=> ( ord_less_eq_set_real @ A2 @ B ) ) ).
% subsetI
thf(fact_269_subsetI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_270_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_271_insertCI,axiom,
! [A: a,B: set_a,B4: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = B4 ) )
=> ( member_a @ A @ ( insert_a @ B4 @ B ) ) ) ).
% insertCI
thf(fact_272_insertCI,axiom,
! [A: real,B: set_real,B4: real] :
( ( ~ ( member_real @ A @ B )
=> ( A = B4 ) )
=> ( member_real @ A @ ( insert_real @ B4 @ B ) ) ) ).
% insertCI
thf(fact_273_insertCI,axiom,
! [A: set_a,B: set_set_a,B4: set_a] :
( ( ~ ( member_set_a @ A @ B )
=> ( A = B4 ) )
=> ( member_set_a @ A @ ( insert_set_a @ B4 @ B ) ) ) ).
% insertCI
thf(fact_274_insert__iff,axiom,
! [A: a,B4: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B4 @ A2 ) )
= ( ( A = B4 )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_275_insert__iff,axiom,
! [A: real,B4: real,A2: set_real] :
( ( member_real @ A @ ( insert_real @ B4 @ A2 ) )
= ( ( A = B4 )
| ( member_real @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_276_insert__iff,axiom,
! [A: set_a,B4: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B4 @ A2 ) )
= ( ( A = B4 )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_277_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_278_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_279_Int__iff,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B ) )
= ( ( member_real @ C2 @ A2 )
& ( member_real @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_280_Int__iff,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
= ( ( member_set_a @ C2 @ A2 )
& ( member_set_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_281_Int__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
& ( member_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_282_IntI,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ A2 )
=> ( ( member_real @ C2 @ B )
=> ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_283_IntI,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_284_IntI,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_285_DiffI,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ A2 )
=> ( ~ ( member_real @ C2 @ B )
=> ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_286_DiffI,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_287_DiffI,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( ~ ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_288_Diff__iff,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B ) )
= ( ( member_real @ C2 @ A2 )
& ~ ( member_real @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_289_Diff__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
& ~ ( member_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_290_Diff__iff,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= ( ( member_set_a @ C2 @ A2 )
& ~ ( member_set_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_291_Diff__idemp,axiom,
! [A2: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_292_Diff__idemp,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ B )
= ( minus_5736297505244876581_set_a @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_293_A_I1_J,axiom,
ord_less_eq_set_a @ a3 @ a0 ).
% A(1)
thf(fact_294_sumset__empty_H_I2_J,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= bot_bot_set_a ) ) ).
% sumset_empty'(2)
thf(fact_295_sumset__empty_H_I1_J,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A2 )
= bot_bot_set_a ) ) ).
% sumset_empty'(1)
thf(fact_296_finite__sumset_H,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B @ g ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% finite_sumset'
thf(fact_297_card__sumset__0__iff_H,axiom,
! [A2: set_a,B: set_a] :
( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) )
= zero_zero_nat )
| ( ( finite_card_a @ ( inf_inf_set_a @ B @ g ) )
= zero_zero_nat ) ) ) ).
% card_sumset_0_iff'
thf(fact_298_infinite__sumset__aux,axiom,
! [A2: set_a,B: set_a] :
( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) )
= ( ( inf_inf_set_a @ B @ g )
!= bot_bot_set_a ) ) ) ).
% infinite_sumset_aux
thf(fact_299_infinite__sumset__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) )
= ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ g ) )
& ( ( inf_inf_set_a @ B @ g )
!= bot_bot_set_a ) )
| ( ( ( inf_inf_set_a @ A2 @ g )
!= bot_bot_set_a )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ B @ g ) ) ) ) ) ).
% infinite_sumset_iff
thf(fact_300_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_301_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_302_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_303_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_304_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_305_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_306_mult__eq__0__iff,axiom,
! [A: real,B4: real] :
( ( ( times_times_real @ A @ B4 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B4 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_307_mult__eq__0__iff,axiom,
! [A: nat,B4: nat] :
( ( ( times_times_nat @ A @ B4 )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B4 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_308_mult__cancel__left,axiom,
! [C2: real,A: real,B4: real] :
( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B4 ) )
= ( ( C2 = zero_zero_real )
| ( A = B4 ) ) ) ).
% mult_cancel_left
thf(fact_309_mult__cancel__left,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B4 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B4 ) ) ) ).
% mult_cancel_left
thf(fact_310_mult__cancel__right,axiom,
! [A: real,C2: real,B4: real] :
( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B4 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A = B4 ) ) ) ).
% mult_cancel_right
thf(fact_311_mult__cancel__right,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B4 @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B4 ) ) ) ).
% mult_cancel_right
thf(fact_312_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_313_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_314_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_315_add__cancel__left__left,axiom,
! [B4: nat,A: nat] :
( ( ( plus_plus_nat @ B4 @ A )
= A )
= ( B4 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_316_add__cancel__left__left,axiom,
! [B4: real,A: real] :
( ( ( plus_plus_real @ B4 @ A )
= A )
= ( B4 = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_317_add__cancel__left__right,axiom,
! [A: nat,B4: nat] :
( ( ( plus_plus_nat @ A @ B4 )
= A )
= ( B4 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_318_add__cancel__left__right,axiom,
! [A: real,B4: real] :
( ( ( plus_plus_real @ A @ B4 )
= A )
= ( B4 = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_319_add__cancel__right__left,axiom,
! [A: nat,B4: nat] :
( ( A
= ( plus_plus_nat @ B4 @ A ) )
= ( B4 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_320_add__cancel__right__left,axiom,
! [A: real,B4: real] :
( ( A
= ( plus_plus_real @ B4 @ A ) )
= ( B4 = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_321_add__cancel__right__right,axiom,
! [A: nat,B4: nat] :
( ( A
= ( plus_plus_nat @ A @ B4 ) )
= ( B4 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_322_add__cancel__right__right,axiom,
! [A: real,B4: real] :
( ( A
= ( plus_plus_real @ A @ B4 ) )
= ( B4 = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_323_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_324_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_325_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_326_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_327_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_328_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_329_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_330_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_331_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_332_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_333_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_334_empty__subsetI,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% empty_subsetI
thf(fact_335_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_336_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_337_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_338_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_339_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_340_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_341_singletonI,axiom,
! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).
% singletonI
thf(fact_342_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_343_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_344_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_345_finite__Int,axiom,
! [F: set_real,G: set_real] :
( ( ( finite_finite_real @ F )
| ( finite_finite_real @ G ) )
=> ( finite_finite_real @ ( inf_inf_set_real @ F @ G ) ) ) ).
% finite_Int
thf(fact_346_finite__Int,axiom,
! [F: set_a,G: set_a] :
( ( ( finite_finite_a @ F )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F @ G ) ) ) ).
% finite_Int
thf(fact_347_Int__subset__iff,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_348_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_349_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_350_Diff__empty,axiom,
! [A2: set_real] :
( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
= A2 ) ).
% Diff_empty
thf(fact_351_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_352_Diff__empty,axiom,
! [A2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ bot_bot_set_set_a )
= A2 ) ).
% Diff_empty
thf(fact_353_empty__Diff,axiom,
! [A2: set_real] :
( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
= bot_bot_set_real ) ).
% empty_Diff
thf(fact_354_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_355_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_356_Diff__cancel,axiom,
! [A2: set_real] :
( ( minus_minus_set_real @ A2 @ A2 )
= bot_bot_set_real ) ).
% Diff_cancel
thf(fact_357_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_358_Diff__cancel,axiom,
! [A2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ A2 )
= bot_bot_set_set_a ) ).
% Diff_cancel
thf(fact_359_Int__insert__left__if0,axiom,
! [A: real,C: set_real,B: set_real] :
( ~ ( member_real @ A @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A @ B ) @ C )
= ( inf_inf_set_real @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_360_Int__insert__left__if0,axiom,
! [A: set_a,C: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C )
= ( inf_inf_set_set_a @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_361_Int__insert__left__if0,axiom,
! [A: a,C: set_a,B: set_a] :
( ~ ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_362_Int__insert__left__if1,axiom,
! [A: real,C: set_real,B: set_real] :
( ( member_real @ A @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A @ B ) @ C )
= ( insert_real @ A @ ( inf_inf_set_real @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_363_Int__insert__left__if1,axiom,
! [A: set_a,C: set_set_a,B: set_set_a] :
( ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_364_Int__insert__left__if1,axiom,
! [A: a,C: set_a,B: set_a] :
( ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_365_insert__inter__insert,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_366_insert__inter__insert,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_367_Int__insert__right__if0,axiom,
! [A: real,A2: set_real,B: set_real] :
( ~ ( member_real @ A @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B ) )
= ( inf_inf_set_real @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_368_Int__insert__right__if0,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_369_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_370_Int__insert__right__if1,axiom,
! [A: real,A2: set_real,B: set_real] :
( ( member_real @ A @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B ) )
= ( insert_real @ A @ ( inf_inf_set_real @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_371_Int__insert__right__if1,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_372_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_373_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_374_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_375_finite__Diff2,axiom,
! [B: set_real,A2: set_real] :
( ( finite_finite_real @ B )
=> ( ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B ) )
= ( finite_finite_real @ A2 ) ) ) ).
% finite_Diff2
thf(fact_376_finite__Diff2,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_377_finite__Diff2,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= ( finite_finite_set_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_378_finite__Diff,axiom,
! [A2: set_real,B: set_real] :
( ( finite_finite_real @ A2 )
=> ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_379_finite__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_380_finite__Diff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_381_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_382_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_383_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_384_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_385_Diff__insert0,axiom,
! [X: real,A2: set_real,B: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ( minus_minus_set_real @ A2 @ ( insert_real @ X @ B ) )
= ( minus_minus_set_real @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_386_Diff__insert0,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_387_Diff__insert0,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ B ) )
= ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_388_insert__Diff1,axiom,
! [X: real,B: set_real,A2: set_real] :
( ( member_real @ X @ B )
=> ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B )
= ( minus_minus_set_real @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_389_insert__Diff1,axiom,
! [X: a,B: set_a,A2: set_a] :
( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_390_insert__Diff1,axiom,
! [X: set_a,B: set_set_a,A2: set_set_a] :
( ( member_set_a @ X @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B )
= ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_391_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_392_add__le__same__cancel1,axiom,
! [B4: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B4 @ A ) @ B4 )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_393_add__le__same__cancel1,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B4 @ A ) @ B4 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_394_add__le__same__cancel2,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B4 ) @ B4 )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_395_add__le__same__cancel2,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B4 ) @ B4 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_396_le__add__same__cancel1,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B4 ) )
= ( ord_less_eq_real @ zero_zero_real @ B4 ) ) ).
% le_add_same_cancel1
thf(fact_397_le__add__same__cancel1,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B4 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B4 ) ) ).
% le_add_same_cancel1
thf(fact_398_le__add__same__cancel2,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B4 @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B4 ) ) ).
% le_add_same_cancel2
thf(fact_399_le__add__same__cancel2,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B4 @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B4 ) ) ).
% le_add_same_cancel2
thf(fact_400_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_401_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_402_diff__ge__0__iff__ge,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B4 ) )
= ( ord_less_eq_real @ B4 @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_403_diff__add__zero,axiom,
! [A: nat,B4: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B4 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_404_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_405_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_406_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_407_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_408_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_409_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_410_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_411_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_412_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_413_singleton__insert__inj__eq_H,axiom,
! [A: real,A2: set_real,B4: real] :
( ( ( insert_real @ A @ A2 )
= ( insert_real @ B4 @ bot_bot_set_real ) )
= ( ( A = B4 )
& ( ord_less_eq_set_real @ A2 @ ( insert_real @ B4 @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_414_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A2: set_set_a,B4: set_a] :
( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B4 @ bot_bot_set_set_a ) )
= ( ( A = B4 )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B4 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_415_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B4: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B4 @ bot_bot_set_a ) )
= ( ( A = B4 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_416_singleton__insert__inj__eq,axiom,
! [B4: real,A: real,A2: set_real] :
( ( ( insert_real @ B4 @ bot_bot_set_real )
= ( insert_real @ A @ A2 ) )
= ( ( A = B4 )
& ( ord_less_eq_set_real @ A2 @ ( insert_real @ B4 @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_417_singleton__insert__inj__eq,axiom,
! [B4: set_a,A: set_a,A2: set_set_a] :
( ( ( insert_set_a @ B4 @ bot_bot_set_set_a )
= ( insert_set_a @ A @ A2 ) )
= ( ( A = B4 )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B4 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_418_singleton__insert__inj__eq,axiom,
! [B4: a,A: a,A2: set_a] :
( ( ( insert_a @ B4 @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B4 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_419_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_420_card_Oempty,axiom,
( ( finite_card_real @ bot_bot_set_real )
= zero_zero_nat ) ).
% card.empty
thf(fact_421_card_Oempty,axiom,
( ( finite_card_set_a @ bot_bot_set_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_422_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_423_insert__disjoint_I1_J,axiom,
! [A: real,A2: set_real,B: set_real] :
( ( ( inf_inf_set_real @ ( insert_real @ A @ A2 ) @ B )
= bot_bot_set_real )
= ( ~ ( member_real @ A @ B )
& ( ( inf_inf_set_real @ A2 @ B )
= bot_bot_set_real ) ) ) ).
% insert_disjoint(1)
thf(fact_424_insert__disjoint_I1_J,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B )
& ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_425_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B ) )
= ( ~ ( member_a @ A @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_426_insert__disjoint_I2_J,axiom,
! [A: real,A2: set_real,B: set_real] :
( ( bot_bot_set_real
= ( inf_inf_set_real @ ( insert_real @ A @ A2 ) @ B ) )
= ( ~ ( member_real @ A @ B )
& ( bot_bot_set_real
= ( inf_inf_set_real @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_427_insert__disjoint_I2_J,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B ) )
= ( ~ ( member_set_a @ A @ B )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_428_disjoint__insert_I1_J,axiom,
! [B: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_429_disjoint__insert_I1_J,axiom,
! [B: set_real,A: real,A2: set_real] :
( ( ( inf_inf_set_real @ B @ ( insert_real @ A @ A2 ) )
= bot_bot_set_real )
= ( ~ ( member_real @ A @ B )
& ( ( inf_inf_set_real @ B @ A2 )
= bot_bot_set_real ) ) ) ).
% disjoint_insert(1)
thf(fact_430_disjoint__insert_I1_J,axiom,
! [B: set_set_a,A: set_a,A2: set_set_a] :
( ( ( inf_inf_set_set_a @ B @ ( insert_set_a @ A @ A2 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B )
& ( ( inf_inf_set_set_a @ B @ A2 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_431_disjoint__insert_I2_J,axiom,
! [A2: set_a,B4: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B4 @ B ) ) )
= ( ~ ( member_a @ B4 @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_432_disjoint__insert_I2_J,axiom,
! [A2: set_real,B4: real,B: set_real] :
( ( bot_bot_set_real
= ( inf_inf_set_real @ A2 @ ( insert_real @ B4 @ B ) ) )
= ( ~ ( member_real @ B4 @ A2 )
& ( bot_bot_set_real
= ( inf_inf_set_real @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_433_disjoint__insert_I2_J,axiom,
! [A2: set_set_a,B4: set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ B4 @ B ) ) )
= ( ~ ( member_set_a @ B4 @ A2 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_434_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_435_card_Oinfinite,axiom,
! [A2: set_real] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite_card_real @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_436_Diff__eq__empty__iff,axiom,
! [A2: set_real,B: set_real] :
( ( ( minus_minus_set_real @ A2 @ B )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_437_Diff__eq__empty__iff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_438_Diff__eq__empty__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_439_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_440_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_441_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_442_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_443_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_444_Diff__disjoint,axiom,
! [A2: set_real,B: set_real] :
( ( inf_inf_set_real @ A2 @ ( minus_minus_set_real @ B @ A2 ) )
= bot_bot_set_real ) ).
% Diff_disjoint
thf(fact_445_Diff__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_446_Diff__disjoint,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B @ A2 ) )
= bot_bot_set_set_a ) ).
% Diff_disjoint
thf(fact_447_K__cardA,axiom,
( ( times_times_real @ k @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a3 ) ) )
= ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ b ) ) ) ) ).
% K_cardA
thf(fact_448_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_449_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_450_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_451_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_452_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_453_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_454__C0_C,axiom,
( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ca ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( minus_minus_set_a @ a3 @ a2 ) @ ( insert_a @ x @ bot_bot_set_a ) ) )
= bot_bot_set_a ) ).
% "0"
thf(fact_455__092_060open_062sumset_AA_H_A_Isumset_AB_A_123x_125_J_A_092_060subseteq_062_Asumset_AA_A_Isumset_AB_AC_J_092_060close_062,axiom,
ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ( insert_a @ x @ bot_bot_set_a ) ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ca ) ) ).
% \<open>sumset A' (sumset B {x}) \<subseteq> sumset A (sumset B C)\<close>
thf(fact_456__092_060open_062card_A_Isumset_AA_AC_J_A_L_Acard_A_Isumset_A_IA_A_N_AA_H_J_A_123x_125_J_A_061_Acard_A_Isumset_AA_AC_J_A_L_Acard_A_I_IA_A_N_AA_H_J_A_092_060inter_062_AG_J_092_060close_062,axiom,
( ( plus_plus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ca ) ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( minus_minus_set_a @ a3 @ a2 ) @ ( insert_a @ x @ bot_bot_set_a ) ) ) )
= ( plus_plus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ca ) ) @ ( finite_card_a @ ( inf_inf_set_a @ ( minus_minus_set_a @ a3 @ a2 ) @ g ) ) ) ) ).
% \<open>card (sumset A C) + card (sumset (A - A') {x}) = card (sumset A C) + card ((A - A') \<inter> G)\<close>
thf(fact_457__092_060open_062card_A_Isumset_AA_A_Isumset_AB_AC_J_J_A_L_A_Icard_A_Isumset_AA_A_Isumset_AB_A_123x_125_J_J_A_N_Acard_A_Isumset_AA_H_A_Isumset_AB_A_123x_125_J_J_J_A_092_060le_062_Acard_A_Isumset_AA_A_Isumset_AB_AC_J_J_A_L_A_Icard_A_Isumset_AA_AB_J_A_N_Acard_A_Isumset_AA_H_AB_J_J_092_060close_062,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ca ) ) ) @ ( minus_minus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ( insert_a @ x @ bot_bot_set_a ) ) ) ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ( insert_a @ x @ bot_bot_set_a ) ) ) ) ) ) @ ( plus_plus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ca ) ) ) @ ( minus_minus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ b ) ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ b ) ) ) ) ).
% \<open>card (sumset A (sumset B C)) + (card (sumset A (sumset B {x})) - card (sumset A' (sumset B {x}))) \<le> card (sumset A (sumset B C)) + (card (sumset A B) - card (sumset A' B))\<close>
thf(fact_458__092_060open_062card_A_Isumset_AA_A_Isumset_AB_A_Iinsert_Ax_AC_J_J_J_A_092_060le_062_Acard_A_Isumset_AA_A_Isumset_AB_AC_J_J_A_L_Acard_A_Isumset_AA_A_Isumset_AB_A_123x_125_J_A_N_Asumset_AA_H_A_Isumset_AB_A_123x_125_J_J_092_060close_062,axiom,
ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ( insert_a @ x @ ca ) ) ) ) @ ( plus_plus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ca ) ) ) @ ( finite_card_a @ ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ( insert_a @ x @ bot_bot_set_a ) ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ( insert_a @ x @ bot_bot_set_a ) ) ) ) ) ) ).
% \<open>card (sumset A (sumset B (insert x C))) \<le> card (sumset A (sumset B C)) + card (sumset A (sumset B {x}) - sumset A' (sumset B {x}))\<close>
thf(fact_459__092_060open_062card_A_Isumset_AA_A_Isumset_AB_AC_J_J_A_L_Acard_A_Isumset_AA_A_Isumset_AB_A_123x_125_J_A_N_Asumset_AA_H_A_Isumset_AB_A_123x_125_J_J_A_061_Acard_A_Isumset_AA_A_Isumset_AB_AC_J_J_A_L_A_Icard_A_Isumset_AA_A_Isumset_AB_A_123x_125_J_J_A_N_Acard_A_Isumset_AA_H_A_Isumset_AB_A_123x_125_J_J_J_092_060close_062,axiom,
( ( plus_plus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ca ) ) ) @ ( finite_card_a @ ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ( insert_a @ x @ bot_bot_set_a ) ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ( insert_a @ x @ bot_bot_set_a ) ) ) ) ) )
= ( plus_plus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ca ) ) ) @ ( minus_minus_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ( insert_a @ x @ bot_bot_set_a ) ) ) ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ( insert_a @ x @ bot_bot_set_a ) ) ) ) ) ) ) ).
% \<open>card (sumset A (sumset B C)) + card (sumset A (sumset B {x}) - sumset A' (sumset B {x})) = card (sumset A (sumset B C)) + (card (sumset A (sumset B {x})) - card (sumset A' (sumset B {x})))\<close>
thf(fact_460_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_461_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_462_sumset__Int__carrier__eq_I2_J,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( inf_inf_set_a @ A2 @ g ) @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ).
% sumset_Int_carrier_eq(2)
thf(fact_463_sumset__Int__carrier__eq_I1_J,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( inf_inf_set_a @ B @ g ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ).
% sumset_Int_carrier_eq(1)
thf(fact_464_sumset__Int__carrier,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ g )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ).
% sumset_Int_carrier
thf(fact_465_sumset__is__empty__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ g )
= bot_bot_set_a ) ) ) ).
% sumset_is_empty_iff
thf(fact_466_Keq,axiom,
( k
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a3 ) ) ) ) ).
% Keq
thf(fact_467_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_468_DiffE,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B ) )
=> ~ ( ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ B ) ) ) ).
% DiffE
thf(fact_469_DiffE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_470_DiffE,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
=> ~ ( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_471_DiffD1,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B ) )
=> ( member_real @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_472_DiffD1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_473_DiffD1,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
=> ( member_set_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_474_DiffD2,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A2 @ B ) )
=> ~ ( member_real @ C2 @ B ) ) ).
% DiffD2
thf(fact_475_DiffD2,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( member_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_476_DiffD2,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
=> ~ ( member_set_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_477_Int__Diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Int_Diff
thf(fact_478_Int__Diff,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( minus_5736297505244876581_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B @ C ) ) ) ).
% Int_Diff
thf(fact_479_Diff__Int2,axiom,
! [A2: set_a,C: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B @ C ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ B ) ) ).
% Diff_Int2
thf(fact_480_Diff__Int2,axiom,
! [A2: set_set_a,C: set_set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ ( inf_inf_set_set_a @ A2 @ C ) @ ( inf_inf_set_set_a @ B @ C ) )
= ( minus_5736297505244876581_set_a @ ( inf_inf_set_set_a @ A2 @ C ) @ B ) ) ).
% Diff_Int2
thf(fact_481_Diff__triv,axiom,
! [A2: set_real,B: set_real] :
( ( ( inf_inf_set_real @ A2 @ B )
= bot_bot_set_real )
=> ( ( minus_minus_set_real @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_482_Diff__triv,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_483_Diff__triv,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
=> ( ( minus_5736297505244876581_set_a @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_484_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_485_Diff__Diff__Int,axiom,
! [A2: set_a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_486_Diff__Diff__Int,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_487_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_488_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_489_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_490_Diff__Int__distrib,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C @ A2 ) @ ( inf_inf_set_a @ C @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_491_Diff__Int__distrib,axiom,
! [C: set_set_a,A2: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= ( minus_5736297505244876581_set_a @ ( inf_inf_set_set_a @ C @ A2 ) @ ( inf_inf_set_set_a @ C @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_492_Diff__Int__distrib2,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_493_Diff__Int__distrib2,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ C )
= ( minus_5736297505244876581_set_a @ ( inf_inf_set_set_a @ A2 @ C ) @ ( inf_inf_set_set_a @ B @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_494_Int__Diff__disjoint,axiom,
! [A2: set_real,B: set_real] :
( ( inf_inf_set_real @ ( inf_inf_set_real @ A2 @ B ) @ ( minus_minus_set_real @ A2 @ B ) )
= bot_bot_set_real ) ).
% Int_Diff_disjoint
thf(fact_495_Int__Diff__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_496_Int__Diff__disjoint,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= bot_bot_set_set_a ) ).
% Int_Diff_disjoint
thf(fact_497_disjoint__iff__not__equal,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ B )
=> ( X2 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_498_disjoint__iff__not__equal,axiom,
! [A2: set_real,B: set_real] :
( ( ( inf_inf_set_real @ A2 @ B )
= bot_bot_set_real )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ! [Y4: real] :
( ( member_real @ Y4 @ B )
=> ( X2 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_499_disjoint__iff__not__equal,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ B )
=> ( X2 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_500_Int__left__commute,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_left_commute
thf(fact_501_Int__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_502_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_503_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_504_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_505_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_506_Int__empty__left,axiom,
! [B: set_real] :
( ( inf_inf_set_real @ bot_bot_set_real @ B )
= bot_bot_set_real ) ).
% Int_empty_left
thf(fact_507_Int__empty__left,axiom,
! [B: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B )
= bot_bot_set_set_a ) ).
% Int_empty_left
thf(fact_508_disjoint__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_509_disjoint__iff,axiom,
! [A2: set_real,B: set_real] :
( ( ( inf_inf_set_real @ A2 @ B )
= bot_bot_set_real )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ~ ( member_real @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_510_disjoint__iff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ~ ( member_set_a @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_511_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] : ( inf_inf_set_a @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_512_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_513_ex__in__conv,axiom,
! [A2: set_real] :
( ( ? [X2: real] : ( member_real @ X2 @ A2 ) )
= ( A2 != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_514_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_515_Int__emptyI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_516_Int__emptyI,axiom,
! [A2: set_real,B: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ~ ( member_real @ X3 @ B ) )
=> ( ( inf_inf_set_real @ A2 @ B )
= bot_bot_set_real ) ) ).
% Int_emptyI
thf(fact_517_Int__emptyI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ~ ( member_set_a @ X3 @ B ) )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_518_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_519_Int__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_assoc
thf(fact_520_equals0I,axiom,
! [A2: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_521_equals0I,axiom,
! [A2: set_real] :
( ! [Y2: real] :
~ ( member_real @ Y2 @ A2 )
=> ( A2 = bot_bot_set_real ) ) ).
% equals0I
thf(fact_522_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_523_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_524_equals0D,axiom,
! [A2: set_real,A: real] :
( ( A2 = bot_bot_set_real )
=> ~ ( member_real @ A @ A2 ) ) ).
% equals0D
thf(fact_525_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_526_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_527_emptyE,axiom,
! [A: real] :
~ ( member_real @ A @ bot_bot_set_real ) ).
% emptyE
thf(fact_528_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_529_IntD2,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B ) )
=> ( member_real @ C2 @ B ) ) ).
% IntD2
thf(fact_530_IntD2,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C2 @ B ) ) ).
% IntD2
thf(fact_531_IntD2,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ B ) ) ).
% IntD2
thf(fact_532_IntD1,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B ) )
=> ( member_real @ C2 @ A2 ) ) ).
% IntD1
thf(fact_533_IntD1,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_534_IntD1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_535_IntE,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A2 @ B ) )
=> ~ ( ( member_real @ C2 @ A2 )
=> ~ ( member_real @ C2 @ B ) ) ) ).
% IntE
thf(fact_536_IntE,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ~ ( ( member_set_a @ C2 @ A2 )
=> ~ ( member_set_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_537_IntE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ~ ( member_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_538_Int__Collect__mono,axiom,
! [A2: set_real,B: set_real,P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_real @ ( inf_inf_set_real @ A2 @ ( collect_real @ P ) ) @ ( inf_inf_set_real @ B @ ( collect_real @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_539_Int__Collect__mono,axiom,
! [A2: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_540_Int__Collect__mono,axiom,
! [A2: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_541_Int__greatest,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_542_Int__absorb2,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_543_Int__absorb1,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_544_Int__lower2,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_545_Int__lower1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_546_Int__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_547_Int__insert__left,axiom,
! [A: real,C: set_real,B: set_real] :
( ( ( member_real @ A @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A @ B ) @ C )
= ( insert_real @ A @ ( inf_inf_set_real @ B @ C ) ) ) )
& ( ~ ( member_real @ A @ C )
=> ( ( inf_inf_set_real @ ( insert_real @ A @ B ) @ C )
= ( inf_inf_set_real @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_548_Int__insert__left,axiom,
! [A: set_a,C: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B @ C ) ) ) )
& ( ~ ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C )
= ( inf_inf_set_set_a @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_549_Int__insert__left,axiom,
! [A: a,C: set_a,B: set_a] :
( ( ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) )
& ( ~ ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_550_Int__insert__right,axiom,
! [A: real,A2: set_real,B: set_real] :
( ( ( member_real @ A @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B ) )
= ( insert_real @ A @ ( inf_inf_set_real @ A2 @ B ) ) ) )
& ( ~ ( member_real @ A @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B ) )
= ( inf_inf_set_real @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_551_Int__insert__right,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_552_Int__insert__right,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_553_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_554_infinite__imp__nonempty,axiom,
! [S: set_real] :
( ~ ( finite_finite_real @ S )
=> ( S != bot_bot_set_real ) ) ).
% infinite_imp_nonempty
thf(fact_555_infinite__imp__nonempty,axiom,
! [S: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ( S != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_556_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_557_finite_OemptyI,axiom,
finite_finite_real @ bot_bot_set_real ).
% finite.emptyI
thf(fact_558_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_559_singletonD,axiom,
! [B4: a,A: a] :
( ( member_a @ B4 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B4 = A ) ) ).
% singletonD
thf(fact_560_singletonD,axiom,
! [B4: real,A: real] :
( ( member_real @ B4 @ ( insert_real @ A @ bot_bot_set_real ) )
=> ( B4 = A ) ) ).
% singletonD
thf(fact_561_singletonD,axiom,
! [B4: set_a,A: set_a] :
( ( member_set_a @ B4 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B4 = A ) ) ).
% singletonD
thf(fact_562_singleton__iff,axiom,
! [B4: a,A: a] :
( ( member_a @ B4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B4 = A ) ) ).
% singleton_iff
thf(fact_563_singleton__iff,axiom,
! [B4: real,A: real] :
( ( member_real @ B4 @ ( insert_real @ A @ bot_bot_set_real ) )
= ( B4 = A ) ) ).
% singleton_iff
thf(fact_564_singleton__iff,axiom,
! [B4: set_a,A: set_a] :
( ( member_set_a @ B4 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B4 = A ) ) ).
% singleton_iff
thf(fact_565_doubleton__eq__iff,axiom,
! [A: a,B4: a,C2: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B4 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C2 )
& ( B4 = D ) )
| ( ( A = D )
& ( B4 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_566_doubleton__eq__iff,axiom,
! [A: real,B4: real,C2: real,D: real] :
( ( ( insert_real @ A @ ( insert_real @ B4 @ bot_bot_set_real ) )
= ( insert_real @ C2 @ ( insert_real @ D @ bot_bot_set_real ) ) )
= ( ( ( A = C2 )
& ( B4 = D ) )
| ( ( A = D )
& ( B4 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_567_doubleton__eq__iff,axiom,
! [A: set_a,B4: set_a,C2: set_a,D: set_a] :
( ( ( insert_set_a @ A @ ( insert_set_a @ B4 @ bot_bot_set_set_a ) )
= ( insert_set_a @ C2 @ ( insert_set_a @ D @ bot_bot_set_set_a ) ) )
= ( ( ( A = C2 )
& ( B4 = D ) )
| ( ( A = D )
& ( B4 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_568_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_569_insert__not__empty,axiom,
! [A: real,A2: set_real] :
( ( insert_real @ A @ A2 )
!= bot_bot_set_real ) ).
% insert_not_empty
thf(fact_570_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_571_singleton__inject,axiom,
! [A: a,B4: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B4 @ bot_bot_set_a ) )
=> ( A = B4 ) ) ).
% singleton_inject
thf(fact_572_singleton__inject,axiom,
! [A: real,B4: real] :
( ( ( insert_real @ A @ bot_bot_set_real )
= ( insert_real @ B4 @ bot_bot_set_real ) )
=> ( A = B4 ) ) ).
% singleton_inject
thf(fact_573_singleton__inject,axiom,
! [A: set_a,B4: set_a] :
( ( ( insert_set_a @ A @ bot_bot_set_set_a )
= ( insert_set_a @ B4 @ bot_bot_set_set_a ) )
=> ( A = B4 ) ) ).
% singleton_inject
thf(fact_574_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_575_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_576_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_577_Diff__insert2,axiom,
! [A2: set_real,A: real,B: set_real] :
( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B ) )
= ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B ) ) ).
% Diff_insert2
thf(fact_578_Diff__insert2,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_579_Diff__insert2,axiom,
! [A2: set_set_a,A: set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_580_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_581_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_582_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_583_Diff__insert,axiom,
! [A2: set_real,A: real,B: set_real] :
( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B ) )
= ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).
% Diff_insert
thf(fact_584_Diff__insert,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_585_Diff__insert,axiom,
! [A2: set_set_a,A: set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ).
% Diff_insert
thf(fact_586_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_587_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_588_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_589_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_590_mult__not__zero,axiom,
! [A: real,B4: real] :
( ( ( times_times_real @ A @ B4 )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B4 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_591_mult__not__zero,axiom,
! [A: nat,B4: nat] :
( ( ( times_times_nat @ A @ B4 )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B4 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_592_divisors__zero,axiom,
! [A: real,B4: real] :
( ( ( times_times_real @ A @ B4 )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B4 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_593_divisors__zero,axiom,
! [A: nat,B4: nat] :
( ( ( times_times_nat @ A @ B4 )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B4 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_594_no__zero__divisors,axiom,
! [A: real,B4: real] :
( ( A != zero_zero_real )
=> ( ( B4 != zero_zero_real )
=> ( ( times_times_real @ A @ B4 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_595_no__zero__divisors,axiom,
! [A: nat,B4: nat] :
( ( A != zero_zero_nat )
=> ( ( B4 != zero_zero_nat )
=> ( ( times_times_nat @ A @ B4 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_596_mult__left__cancel,axiom,
! [C2: real,A: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B4 ) )
= ( A = B4 ) ) ) ).
% mult_left_cancel
thf(fact_597_mult__left__cancel,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B4 ) )
= ( A = B4 ) ) ) ).
% mult_left_cancel
thf(fact_598_mult__right__cancel,axiom,
! [C2: real,A: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B4 @ C2 ) )
= ( A = B4 ) ) ) ).
% mult_right_cancel
thf(fact_599_mult__right__cancel,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B4 @ C2 ) )
= ( A = B4 ) ) ) ).
% mult_right_cancel
thf(fact_600_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_601_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_602_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_603_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_604_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_605_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
= ( ^ [A4: real,B3: real] :
( ( minus_minus_real @ A4 @ B3 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_606_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_607_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_608_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_609_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_610_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_611_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_612_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_613_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_614_card__Diff__subset__Int,axiom,
! [A2: set_real,B: set_real] :
( ( finite_finite_real @ ( inf_inf_set_real @ A2 @ B ) )
=> ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ ( inf_inf_set_real @ A2 @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_615_card__Diff__subset__Int,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ B ) )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_616_card__Diff__subset__Int,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_617_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_618_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_619_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_620_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_621_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_622_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_623_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A7: set_a] :
( ~ ( finite_finite_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X3 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_624_infinite__finite__induct,axiom,
! [P: set_real > $o,A2: set_real] :
( ! [A7: set_real] :
( ~ ( finite_finite_real @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,F2: set_real] :
( ( finite_finite_real @ F2 )
=> ( ~ ( member_real @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_real @ X3 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_625_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A2: set_set_a] :
( ! [A7: set_set_a] :
( ~ ( finite_finite_set_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ~ ( member_set_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ X3 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_626_finite__empty__induct,axiom,
! [A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: real,A7: set_real] :
( ( finite_finite_real @ A7 )
=> ( ( member_real @ A3 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ A3 @ bot_bot_set_real ) ) ) ) ) )
=> ( P @ bot_bot_set_real ) ) ) ) ).
% finite_empty_induct
thf(fact_627_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: a,A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( member_a @ A3 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_628_finite__empty__induct,axiom,
! [A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: set_a,A7: set_set_a] :
( ( finite_finite_set_a @ A7 )
=> ( ( member_set_a @ A3 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ A3 @ bot_bot_set_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_629_infinite__coinduct,axiom,
! [X4: set_real > $o,A2: set_real] :
( ( X4 @ A2 )
=> ( ! [A7: set_real] :
( ( X4 @ A7 )
=> ? [X5: real] :
( ( member_real @ X5 @ A7 )
& ( ( X4 @ ( minus_minus_set_real @ A7 @ ( insert_real @ X5 @ bot_bot_set_real ) ) )
| ~ ( finite_finite_real @ ( minus_minus_set_real @ A7 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) ) ) )
=> ~ ( finite_finite_real @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_630_infinite__coinduct,axiom,
! [X4: set_a > $o,A2: set_a] :
( ( X4 @ A2 )
=> ( ! [A7: set_a] :
( ( X4 @ A7 )
=> ? [X5: a] :
( ( member_a @ X5 @ A7 )
& ( ( X4 @ ( minus_minus_set_a @ A7 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A7 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_631_infinite__coinduct,axiom,
! [X4: set_set_a > $o,A2: set_set_a] :
( ( X4 @ A2 )
=> ( ! [A7: set_set_a] :
( ( X4 @ A7 )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A7 )
& ( ( X4 @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) )
| ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) ) ) )
=> ~ ( finite_finite_set_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_632_finite__ne__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( F != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_633_finite__ne__induct,axiom,
! [F: set_real,P: set_real > $o] :
( ( finite_finite_real @ F )
=> ( ( F != bot_bot_set_real )
=> ( ! [X3: real] : ( P @ ( insert_real @ X3 @ bot_bot_set_real ) )
=> ( ! [X3: real,F2: set_real] :
( ( finite_finite_real @ F2 )
=> ( ( F2 != bot_bot_set_real )
=> ( ~ ( member_real @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_real @ X3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_634_finite__ne__induct,axiom,
! [F: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( F != bot_bot_set_set_a )
=> ( ! [X3: set_a] : ( P @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
=> ( ! [X3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( F2 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ X3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_635_infinite__remove,axiom,
! [S: set_real,A: real] :
( ~ ( finite_finite_real @ S )
=> ~ ( finite_finite_real @ ( minus_minus_set_real @ S @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% infinite_remove
thf(fact_636_infinite__remove,axiom,
! [S: set_a,A: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_637_infinite__remove,axiom,
! [S: set_set_a,A: set_a] :
( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% infinite_remove
thf(fact_638_finite__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X3 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_639_finite__induct,axiom,
! [F: set_real,P: set_real > $o] :
( ( finite_finite_real @ F )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,F2: set_real] :
( ( finite_finite_real @ F2 )
=> ( ~ ( member_real @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_real @ X3 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_640_finite__induct,axiom,
! [F: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ~ ( member_set_a @ X3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ X3 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_641_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A6: set_a,B3: a] :
( ( A4
= ( insert_a @ B3 @ A6 ) )
& ( finite_finite_a @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_642_finite_Osimps,axiom,
( finite_finite_real
= ( ^ [A4: set_real] :
( ( A4 = bot_bot_set_real )
| ? [A6: set_real,B3: real] :
( ( A4
= ( insert_real @ B3 @ A6 ) )
& ( finite_finite_real @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_643_finite_Osimps,axiom,
( finite_finite_set_a
= ( ^ [A4: set_set_a] :
( ( A4 = bot_bot_set_set_a )
| ? [A6: set_set_a,B3: set_a] :
( ( A4
= ( insert_set_a @ B3 @ A6 ) )
& ( finite_finite_set_a @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_644_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A7: set_a] :
( ? [A3: a] :
( A
= ( insert_a @ A3 @ A7 ) )
=> ~ ( finite_finite_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_645_finite_Ocases,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ~ ! [A7: set_real] :
( ? [A3: real] :
( A
= ( insert_real @ A3 @ A7 ) )
=> ~ ( finite_finite_real @ A7 ) ) ) ) ).
% finite.cases
thf(fact_646_finite_Ocases,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ~ ! [A7: set_set_a] :
( ? [A3: set_a] :
( A
= ( insert_set_a @ A3 @ A7 ) )
=> ~ ( finite_finite_set_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_647_subset__singleton__iff,axiom,
! [X4: set_real,A: real] :
( ( ord_less_eq_set_real @ X4 @ ( insert_real @ A @ bot_bot_set_real ) )
= ( ( X4 = bot_bot_set_real )
| ( X4
= ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% subset_singleton_iff
thf(fact_648_subset__singleton__iff,axiom,
! [X4: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( ( X4 = bot_bot_set_set_a )
| ( X4
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_649_subset__singleton__iff,axiom,
! [X4: set_a,A: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_650_Diff__single__insert,axiom,
! [A2: set_real,X: real,B: set_real] :
( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B )
=> ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_651_Diff__single__insert,axiom,
! [A2: set_set_a,X: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_652_Diff__single__insert,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_653_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_654_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_655_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_656_subset__insert__iff,axiom,
! [A2: set_real,X: real,B: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B ) )
= ( ( ( member_real @ X @ A2 )
=> ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B ) )
& ( ~ ( member_real @ X @ A2 )
=> ( ord_less_eq_set_real @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_657_subset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B ) )
= ( ( ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_658_subset__insert__iff,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_659_finite__subset__induct_H,axiom,
! [F: set_real,A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F )
=> ( ( ord_less_eq_set_real @ F @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A3: real,F2: set_real] :
( ( finite_finite_real @ F2 )
=> ( ( member_real @ A3 @ A2 )
=> ( ( ord_less_eq_set_real @ F2 @ A2 )
=> ( ~ ( member_real @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_real @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_660_finite__subset__induct_H,axiom,
! [F: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( member_set_a @ A3 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ~ ( member_set_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_661_finite__subset__induct_H,axiom,
! [F: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A3 @ A2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ~ ( member_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_662_finite__subset__induct,axiom,
! [F: set_real,A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F )
=> ( ( ord_less_eq_set_real @ F @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A3: real,F2: set_real] :
( ( finite_finite_real @ F2 )
=> ( ( member_real @ A3 @ A2 )
=> ( ~ ( member_real @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_real @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_663_finite__subset__induct,axiom,
! [F: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( member_set_a @ A3 @ A2 )
=> ( ~ ( member_set_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_664_finite__subset__induct,axiom,
! [F: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A3 @ A2 )
=> ( ~ ( member_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_665_finite__remove__induct,axiom,
! [B: set_real,P: set_real > $o] :
( ( finite_finite_real @ B )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A7: set_real] :
( ( finite_finite_real @ A7 )
=> ( ( A7 != bot_bot_set_real )
=> ( ( ord_less_eq_set_real @ A7 @ B )
=> ( ! [X5: real] :
( ( member_real @ X5 @ A7 )
=> ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_666_finite__remove__induct,axiom,
! [B: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ B )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A7: set_set_a] :
( ( finite_finite_set_a @ A7 )
=> ( ( A7 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A7 @ B )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A7 )
=> ( P @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_667_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( A7 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A7 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_668_remove__induct,axiom,
! [P: set_real > $o,B: set_real] :
( ( P @ bot_bot_set_real )
=> ( ( ~ ( finite_finite_real @ B )
=> ( P @ B ) )
=> ( ! [A7: set_real] :
( ( finite_finite_real @ A7 )
=> ( ( A7 != bot_bot_set_real )
=> ( ( ord_less_eq_set_real @ A7 @ B )
=> ( ! [X5: real] :
( ( member_real @ X5 @ A7 )
=> ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_669_remove__induct,axiom,
! [P: set_set_a > $o,B: set_set_a] :
( ( P @ bot_bot_set_set_a )
=> ( ( ~ ( finite_finite_set_a @ B )
=> ( P @ B ) )
=> ( ! [A7: set_set_a] :
( ( finite_finite_set_a @ A7 )
=> ( ( A7 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A7 @ B )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A7 )
=> ( P @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_670_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( A7 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A7 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_671_mult__mono,axiom,
! [A: real,B4: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_672_mult__mono,axiom,
! [A: nat,B4: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_673_mult__mono_H,axiom,
! [A: real,B4: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_674_mult__mono_H,axiom,
! [A: nat,B4: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_675_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_676_split__mult__pos__le,axiom,
! [A: real,B4: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B4 ) ) ) ).
% split_mult_pos_le
thf(fact_677_mult__left__mono__neg,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_678_mult__nonpos__nonpos,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B4 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B4 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_679_mult__left__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) ) ) ) ).
% mult_left_mono
thf(fact_680_mult__left__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% mult_left_mono
thf(fact_681_mult__right__mono__neg,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_682_mult__right__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_683_mult__right__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_684_mult__le__0__iff,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B4 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) ) ) ) ).
% mult_le_0_iff
thf(fact_685_split__mult__neg__le,axiom,
! [A: real,B4: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B4 ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_686_split__mult__neg__le,axiom,
! [A: nat,B4: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B4 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B4 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_687_mult__nonneg__nonneg,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B4 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_688_mult__nonneg__nonneg,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B4 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_689_mult__nonneg__nonpos,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B4 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B4 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_690_mult__nonneg__nonpos,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B4 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_691_mult__nonpos__nonneg,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B4 ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_692_mult__nonpos__nonneg,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_693_mult__nonneg__nonpos2,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B4 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B4 @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_694_mult__nonneg__nonpos2,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B4 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B4 @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_695_zero__le__mult__iff,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B4 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_696_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_697_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_698_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_699_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_700_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_701_add__decreasing,axiom,
! [A: real,C2: real,B4: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C2 @ B4 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ B4 ) ) ) ).
% add_decreasing
thf(fact_702_add__decreasing,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B4 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B4 ) ) ) ).
% add_decreasing
thf(fact_703_add__increasing,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( ord_less_eq_real @ B4 @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_704_add__increasing,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ B4 @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_705_add__decreasing2,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B4 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ B4 ) ) ) ).
% add_decreasing2
thf(fact_706_add__decreasing2,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B4 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B4 ) ) ) ).
% add_decreasing2
thf(fact_707_add__increasing2,axiom,
! [C2: real,B4: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ B4 @ A )
=> ( ord_less_eq_real @ B4 @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_708_add__increasing2,axiom,
! [C2: nat,B4: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ A )
=> ( ord_less_eq_nat @ B4 @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_709_add__nonneg__nonneg,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B4 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_710_add__nonneg__nonneg,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B4 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_711_add__nonpos__nonpos,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B4 @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B4 ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_712_add__nonpos__nonpos,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B4 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B4 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_713_add__nonneg__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_714_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_715_add__nonpos__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_716_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_717_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B3 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_718_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_719_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_720_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_721_sum__squares__ge__zero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_722_Diff__infinite__finite,axiom,
! [T: set_real,S: set_real] :
( ( finite_finite_real @ T )
=> ( ~ ( finite_finite_real @ S )
=> ~ ( finite_finite_real @ ( minus_minus_set_real @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_723_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_724_Diff__infinite__finite,axiom,
! [T: set_set_a,S: set_set_a] :
( ( finite_finite_set_a @ T )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_725_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_726_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_727_subset__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_728_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_729_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_730_double__diff,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ( minus_5736297505244876581_set_a @ B @ ( minus_5736297505244876581_set_a @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_731_double__diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_732_Diff__subset,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_733_Diff__subset,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_734_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [T2: real] :
( ( member_real @ T2 @ A6 )
=> ( member_real @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_735_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A6: set_set_a,B6: set_set_a] :
! [T2: set_a] :
( ( member_set_a @ T2 @ A6 )
=> ( member_set_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_736_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A6 )
=> ( member_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_737_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_738_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_739_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [X2: real] :
( ( member_real @ X2 @ A6 )
=> ( member_real @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_740_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A6: set_set_a,B6: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A6 )
=> ( member_set_a @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_741_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( member_a @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_742_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_743_Diff__mono,axiom,
! [A2: set_set_a,C: set_set_a,D2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ D2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ ( minus_5736297505244876581_set_a @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_744_Diff__mono,axiom,
! [A2: set_a,C: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_745_subsetD,axiom,
! [A2: set_real,B: set_real,C2: real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ B ) ) ) ).
% subsetD
thf(fact_746_subsetD,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_747_subsetD,axiom,
! [A2: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_748_in__mono,axiom,
! [A2: set_real,B: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B ) ) ) ).
% in_mono
thf(fact_749_in__mono,axiom,
! [A2: set_set_a,B: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_750_in__mono,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_751_insertE,axiom,
! [A: a,B4: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B4 @ A2 ) )
=> ( ( A != B4 )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_752_insertE,axiom,
! [A: real,B4: real,A2: set_real] :
( ( member_real @ A @ ( insert_real @ B4 @ A2 ) )
=> ( ( A != B4 )
=> ( member_real @ A @ A2 ) ) ) ).
% insertE
thf(fact_753_insertE,axiom,
! [A: set_a,B4: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B4 @ A2 ) )
=> ( ( A != B4 )
=> ( member_set_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_754_insertI1,axiom,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).
% insertI1
thf(fact_755_insertI1,axiom,
! [A: real,B: set_real] : ( member_real @ A @ ( insert_real @ A @ B ) ) ).
% insertI1
thf(fact_756_insertI1,axiom,
! [A: set_a,B: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B ) ) ).
% insertI1
thf(fact_757_insertI2,axiom,
! [A: a,B: set_a,B4: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ B4 @ B ) ) ) ).
% insertI2
thf(fact_758_insertI2,axiom,
! [A: real,B: set_real,B4: real] :
( ( member_real @ A @ B )
=> ( member_real @ A @ ( insert_real @ B4 @ B ) ) ) ).
% insertI2
thf(fact_759_insertI2,axiom,
! [A: set_a,B: set_set_a,B4: set_a] :
( ( member_set_a @ A @ B )
=> ( member_set_a @ A @ ( insert_set_a @ B4 @ B ) ) ) ).
% insertI2
thf(fact_760_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_761_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_762_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_763_insert__ident,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_764_insert__ident,axiom,
! [X: real,A2: set_real,B: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ~ ( member_real @ X @ B )
=> ( ( ( insert_real @ X @ A2 )
= ( insert_real @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_765_insert__ident,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ~ ( member_set_a @ X @ B )
=> ( ( ( insert_set_a @ X @ A2 )
= ( insert_set_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_766_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_767_insert__absorb,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ( ( insert_real @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_768_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_769_insert__eq__iff,axiom,
! [A: a,A2: set_a,B4: a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B4 @ B )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B4 @ B ) )
= ( ( ( A = B4 )
=> ( A2 = B ) )
& ( ( A != B4 )
=> ? [C5: set_a] :
( ( A2
= ( insert_a @ B4 @ C5 ) )
& ~ ( member_a @ B4 @ C5 )
& ( B
= ( insert_a @ A @ C5 ) )
& ~ ( member_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_770_insert__eq__iff,axiom,
! [A: real,A2: set_real,B4: real,B: set_real] :
( ~ ( member_real @ A @ A2 )
=> ( ~ ( member_real @ B4 @ B )
=> ( ( ( insert_real @ A @ A2 )
= ( insert_real @ B4 @ B ) )
= ( ( ( A = B4 )
=> ( A2 = B ) )
& ( ( A != B4 )
=> ? [C5: set_real] :
( ( A2
= ( insert_real @ B4 @ C5 ) )
& ~ ( member_real @ B4 @ C5 )
& ( B
= ( insert_real @ A @ C5 ) )
& ~ ( member_real @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_771_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B4: set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B4 @ B )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B4 @ B ) )
= ( ( ( A = B4 )
=> ( A2 = B ) )
& ( ( A != B4 )
=> ? [C5: set_set_a] :
( ( A2
= ( insert_set_a @ B4 @ C5 ) )
& ~ ( member_set_a @ B4 @ C5 )
& ( B
= ( insert_set_a @ A @ C5 ) )
& ~ ( member_set_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_772_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_773_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_774_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_775_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_776_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_777_insert__Diff__if,axiom,
! [X: real,B: set_real,A2: set_real] :
( ( ( member_real @ X @ B )
=> ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B )
= ( minus_minus_set_real @ A2 @ B ) ) )
& ( ~ ( member_real @ X @ B )
=> ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B )
= ( insert_real @ X @ ( minus_minus_set_real @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_778_insert__Diff__if,axiom,
! [X: a,B: set_a,A2: set_a] :
( ( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) )
& ( ~ ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_779_insert__Diff__if,axiom,
! [X: set_a,B: set_set_a,A2: set_set_a] :
( ( ( member_set_a @ X @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B )
= ( minus_5736297505244876581_set_a @ A2 @ B ) ) )
& ( ~ ( member_set_a @ X @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B )
= ( insert_set_a @ X @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_780_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_781_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_782_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_783_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_784_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_785_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_786_rev__finite__subset,axiom,
! [B: set_real,A2: set_real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A2 @ B )
=> ( finite_finite_real @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_787_rev__finite__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_788_infinite__super,axiom,
! [S: set_real,T: set_real] :
( ( ord_less_eq_set_real @ S @ T )
=> ( ~ ( finite_finite_real @ S )
=> ~ ( finite_finite_real @ T ) ) ) ).
% infinite_super
thf(fact_789_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_790_finite__subset,axiom,
! [A2: set_real,B: set_real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( finite_finite_real @ B )
=> ( finite_finite_real @ A2 ) ) ) ).
% finite_subset
thf(fact_791_finite__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_792_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_793_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_794_finite_OinsertI,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( finite_finite_real @ ( insert_real @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_795_insert__mono,axiom,
! [C: set_set_a,D2: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ C @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A @ C ) @ ( insert_set_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_796_insert__mono,axiom,
! [C: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_797_subset__insert,axiom,
! [X: real,A2: set_real,B: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B ) )
= ( ord_less_eq_set_real @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_798_subset__insert,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_799_subset__insert,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_800_subset__insertI,axiom,
! [B: set_set_a,A: set_a] : ( ord_le3724670747650509150_set_a @ B @ ( insert_set_a @ A @ B ) ) ).
% subset_insertI
thf(fact_801_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).
% subset_insertI
thf(fact_802_subset__insertI2,axiom,
! [A2: set_set_a,B: set_set_a,B4: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B4 @ B ) ) ) ).
% subset_insertI2
thf(fact_803_subset__insertI2,axiom,
! [A2: set_a,B: set_a,B4: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ B ) ) ) ).
% subset_insertI2
thf(fact_804_subset__Diff__insert,axiom,
! [A2: set_real,B: set_real,X: real,C: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B @ ( insert_real @ X @ C ) ) )
= ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B @ C ) )
& ~ ( member_real @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_805_subset__Diff__insert,axiom,
! [A2: set_set_a,B: set_set_a,X: set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B @ ( insert_set_a @ X @ C ) ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B @ C ) )
& ~ ( member_set_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_806_subset__Diff__insert,axiom,
! [A2: set_a,B: set_a,X: a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ ( insert_a @ X @ C ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ C ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_807_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_808_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_809_card__Diff__subset,axiom,
! [B: set_real,A2: set_real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ B @ A2 )
=> ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_810_card__Diff__subset,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_811_card__Diff__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_812_card__subset__eq,axiom,
! [B: set_real,A2: set_real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( ( finite_card_real @ A2 )
= ( finite_card_real @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_813_card__subset__eq,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_814_diff__card__le__card__Diff,axiom,
! [B: set_real,A2: set_real] :
( ( finite_finite_real @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B ) ) @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_815_diff__card__le__card__Diff,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_816_diff__card__le__card__Diff,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_817_card__le__sym__Diff,axiom,
! [A2: set_real,B: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_real @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ B ) ) @ ( finite_card_real @ ( minus_minus_set_real @ B @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_818_card__le__sym__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_819_card__le__sym__Diff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_820_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_821_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_822_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_real,C: nat] :
( ! [G2: set_real] :
( ( ord_less_eq_set_real @ G2 @ F )
=> ( ( finite_finite_real @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_real @ G2 ) @ C ) ) )
=> ( ( finite_finite_real @ F )
& ( ord_less_eq_nat @ ( finite_card_real @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_823_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_a,C: nat] :
( ! [G2: set_a] :
( ( ord_less_eq_set_a @ G2 @ F )
=> ( ( finite_finite_a @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G2 ) @ C ) ) )
=> ( ( finite_finite_a @ F )
& ( ord_less_eq_nat @ ( finite_card_a @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_824_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_real] :
( ( ord_less_eq_nat @ N @ ( finite_card_real @ S ) )
=> ~ ! [T3: set_real] :
( ( ord_less_eq_set_real @ T3 @ S )
=> ( ( ( finite_card_real @ T3 )
= N )
=> ~ ( finite_finite_real @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_825_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S )
=> ( ( ( finite_card_a @ T3 )
= N )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_826_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_827_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_828_card__seteq,axiom,
! [B: set_real,A2: set_real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_real @ B ) @ ( finite_card_real @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_829_card__seteq,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_830_card__mono,axiom,
! [B: set_real,A2: set_real] :
( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B ) ) ) ) ).
% card_mono
thf(fact_831_card__mono,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ).
% card_mono
thf(fact_832__092_060open_062sumset_AA_A_Isumset_AB_A_Iinsert_Ax_AC_J_J_A_092_060subseteq_062_Asumset_AA_A_Isumset_AB_AC_J_A_092_060union_062_A_Isumset_AA_A_Isumset_AB_A_123x_125_J_A_N_Asumset_AA_H_A_Isumset_AB_A_123x_125_J_J_092_060close_062,axiom,
ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ( insert_a @ x @ ca ) ) ) @ ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ca ) ) @ ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ( insert_a @ x @ bot_bot_set_a ) ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ a2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ b @ ( insert_a @ x @ bot_bot_set_a ) ) ) ) ) ).
% \<open>sumset A (sumset B (insert x C)) \<subseteq> sumset A (sumset B C) \<union> (sumset A (sumset B {x}) - sumset A' (sumset B {x}))\<close>
thf(fact_833_K__le,axiom,
! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ a3 )
=> ( ( A5 != bot_bot_set_a )
=> ( ord_less_eq_real @ k @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A5 ) ) ) ) ) ) ).
% K_le
thf(fact_834__C1_C,axiom,
( ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( insert_a @ x @ ca ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ca ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( minus_minus_set_a @ a3 @ a2 ) @ ( insert_a @ x @ bot_bot_set_a ) ) ) ) ).
% "1"
thf(fact_835_sum__squares__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_836_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_837_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_838_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_839_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_840_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_841_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_842_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_843_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_844_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_845_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_846_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_847_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_848_KS_I2_J,axiom,
ks != bot_bot_set_real ).
% KS(2)
thf(fact_849__092_060open_062K_A_092_060in_062_AKS_092_060close_062,axiom,
member_real @ k @ ks ).
% \<open>K \<in> KS\<close>
thf(fact_850__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_851_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_852_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_853_inf_Oleft__idem,axiom,
! [A: set_a,B4: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B4 ) )
= ( inf_inf_set_a @ A @ B4 ) ) ).
% inf.left_idem
thf(fact_854_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_855_inf_Oright__idem,axiom,
! [A: set_a,B4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B4 ) @ B4 )
= ( inf_inf_set_a @ A @ B4 ) ) ).
% inf.right_idem
thf(fact_856_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_857_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_858_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_859_sup_Oleft__idem,axiom,
! [A: set_a,B4: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B4 ) )
= ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.left_idem
thf(fact_860_sup__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_861_sup_Oright__idem,axiom,
! [A: set_a,B4: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B4 ) @ B4 )
= ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.right_idem
thf(fact_862_Un__iff,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( sup_sup_set_real @ A2 @ B ) )
= ( ( member_real @ C2 @ A2 )
| ( member_real @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_863_Un__iff,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( ( member_set_a @ C2 @ A2 )
| ( member_set_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_864_Un__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
| ( member_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_865_UnCI,axiom,
! [C2: real,B: set_real,A2: set_real] :
( ( ~ ( member_real @ C2 @ B )
=> ( member_real @ C2 @ A2 ) )
=> ( member_real @ C2 @ ( sup_sup_set_real @ A2 @ B ) ) ) ).
% UnCI
thf(fact_866_UnCI,axiom,
! [C2: set_a,B: set_set_a,A2: set_set_a] :
( ( ~ ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ A2 ) )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_867_UnCI,axiom,
! [C2: a,B: set_a,A2: set_a] :
( ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ A2 ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_868_sumset__subset__Un2,axiom,
! [A2: set_a,B: set_a,B5: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B @ B5 ) )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B5 ) ) ) ).
% sumset_subset_Un2
thf(fact_869_sumset__subset__Un1,axiom,
! [A2: set_a,A5: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ A5 ) @ B )
= ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A5 @ B ) ) ) ).
% sumset_subset_Un1
thf(fact_870_sumset__subset__Un_I1_J,axiom,
! [A2: set_a,B: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sumset_subset_Un(1)
thf(fact_871_sumset__subset__Un_I2_J,axiom,
! [A2: set_a,B: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A2 @ C ) @ B ) ) ).
% sumset_subset_Un(2)
thf(fact_872_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_873_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_874_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_875_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_876_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_877_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_878_inf_Obounded__iff,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B4 @ C2 ) )
= ( ( ord_less_eq_real @ A @ B4 )
& ( ord_less_eq_real @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_879_inf_Obounded__iff,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) )
= ( ( ord_less_eq_set_a @ A @ B4 )
& ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_880_inf_Obounded__iff,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B4 )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_881_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_882_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_883_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_884_le__sup__iff,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_real @ X @ Z2 )
& ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_885_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_set_a @ X @ Z2 )
& ( ord_less_eq_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_886_le__sup__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_887_sup_Obounded__iff,axiom,
! [B4: real,C2: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B4 @ C2 ) @ A )
= ( ( ord_less_eq_real @ B4 @ A )
& ( ord_less_eq_real @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_888_sup_Obounded__iff,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A )
= ( ( ord_less_eq_set_a @ B4 @ A )
& ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_889_sup_Obounded__iff,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A )
= ( ( ord_less_eq_nat @ B4 @ A )
& ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_890_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_891_sup__bot_Oright__neutral,axiom,
! [A: set_real] :
( ( sup_sup_set_real @ A @ bot_bot_set_real )
= A ) ).
% sup_bot.right_neutral
thf(fact_892_sup__bot_Oright__neutral,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_893_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B4: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B4 ) )
= ( ( A = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_894_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_real,B4: set_real] :
( ( bot_bot_set_real
= ( sup_sup_set_real @ A @ B4 ) )
= ( ( A = bot_bot_set_real )
& ( B4 = bot_bot_set_real ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_895_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_set_a,B4: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ A @ B4 ) )
= ( ( A = bot_bot_set_set_a )
& ( B4 = bot_bot_set_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_896_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_897_sup__bot_Oleft__neutral,axiom,
! [A: set_real] :
( ( sup_sup_set_real @ bot_bot_set_real @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_898_sup__bot_Oleft__neutral,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_899_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B4: set_a] :
( ( ( sup_sup_set_a @ A @ B4 )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_900_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_real,B4: set_real] :
( ( ( sup_sup_set_real @ A @ B4 )
= bot_bot_set_real )
= ( ( A = bot_bot_set_real )
& ( B4 = bot_bot_set_real ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_901_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_set_a,B4: set_set_a] :
( ( ( sup_sup_set_set_a @ A @ B4 )
= bot_bot_set_set_a )
= ( ( A = bot_bot_set_set_a )
& ( B4 = bot_bot_set_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_902_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_903_sup__eq__bot__iff,axiom,
! [X: set_real,Y: set_real] :
( ( ( sup_sup_set_real @ X @ Y )
= bot_bot_set_real )
= ( ( X = bot_bot_set_real )
& ( Y = bot_bot_set_real ) ) ) ).
% sup_eq_bot_iff
thf(fact_904_sup__eq__bot__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( sup_sup_set_set_a @ X @ Y )
= bot_bot_set_set_a )
= ( ( X = bot_bot_set_set_a )
& ( Y = bot_bot_set_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_905_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_906_bot__eq__sup__iff,axiom,
! [X: set_real,Y: set_real] :
( ( bot_bot_set_real
= ( sup_sup_set_real @ X @ Y ) )
= ( ( X = bot_bot_set_real )
& ( Y = bot_bot_set_real ) ) ) ).
% bot_eq_sup_iff
thf(fact_907_bot__eq__sup__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_set_a )
& ( Y = bot_bot_set_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_908_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_909_sup__bot__right,axiom,
! [X: set_real] :
( ( sup_sup_set_real @ X @ bot_bot_set_real )
= X ) ).
% sup_bot_right
thf(fact_910_sup__bot__right,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ bot_bot_set_set_a )
= X ) ).
% sup_bot_right
thf(fact_911_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_912_sup__bot__left,axiom,
! [X: set_real] :
( ( sup_sup_set_real @ bot_bot_set_real @ X )
= X ) ).
% sup_bot_left
thf(fact_913_sup__bot__left,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_914_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_915_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_916_Un__empty,axiom,
! [A2: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_917_Un__empty,axiom,
! [A2: set_real,B: set_real] :
( ( ( sup_sup_set_real @ A2 @ B )
= bot_bot_set_real )
= ( ( A2 = bot_bot_set_real )
& ( B = bot_bot_set_real ) ) ) ).
% Un_empty
thf(fact_918_Un__empty,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( sup_sup_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ( A2 = bot_bot_set_set_a )
& ( B = bot_bot_set_set_a ) ) ) ).
% Un_empty
thf(fact_919_finite__Un,axiom,
! [F: set_real,G: set_real] :
( ( finite_finite_real @ ( sup_sup_set_real @ F @ G ) )
= ( ( finite_finite_real @ F )
& ( finite_finite_real @ G ) ) ) ).
% finite_Un
thf(fact_920_finite__Un,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) )
= ( ( finite_finite_a @ F )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_921_Un__subset__iff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( ( ord_less_eq_set_a @ A2 @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_922_Un__insert__right,axiom,
! [A2: set_set_a,A: set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_923_Un__insert__right,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_924_Un__insert__left,axiom,
! [A: set_a,B: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ ( insert_set_a @ A @ B ) @ C )
= ( insert_set_a @ A @ ( sup_sup_set_set_a @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_925_Un__insert__left,axiom,
! [A: a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_926_Int__Un__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( sup_sup_set_a @ T @ ( inf_inf_set_a @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_927_Int__Un__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_928_Int__Un__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_929_Int__Un__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_930_Un__Int__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( inf_inf_set_a @ T @ ( sup_sup_set_a @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_931_Un__Int__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_932_Un__Int__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_933_Un__Int__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_934_Un__Diff__cancel2,axiom,
! [B: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B @ A2 ) @ A2 )
= ( sup_sup_set_a @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_935_Un__Diff__cancel2,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( sup_sup_set_set_a @ ( minus_5736297505244876581_set_a @ B @ A2 ) @ A2 )
= ( sup_sup_set_set_a @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_936_Un__Diff__cancel,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_937_Un__Diff__cancel,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B @ A2 ) )
= ( sup_sup_set_set_a @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_938_nonzero__mult__div__cancel__left,axiom,
! [A: real,B4: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B4 ) @ A )
= B4 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_939_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B4: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B4 ) @ A )
= B4 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_940_nonzero__mult__div__cancel__left,axiom,
! [A: int,B4: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B4 ) @ A )
= B4 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_941_nonzero__mult__div__cancel__right,axiom,
! [B4: real,A: real] :
( ( B4 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B4 ) @ B4 )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_942_nonzero__mult__div__cancel__right,axiom,
! [B4: nat,A: nat] :
( ( B4 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B4 ) @ B4 )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_943_nonzero__mult__div__cancel__right,axiom,
! [B4: int,A: int] :
( ( B4 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B4 ) @ B4 )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_944__092_060open_062real_A_Icard_A_Isumset_AA0_AB_J_J_A_P_Areal_A_Icard_AA0_J_A_092_060in_062_AKS_092_060close_062,axiom,
member_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a0 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a0 ) ) ) @ ks ).
% \<open>real (card (sumset A0 B)) / real (card A0) \<in> KS\<close>
thf(fact_945_K__def,axiom,
( k
= ( lattic3629708407755379051n_real @ ks ) ) ).
% K_def
thf(fact_946__092_060open_062A_A_092_060in_062_APow_AA0_A_N_A_123_123_125_125_092_060close_062,axiom,
member_set_a @ a3 @ ( 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_947_Un__left__commute,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_948_Un__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_949_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] : ( sup_sup_set_a @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_950_Un__absorb,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_951_Un__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_assoc
thf(fact_952_ball__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_953_bex__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: a] :
( ( member_a @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_954_UnI2,axiom,
! [C2: real,B: set_real,A2: set_real] :
( ( member_real @ C2 @ B )
=> ( member_real @ C2 @ ( sup_sup_set_real @ A2 @ B ) ) ) ).
% UnI2
thf(fact_955_UnI2,axiom,
! [C2: set_a,B: set_set_a,A2: set_set_a] :
( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_956_UnI2,axiom,
! [C2: a,B: set_a,A2: set_a] :
( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_957_UnI1,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ ( sup_sup_set_real @ A2 @ B ) ) ) ).
% UnI1
thf(fact_958_UnI1,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_959_UnI1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_960_UnE,axiom,
! [C2: real,A2: set_real,B: set_real] :
( ( member_real @ C2 @ ( sup_sup_set_real @ A2 @ B ) )
=> ( ~ ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ B ) ) ) ).
% UnE
thf(fact_961_UnE,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) )
=> ( ~ ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_962_UnE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( ~ ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_963_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z2 ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z2 @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_964_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z2 @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_965_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_966_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_967_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_968_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_real] :
( ( sup_sup_set_real @ X @ bot_bot_set_real )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_969_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ bot_bot_set_set_a )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_970_sup__inf__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z2 ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_971_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_972_inf__sup__distrib2,axiom,
! [Y: set_a,Z2: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_973_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_974_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ! [X3: set_a,Y2: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y2 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ ( sup_sup_set_a @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_975_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ! [X3: set_a,Y2: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y2 ) @ ( inf_inf_set_a @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_976_inf__sup__ord_I4_J,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_977_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_978_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_979_inf__sup__ord_I3_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sup_sup_real @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_980_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_981_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_982_le__supE,axiom,
! [A: real,B4: real,X: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ A @ B4 ) @ X )
=> ~ ( ( ord_less_eq_real @ A @ X )
=> ~ ( ord_less_eq_real @ B4 @ X ) ) ) ).
% le_supE
thf(fact_983_le__supE,axiom,
! [A: set_a,B4: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A @ X )
=> ~ ( ord_less_eq_set_a @ B4 @ X ) ) ) ).
% le_supE
thf(fact_984_le__supE,axiom,
! [A: nat,B4: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B4 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B4 @ X ) ) ) ).
% le_supE
thf(fact_985_le__supI,axiom,
! [A: real,X: real,B4: real] :
( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ B4 @ X )
=> ( ord_less_eq_real @ ( sup_sup_real @ A @ B4 ) @ X ) ) ) ).
% le_supI
thf(fact_986_le__supI,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ( ord_less_eq_set_a @ B4 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ X ) ) ) ).
% le_supI
thf(fact_987_le__supI,axiom,
! [A: nat,X: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B4 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B4 ) @ X ) ) ) ).
% le_supI
thf(fact_988_sup__ge1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sup_sup_real @ X @ Y ) ) ).
% sup_ge1
thf(fact_989_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_990_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_991_sup__ge2,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X @ Y ) ) ).
% sup_ge2
thf(fact_992_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_993_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_994_le__supI1,axiom,
! [X: real,A: real,B4: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ord_less_eq_real @ X @ ( sup_sup_real @ A @ B4 ) ) ) ).
% le_supI1
thf(fact_995_le__supI1,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% le_supI1
thf(fact_996_le__supI1,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% le_supI1
thf(fact_997_le__supI2,axiom,
! [X: real,B4: real,A: real] :
( ( ord_less_eq_real @ X @ B4 )
=> ( ord_less_eq_real @ X @ ( sup_sup_real @ A @ B4 ) ) ) ).
% le_supI2
thf(fact_998_le__supI2,axiom,
! [X: set_a,B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ B4 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% le_supI2
thf(fact_999_le__supI2,axiom,
! [X: nat,B4: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B4 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% le_supI2
thf(fact_1000_sup_Omono,axiom,
! [C2: real,A: real,D: real,B4: real] :
( ( ord_less_eq_real @ C2 @ A )
=> ( ( ord_less_eq_real @ D @ B4 )
=> ( ord_less_eq_real @ ( sup_sup_real @ C2 @ D ) @ ( sup_sup_real @ A @ B4 ) ) ) ) ).
% sup.mono
thf(fact_1001_sup_Omono,axiom,
! [C2: set_a,A: set_a,D: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ D @ B4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C2 @ D ) @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% sup.mono
thf(fact_1002_sup_Omono,axiom,
! [C2: nat,A: nat,D: nat,B4: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ( ord_less_eq_nat @ D @ B4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A @ B4 ) ) ) ) ).
% sup.mono
thf(fact_1003_sup__mono,axiom,
! [A: real,C2: real,B4: real,D: real] :
( ( ord_less_eq_real @ A @ C2 )
=> ( ( ord_less_eq_real @ B4 @ D )
=> ( ord_less_eq_real @ ( sup_sup_real @ A @ B4 ) @ ( sup_sup_real @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_1004_sup__mono,axiom,
! [A: set_a,C2: set_a,B4: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_1005_sup__mono,axiom,
! [A: nat,C2: nat,B4: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B4 ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_1006_sup__least,axiom,
! [Y: real,X: real,Z2: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ Z2 @ X )
=> ( ord_less_eq_real @ ( sup_sup_real @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1007_sup__least,axiom,
! [Y: set_a,X: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1008_sup__least,axiom,
! [Y: nat,X: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1009_le__iff__sup,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y4: real] :
( ( sup_sup_real @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_1010_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_1011_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( sup_sup_nat @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_1012_sup_OorderE,axiom,
! [B4: real,A: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( A
= ( sup_sup_real @ A @ B4 ) ) ) ).
% sup.orderE
thf(fact_1013_sup_OorderE,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( A
= ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.orderE
thf(fact_1014_sup_OorderE,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( A
= ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.orderE
thf(fact_1015_sup_OorderI,axiom,
! [A: real,B4: real] :
( ( A
= ( sup_sup_real @ A @ B4 ) )
=> ( ord_less_eq_real @ B4 @ A ) ) ).
% sup.orderI
thf(fact_1016_sup_OorderI,axiom,
! [A: set_a,B4: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B4 ) )
=> ( ord_less_eq_set_a @ B4 @ A ) ) ).
% sup.orderI
thf(fact_1017_sup_OorderI,axiom,
! [A: nat,B4: nat] :
( ( A
= ( sup_sup_nat @ A @ B4 ) )
=> ( ord_less_eq_nat @ B4 @ A ) ) ).
% sup.orderI
thf(fact_1018_sup__unique,axiom,
! [F3: real > real > real,X: real,Y: real] :
( ! [X3: real,Y2: real] : ( ord_less_eq_real @ X3 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: real,Y2: real] : ( ord_less_eq_real @ Y2 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ Y2 @ X3 )
=> ( ( ord_less_eq_real @ Z3 @ X3 )
=> ( ord_less_eq_real @ ( F3 @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_real @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1019_sup__unique,axiom,
! [F3: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X3 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ( ( ord_less_eq_set_a @ Z3 @ X3 )
=> ( ord_less_eq_set_a @ ( F3 @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1020_sup__unique,axiom,
! [F3: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ ( F3 @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1021_sup_Oabsorb1,axiom,
! [B4: real,A: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( sup_sup_real @ A @ B4 )
= A ) ) ).
% sup.absorb1
thf(fact_1022_sup_Oabsorb1,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( sup_sup_set_a @ A @ B4 )
= A ) ) ).
% sup.absorb1
thf(fact_1023_sup_Oabsorb1,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( sup_sup_nat @ A @ B4 )
= A ) ) ).
% sup.absorb1
thf(fact_1024_sup_Oabsorb2,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( sup_sup_real @ A @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_1025_sup_Oabsorb2,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( sup_sup_set_a @ A @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_1026_sup_Oabsorb2,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( sup_sup_nat @ A @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_1027_sup__absorb1,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( sup_sup_real @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1028_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1029_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1030_sup__absorb2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( sup_sup_real @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1031_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1032_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1033_sup_OboundedE,axiom,
! [B4: real,C2: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B4 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_real @ B4 @ A )
=> ~ ( ord_less_eq_real @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_1034_sup_OboundedE,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B4 @ A )
=> ~ ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_1035_sup_OboundedE,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_nat @ B4 @ A )
=> ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_1036_sup_OboundedI,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( ord_less_eq_real @ C2 @ A )
=> ( ord_less_eq_real @ ( sup_sup_real @ B4 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1037_sup_OboundedI,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1038_sup_OboundedI,axiom,
! [B4: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1039_sup_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( A4
= ( sup_sup_real @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1040_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1041_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1042_sup_Ocobounded1,axiom,
! [A: real,B4: real] : ( ord_less_eq_real @ A @ ( sup_sup_real @ A @ B4 ) ) ).
% sup.cobounded1
thf(fact_1043_sup_Ocobounded1,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.cobounded1
thf(fact_1044_sup_Ocobounded1,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B4 ) ) ).
% sup.cobounded1
thf(fact_1045_sup_Ocobounded2,axiom,
! [B4: real,A: real] : ( ord_less_eq_real @ B4 @ ( sup_sup_real @ A @ B4 ) ) ).
% sup.cobounded2
thf(fact_1046_sup_Ocobounded2,axiom,
! [B4: set_a,A: set_a] : ( ord_less_eq_set_a @ B4 @ ( sup_sup_set_a @ A @ B4 ) ) ).
% sup.cobounded2
thf(fact_1047_sup_Ocobounded2,axiom,
! [B4: nat,A: nat] : ( ord_less_eq_nat @ B4 @ ( sup_sup_nat @ A @ B4 ) ) ).
% sup.cobounded2
thf(fact_1048_sup_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( sup_sup_real @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1049_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1050_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1051_sup_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( sup_sup_real @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1052_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1053_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1054_sup_OcoboundedI1,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_eq_real @ C2 @ A )
=> ( ord_less_eq_real @ C2 @ ( sup_sup_real @ A @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_1055_sup_OcoboundedI1,axiom,
! [C2: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_1056_sup_OcoboundedI1,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_1057_sup_OcoboundedI2,axiom,
! [C2: real,B4: real,A: real] :
( ( ord_less_eq_real @ C2 @ B4 )
=> ( ord_less_eq_real @ C2 @ ( sup_sup_real @ A @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_1058_sup_OcoboundedI2,axiom,
! [C2: set_a,B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C2 @ B4 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_1059_sup_OcoboundedI2,axiom,
! [C2: nat,B4: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ B4 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_1060_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_1061_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_1062_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_1063_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_1064_sup_Oassoc,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B4 ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% sup.assoc
thf(fact_1065_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_1066_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A4 ) ) ) ).
% sup.commute
thf(fact_1067_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X2 ) ) ) ).
% sup_commute
thf(fact_1068_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_a,K: set_a,A: set_a,B4: set_a] :
( ( A2
= ( sup_sup_set_a @ K @ A ) )
=> ( ( sup_sup_set_a @ A2 @ B4 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1069_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B4: set_a,A: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B4 ) )
=> ( ( sup_sup_set_a @ A @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1070_sup_Oleft__commute,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( sup_sup_set_a @ B4 @ ( sup_sup_set_a @ A @ C2 ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_1071_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_1072_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_1073_bot__set__def,axiom,
( bot_bot_set_real
= ( collect_real @ bot_bot_real_o ) ) ).
% bot_set_def
thf(fact_1074_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_1075_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1076_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_1077_Un__empty__left,axiom,
! [B: set_real] :
( ( sup_sup_set_real @ bot_bot_set_real @ B )
= B ) ).
% Un_empty_left
thf(fact_1078_Un__empty__left,axiom,
! [B: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_1079_Un__empty__right,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_1080_Un__empty__right,axiom,
! [A2: set_real] :
( ( sup_sup_set_real @ A2 @ bot_bot_set_real )
= A2 ) ).
% Un_empty_right
thf(fact_1081_Un__empty__right,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ bot_bot_set_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_1082_finite__UnI,axiom,
! [F: set_real,G: set_real] :
( ( finite_finite_real @ F )
=> ( ( finite_finite_real @ G )
=> ( finite_finite_real @ ( sup_sup_set_real @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_1083_finite__UnI,axiom,
! [F: set_a,G: set_a] :
( ( finite_finite_a @ F )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_1084_Un__infinite,axiom,
! [S: set_real,T: set_real] :
( ~ ( finite_finite_real @ S )
=> ~ ( finite_finite_real @ ( sup_sup_set_real @ S @ T ) ) ) ).
% Un_infinite
thf(fact_1085_Un__infinite,axiom,
! [S: set_a,T: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_1086_infinite__Un,axiom,
! [S: set_real,T: set_real] :
( ( ~ ( finite_finite_real @ ( sup_sup_set_real @ S @ T ) ) )
= ( ~ ( finite_finite_real @ S )
| ~ ( finite_finite_real @ T ) ) ) ).
% infinite_Un
thf(fact_1087_infinite__Un,axiom,
! [S: set_a,T: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T ) ) ) ).
% infinite_Un
thf(fact_1088_Un__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1089_Un__least,axiom,
! [A2: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_1090_Un__upper1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper1
thf(fact_1091_Un__upper2,axiom,
! [B: set_a,A2: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper2
thf(fact_1092_Un__absorb1,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_1093_Un__absorb2,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_1094_subset__UnE,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
=> ~ ! [A8: set_a] :
( ( ord_less_eq_set_a @ A8 @ A2 )
=> ! [B8: set_a] :
( ( ord_less_eq_set_a @ B8 @ B )
=> ( C
!= ( sup_sup_set_a @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_1095_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( sup_sup_set_a @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_1096_Un__Int__crazy,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ B @ C ) ) @ ( inf_inf_set_a @ C @ A2 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ B @ C ) ) @ ( sup_sup_set_a @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_1097_Int__Un__distrib,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_1098_Un__Int__distrib,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_1099_Int__Un__distrib2,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C ) @ A2 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A2 ) @ ( inf_inf_set_a @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_1100_Un__Int__distrib2,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C ) @ A2 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A2 ) @ ( sup_sup_set_a @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_1101_Un__Diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ C ) @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Un_Diff
thf(fact_1102_Un__Diff,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( minus_5736297505244876581_set_a @ ( sup_sup_set_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_set_a @ ( minus_5736297505244876581_set_a @ A2 @ C ) @ ( minus_5736297505244876581_set_a @ B @ C ) ) ) ).
% Un_Diff
thf(fact_1103_distrib__sup__le,axiom,
! [X: real,Y: real,Z2: real] : ( ord_less_eq_real @ ( sup_sup_real @ X @ ( inf_inf_real @ Y @ Z2 ) ) @ ( inf_inf_real @ ( sup_sup_real @ X @ Y ) @ ( sup_sup_real @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1104_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1105_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1106_distrib__inf__le,axiom,
! [X: real,Y: real,Z2: real] : ( ord_less_eq_real @ ( sup_sup_real @ ( inf_inf_real @ X @ Y ) @ ( inf_inf_real @ X @ Z2 ) ) @ ( inf_inf_real @ X @ ( sup_sup_real @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1107_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1108_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z2 ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1109_insert__is__Un,axiom,
( insert_a
= ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_1110_insert__is__Un,axiom,
( insert_real
= ( ^ [A4: real] : ( sup_sup_set_real @ ( insert_real @ A4 @ bot_bot_set_real ) ) ) ) ).
% insert_is_Un
thf(fact_1111_insert__is__Un,axiom,
( insert_set_a
= ( ^ [A4: set_a] : ( sup_sup_set_set_a @ ( insert_set_a @ A4 @ bot_bot_set_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_1112_Un__singleton__iff,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ( sup_sup_set_a @ A2 @ B )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1113_Un__singleton__iff,axiom,
! [A2: set_real,B: set_real,X: real] :
( ( ( sup_sup_set_real @ A2 @ B )
= ( insert_real @ X @ bot_bot_set_real ) )
= ( ( ( A2 = bot_bot_set_real )
& ( B
= ( insert_real @ X @ bot_bot_set_real ) ) )
| ( ( A2
= ( insert_real @ X @ bot_bot_set_real ) )
& ( B = bot_bot_set_real ) )
| ( ( A2
= ( insert_real @ X @ bot_bot_set_real ) )
& ( B
= ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1114_Un__singleton__iff,axiom,
! [A2: set_set_a,B: set_set_a,X: set_a] :
( ( ( sup_sup_set_set_a @ A2 @ B )
= ( insert_set_a @ X @ bot_bot_set_set_a ) )
= ( ( ( A2 = bot_bot_set_set_a )
& ( B
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
| ( ( A2
= ( insert_set_a @ X @ bot_bot_set_set_a ) )
& ( B = bot_bot_set_set_a ) )
| ( ( A2
= ( insert_set_a @ X @ bot_bot_set_set_a ) )
& ( B
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1115_singleton__Un__iff,axiom,
! [X: a,A2: set_a,B: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1116_singleton__Un__iff,axiom,
! [X: real,A2: set_real,B: set_real] :
( ( ( insert_real @ X @ bot_bot_set_real )
= ( sup_sup_set_real @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_real )
& ( B
= ( insert_real @ X @ bot_bot_set_real ) ) )
| ( ( A2
= ( insert_real @ X @ bot_bot_set_real ) )
& ( B = bot_bot_set_real ) )
| ( ( A2
= ( insert_real @ X @ bot_bot_set_real ) )
& ( B
= ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1117_singleton__Un__iff,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ( ( insert_set_a @ X @ bot_bot_set_set_a )
= ( sup_sup_set_set_a @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_set_a )
& ( B
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
| ( ( A2
= ( insert_set_a @ X @ bot_bot_set_set_a ) )
& ( B = bot_bot_set_set_a ) )
| ( ( A2
= ( insert_set_a @ X @ bot_bot_set_set_a ) )
& ( B
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1118_Un__Int__assoc__eq,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) )
= ( ord_less_eq_set_a @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_1119_Diff__partition,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( sup_sup_set_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_1120_Diff__partition,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_1121_Diff__subset__conv,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ C )
= ( ord_le3724670747650509150_set_a @ A2 @ ( sup_sup_set_set_a @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_1122_Diff__subset__conv,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C )
= ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_1123_Un__Diff__Int,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( inf_inf_set_a @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_1124_Un__Diff__Int,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ ( inf_inf_set_set_a @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_1125_Int__Diff__Un,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_1126_Int__Diff__Un,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_1127_Diff__Int,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ C ) ) ) ).
% Diff_Int
thf(fact_1128_Diff__Int,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C ) )
= ( sup_sup_set_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ ( minus_5736297505244876581_set_a @ A2 @ C ) ) ) ).
% Diff_Int
thf(fact_1129_Diff__Un,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ C ) ) ) ).
% Diff_Un
thf(fact_1130_Diff__Un,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ ( sup_sup_set_set_a @ B @ C ) )
= ( inf_inf_set_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ ( minus_5736297505244876581_set_a @ A2 @ C ) ) ) ).
% Diff_Un
thf(fact_1131_card__Un__le,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( sup_sup_set_a @ A2 @ B ) ) @ ( plus_plus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ).
% card_Un_le
thf(fact_1132_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_1133_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_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_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_1136_inf_Oassoc,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% inf.assoc
thf(fact_1137_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_1138_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_1139_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X2 ) ) ) ).
% inf_commute
thf(fact_1140_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_a,K: set_a,A: set_a,B4: set_a] :
( ( A2
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A2 @ B4 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_1141_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B4: set_a,A: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B4 ) )
=> ( ( inf_inf_set_a @ A @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_1142_inf_Oleft__commute,axiom,
! [B4: set_a,A: set_a,C2: set_a] :
( ( inf_inf_set_a @ B4 @ ( inf_inf_set_a @ A @ C2 ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_1143_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_1144_card__Un__Int,axiom,
! [A2: set_real,B: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_real @ B )
=> ( ( plus_plus_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B ) )
= ( plus_plus_nat @ ( finite_card_real @ ( sup_sup_set_real @ A2 @ B ) ) @ ( finite_card_real @ ( inf_inf_set_real @ A2 @ B ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_1145_card__Un__Int,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ( plus_plus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) )
= ( plus_plus_nat @ ( finite_card_a @ ( sup_sup_set_a @ A2 @ B ) ) @ ( finite_card_a @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_1146_card__Un__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( finite_card_a @ ( sup_sup_set_a @ A2 @ B ) )
= ( plus_plus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_1147_card__Un__disjoint,axiom,
! [A2: set_real,B: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_real @ B )
=> ( ( ( inf_inf_set_real @ A2 @ B )
= bot_bot_set_real )
=> ( ( finite_card_real @ ( sup_sup_set_real @ A2 @ B ) )
= ( plus_plus_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_1148_card__Un__disjoint,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B )
=> ( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
=> ( ( finite_card_set_a @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( plus_plus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_1149_inf_OcoboundedI2,axiom,
! [B4: real,C2: real,A: real] :
( ( ord_less_eq_real @ B4 @ C2 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_1150_inf_OcoboundedI2,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_1151_inf_OcoboundedI2,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_1152_inf_OcoboundedI1,axiom,
! [A: real,C2: real,B4: real] :
( ( ord_less_eq_real @ A @ C2 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_1153_inf_OcoboundedI1,axiom,
! [A: set_a,C2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_1154_inf_OcoboundedI1,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_1155_inf_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( inf_inf_real @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_1156_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_1157_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_1158_inf_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( inf_inf_real @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1159_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1160_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1161_inf_Ocobounded2,axiom,
! [A: real,B4: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_1162_inf_Ocobounded2,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_1163_inf_Ocobounded2,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_1164_inf_Ocobounded1,axiom,
! [A: real,B4: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_1165_inf_Ocobounded1,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_1166_inf_Ocobounded1,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_1167_inf_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( A4
= ( inf_inf_real @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_1168_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_1169_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( A4
= ( inf_inf_nat @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_1170_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_1171_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_1172_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_1173_inf_OboundedI,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ A @ C2 )
=> ( ord_less_eq_real @ A @ ( inf_inf_real @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1174_inf_OboundedI,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1175_inf_OboundedI,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1176_inf_OboundedE,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_real @ A @ B4 )
=> ~ ( ord_less_eq_real @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1177_inf_OboundedE,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B4 )
=> ~ ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1178_inf_OboundedE,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A @ B4 )
=> ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1179_inf__absorb2,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( inf_inf_real @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1180_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_1181_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1182_inf__absorb1,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( inf_inf_real @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1183_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1184_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1185_inf_Oabsorb2,axiom,
! [B4: real,A: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( inf_inf_real @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_1186_inf_Oabsorb2,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( inf_inf_set_a @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_1187_inf_Oabsorb2,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( inf_inf_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_1188_inf_Oabsorb1,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( inf_inf_real @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_1189_inf_Oabsorb1,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( inf_inf_set_a @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_1190_inf_Oabsorb1,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( inf_inf_nat @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_1191__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 @ a3 @ ( 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 @ a3 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a3 ) ) ) ) ) ).
% \<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_1192__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,
? [A7: set_a] :
( ( member_set_a @ A7 @ ( minus_5736297505244876581_set_a @ ( pow_a @ a0 ) @ ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) )
& ( k
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A7 @ b ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A7 ) ) ) ) ) ).
% \<open>\<exists>A. A \<in> Pow A0 - {{}} \<and> K = real (card (sumset A B)) / real (card A)\<close>
thf(fact_1193_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_1194__092_060open_0620_A_060_AK_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ k ).
% \<open>0 < K\<close>
thf(fact_1195_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_1196_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1197_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_1198_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_1199_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_1200_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_1201_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_1202_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1203_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1204_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1205_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_1206_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1207_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_1208_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_1209_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_1210_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1211_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_1212_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_1213_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_1214_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_1215_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_1216_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_1217_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_1218_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_1219_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_1220_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1221_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1222_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1223_less__not__refl3,axiom,
! [S2: nat,T4: nat] :
( ( ord_less_nat @ S2 @ T4 )
=> ( S2 != T4 ) ) ).
% less_not_refl3
thf(fact_1224_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1225_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1226_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_1227_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1228_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1229_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1230_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1231_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1232_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1233_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1234_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1235_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F3 @ I2 ) @ ( F3 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F3 @ I ) @ ( F3 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1236_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_1237_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_1238_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1239_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_1240_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( M2 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1241_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1242_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1243_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1244_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1245_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1246_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1247_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1248_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1249_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_1250_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_1251_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1252_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_1253_mono__nat__linear__lb,axiom,
! [F3: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F3 @ M4 ) @ ( F3 @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F3 @ M ) @ K ) @ ( F3 @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1254_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_1255_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_1256_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_1257_diff__less__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B4 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1258_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_1259_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1260_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1261_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_1262_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B4: nat] :
( ( P @ ( minus_minus_nat @ A @ B4 ) )
= ( ~ ( ( ( ord_less_nat @ A @ B4 )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B4 @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1263_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B4: nat] :
( ( P @ ( minus_minus_nat @ A @ B4 ) )
= ( ( ( ord_less_nat @ A @ B4 )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B4 @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1264_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_div
thf(fact_1265_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1266_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1267_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1268_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
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ k @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ca ) ) ) ) @ ( minus_minus_real @ ( times_times_real @ k @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a3 ) ) ) @ ( times_times_real @ k @ ( semiri5074537144036343181t_real @ ( finite_card_a @ a2 ) ) ) ) ) @ ( times_times_real @ k @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ a3 @ ( insert_a @ x @ ca ) ) ) ) ) ).
%------------------------------------------------------------------------------