TPTP Problem File: SLH0592^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Pluennecke_Ruzsa_Inequality/0003_Pluennecke_Ruzsa_Inequality/prob_00394_014426__12227524_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1363 ( 574 unt; 90 typ; 0 def)
% Number of atoms : 3579 (1268 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 11564 ( 345 ~; 79 |; 232 &;9442 @)
% ( 0 <=>;1466 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 6 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 372 ( 372 >; 0 *; 0 +; 0 <<)
% Number of symbols : 84 ( 81 usr; 19 con; 0-5 aty)
% Number of variables : 3144 ( 105 ^;2975 !; 64 ?;3144 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:21:16.112
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (81)
thf(sy_c_Finite__Set_Ocard_001t__Real__Oreal,type,
finite_card_real: set_real > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Group__Theory_Oabelian__group_001tf__a,type,
group_201663378560352916roup_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001tf__a,type,
group_4866109990395492029noid_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ogroup_001tf__a,type,
group_group_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001tf__a,type,
group_Units_a: set_a > ( a > a > a ) > a > set_a ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001tf__a,type,
group_invertible_a: set_a > ( a > a > a ) > a > 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_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
inf_inf_int: int > int > int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Real__Oreal,type,
inf_inf_real: real > real > real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Real__Oreal_J,type,
inf_inf_set_real: set_real > set_real > set_real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_NthRoot_Osqrt,type,
sqrt: real > real ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_001t__Real__Oreal,type,
pluenn1014277435162747966p_real: set_real > ( real > real > real ) > real > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_001tf__a,type,
pluenn1164192988769422572roup_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_ORuzsa__distance_001tf__a,type,
pluenn5761198478017115492ance_a: set_a > ( a > a > a ) > a > set_a > set_a > real ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Ominusset_001tf__a,type,
pluenn2534204936789923946sset_a: set_a > ( a > a > a ) > a > set_a > set_a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset_001t__Real__Oreal,type,
pluenn7361685508355272389t_real: set_real > ( real > real > real ) > set_real > set_real > set_real ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset_001tf__a,type,
pluenn3038260743871226533mset_a: set_a > ( a > a > a ) > set_a > set_a > set_a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumset__iterated_001tf__a,type,
pluenn1960970773371692859ated_a: set_a > ( a > a > a ) > a > set_a > nat > set_a ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumsetp_001t__Real__Oreal,type,
pluenn3384280056939765061p_real: set_real > ( real > real > real ) > ( real > $o ) > ( real > $o ) > real > $o ).
thf(sy_c_Pluennecke__Ruzsa__Inequality_Oadditive__abelian__group_Osumsetp_001tf__a,type,
pluenn895083305082786853setp_a: set_a > ( a > a > a ) > ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_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_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_G,type,
g: set_a ).
thf(sy_v_U,type,
u: set_a ).
thf(sy_v_V,type,
v: set_a ).
thf(sy_v_W,type,
w: set_a ).
thf(sy_v_addition,type,
addition: a > a > a ).
thf(sy_v_zero,type,
zero: a ).
% Relevant facts (1269)
thf(fact_0_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( addition @ X @ Y )
= ( addition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_1_sumset_Ocases,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( addition @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ g )
=> ( ( member_a @ B2 @ B )
=> ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).
% sumset.cases
thf(fact_2_sumset_Osimps,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= ( ? [A4: a,B3: a] :
( ( A
= ( addition @ A4 @ B3 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ g )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ g ) ) ) ) ).
% sumset.simps
thf(fact_3_sumset_OsumsetI,axiom,
! [A: a,A2: set_a,B4: a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ g )
=> ( ( member_a @ B4 @ B )
=> ( ( member_a @ B4 @ g )
=> ( member_a @ ( addition @ A @ B4 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ) ) ).
% sumset.sumsetI
thf(fact_4_sumset__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) @ C )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ C ) ) ) ).
% sumset_assoc
thf(fact_5_sumset__commute,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B )
= ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A2 ) ) ).
% sumset_commute
thf(fact_6_local_Oinverse__unique,axiom,
! [U: a,V: a,V2: a] :
( ( ( addition @ U @ V )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( ( member_a @ V @ g )
=> ( V2 = V ) ) ) ) ) ) ).
% local.inverse_unique
thf(fact_7_assms_I3_J,axiom,
u != bot_bot_set_a ).
% assms(3)
thf(fact_8_minusset__distrib__sum,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ).
% minusset_distrib_sum
thf(fact_9_card__differenceset__commute,axiom,
! [B: set_a,A2: set_a] :
( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) )
= ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ) ).
% card_differenceset_commute
thf(fact_10_assms_I1_J,axiom,
finite_finite_a @ u ).
% assms(1)
thf(fact_11_assms_I6_J,axiom,
finite_finite_a @ w ).
% assms(6)
thf(fact_12_assms_I4_J,axiom,
finite_finite_a @ v ).
% assms(4)
thf(fact_13_assms_I2_J,axiom,
ord_less_eq_set_a @ u @ g ).
% assms(2)
thf(fact_14_assms_I7_J,axiom,
ord_less_eq_set_a @ w @ g ).
% assms(7)
thf(fact_15_assms_I5_J,axiom,
ord_less_eq_set_a @ v @ g ).
% assms(5)
thf(fact_16_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_17_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_18_unit__closed,axiom,
member_a @ zero @ g ).
% unit_closed
thf(fact_19_left__unit,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( addition @ zero @ A )
= A ) ) ).
% left_unit
thf(fact_20_right__unit,axiom,
! [A: a] :
( ( member_a @ A @ g )
=> ( ( addition @ A @ zero )
= A ) ) ).
% right_unit
thf(fact_21_differenceset__commute,axiom,
! [B: set_a,A2: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ).
% differenceset_commute
thf(fact_22_Ruzsa__distance__def,axiom,
! [A2: set_a,B: set_a] :
( ( pluenn5761198478017115492ance_a @ g @ addition @ zero @ A2 @ B )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ) @ ( times_times_real @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A2 ) ) ) @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ B ) ) ) ) ) ) ).
% Ruzsa_distance_def
thf(fact_23_additive__abelian__group__axioms,axiom,
pluenn1164192988769422572roup_a @ g @ addition @ zero ).
% additive_abelian_group_axioms
thf(fact_24_commutative__monoid__axioms,axiom,
group_4866109990395492029noid_a @ g @ addition @ zero ).
% commutative_monoid_axioms
thf(fact_25_additive__abelian__group_Osumset_Ocong,axiom,
pluenn3038260743871226533mset_a = pluenn3038260743871226533mset_a ).
% additive_abelian_group.sumset.cong
thf(fact_26_additive__abelian__group_Ominusset_Ocong,axiom,
pluenn2534204936789923946sset_a = pluenn2534204936789923946sset_a ).
% additive_abelian_group.minusset.cong
thf(fact_27_abelian__group__axioms,axiom,
group_201663378560352916roup_a @ g @ addition @ zero ).
% abelian_group_axioms
thf(fact_28__092_060open_062real_A_Icard_AU_A_K_Acard_A_Idifferenceset_AV_AW_J_J_A_P_A_Ireal_A_Icard_AU_J_A_K_Asqrt_A_Ireal_A_Icard_AV_J_J_A_K_Asqrt_A_Ireal_A_Icard_AW_J_J_J_A_092_060le_062_Areal_A_Icard_A_Idifferenceset_AU_AV_J_A_K_Acard_A_Idifferenceset_AU_AW_J_J_A_P_A_Ireal_A_Icard_AU_J_A_K_Asqrt_A_Ireal_A_Icard_AV_J_J_A_K_Asqrt_A_Ireal_A_Icard_AW_J_J_J_092_060close_062,axiom,
ord_less_eq_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( finite_card_a @ u ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ u ) ) @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ v ) ) ) ) @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ w ) ) ) ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ v ) ) ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ u ) ) @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ v ) ) ) ) @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ w ) ) ) ) ) ).
% \<open>real (card U * card (differenceset V W)) / (real (card U) * sqrt (real (card V)) * sqrt (real (card W))) \<le> real (card (differenceset U V) * card (differenceset U W)) / (real (card U) * sqrt (real (card V)) * sqrt (real (card W)))\<close>
thf(fact_29__C_K_C,axiom,
ord_less_eq_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ) ) @ ( times_times_real @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ v ) ) ) @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ w ) ) ) ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ v ) ) ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ u ) ) @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ v ) ) ) ) @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ w ) ) ) ) ) ).
% "*"
thf(fact_30_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_31_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_32_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_33__092_060open_062card_AU_A_K_Acard_A_Idifferenceset_AV_AW_J_A_092_060le_062_Acard_A_Idifferenceset_AU_AV_J_A_K_Acard_A_Idifferenceset_AU_AW_J_092_060close_062,axiom,
ord_less_eq_nat @ ( times_times_nat @ ( finite_card_a @ u ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ) ) @ ( times_times_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ v ) ) ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ) ) ).
% \<open>card U * card (differenceset V W) \<le> card (differenceset U V) * card (differenceset U W)\<close>
thf(fact_34_card__sumset__iterated__minusset,axiom,
! [A2: set_a,K: nat] :
( ( finite_card_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ K ) )
= ( finite_card_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K ) ) ) ).
% card_sumset_iterated_minusset
thf(fact_35_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_36_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_37_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_38_minusset__iterated__minusset,axiom,
! [A2: set_a,K: nat] :
( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ K )
= ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K ) ) ) ).
% minusset_iterated_minusset
thf(fact_39_card__minusset_H,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ g )
=> ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
= ( finite_card_a @ A2 ) ) ) ).
% card_minusset'
thf(fact_40_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_41_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_42_finite__sumset,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ B ) ) ) ) ).
% finite_sumset
thf(fact_43_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_44_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_45_finite__minusset,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) ) ) ).
% finite_minusset
thf(fact_46_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_minusset__subset__carrier,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) @ g ) ).
% minusset_subset_carrier
thf(fact_51_finite__sumset__iterated,axiom,
! [A2: set_a,R: nat] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ R ) ) ) ).
% finite_sumset_iterated
thf(fact_52_sumset__iterated__subset__carrier,axiom,
! [A2: set_a,K: nat] : ( ord_less_eq_set_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ K ) @ g ) ).
% sumset_iterated_subset_carrier
thf(fact_53_finite__differenceset,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ) ) ).
% finite_differenceset
thf(fact_54_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_55_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_56_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_57_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_58_Ruzsa__triangle__ineq1,axiom,
! [U2: set_a,V3: set_a,W: set_a] :
( ( finite_finite_a @ U2 )
=> ( ( ord_less_eq_set_a @ U2 @ g )
=> ( ( finite_finite_a @ V3 )
=> ( ( ord_less_eq_set_a @ V3 @ g )
=> ( ( finite_finite_a @ W )
=> ( ( ord_less_eq_set_a @ W @ g )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( finite_card_a @ U2 ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ V3 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ W ) ) ) ) @ ( times_times_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ U2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ V3 ) ) ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ U2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ W ) ) ) ) ) ) ) ) ) ) ) ).
% Ruzsa_triangle_ineq1
thf(fact_59_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_60_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_61_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_62_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_63_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_64_additive__abelian__group_Ointro,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( group_201663378560352916roup_a @ G @ Addition @ Zero )
=> ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero ) ) ).
% additive_abelian_group.intro
thf(fact_65_additive__abelian__group_Oaxioms,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( group_201663378560352916roup_a @ G @ Addition @ Zero ) ) ).
% additive_abelian_group.axioms
thf(fact_66_additive__abelian__group_Osumsetp_Ocases,axiom,
! [G: set_real,Addition: real > real > real,Zero: real,A2: real > $o,B: real > $o,A: real] :
( ( pluenn1014277435162747966p_real @ G @ Addition @ Zero )
=> ( ( pluenn3384280056939765061p_real @ G @ Addition @ A2 @ B @ A )
=> ~ ! [A3: real,B2: real] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( A2 @ A3 )
=> ( ( member_real @ A3 @ G )
=> ( ( B @ B2 )
=> ~ ( member_real @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.cases
thf(fact_67_additive__abelian__group_Osumsetp_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,B: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ A )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( A2 @ A3 )
=> ( ( member_a @ A3 @ G )
=> ( ( B @ B2 )
=> ~ ( member_a @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.cases
thf(fact_68_additive__abelian__group_Osumsetp_Osimps,axiom,
! [G: set_real,Addition: real > real > real,Zero: real,A2: real > $o,B: real > $o,A: real] :
( ( pluenn1014277435162747966p_real @ G @ Addition @ Zero )
=> ( ( pluenn3384280056939765061p_real @ G @ Addition @ A2 @ B @ A )
= ( ? [A4: real,B3: real] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( A2 @ A4 )
& ( member_real @ A4 @ G )
& ( B @ B3 )
& ( member_real @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumsetp.simps
thf(fact_69_additive__abelian__group_Osumsetp_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,B: a > $o,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ A )
= ( ? [A4: a,B3: a] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( A2 @ A4 )
& ( member_a @ A4 @ G )
& ( B @ B3 )
& ( member_a @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumsetp.simps
thf(fact_70_additive__abelian__group_Osumsetp_OsumsetI,axiom,
! [G: set_real,Addition: real > real > real,Zero: real,A2: real > $o,A: real,B: real > $o,B4: real] :
( ( pluenn1014277435162747966p_real @ G @ Addition @ Zero )
=> ( ( A2 @ A )
=> ( ( member_real @ A @ G )
=> ( ( B @ B4 )
=> ( ( member_real @ B4 @ G )
=> ( pluenn3384280056939765061p_real @ G @ Addition @ A2 @ B @ ( Addition @ A @ B4 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.sumsetI
thf(fact_71_additive__abelian__group_Osumsetp_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: a > $o,A: a,B: a > $o,B4: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( A2 @ A )
=> ( ( member_a @ A @ G )
=> ( ( B @ B4 )
=> ( ( member_a @ B4 @ G )
=> ( pluenn895083305082786853setp_a @ G @ Addition @ A2 @ B @ ( Addition @ A @ B4 ) ) ) ) ) ) ) ).
% additive_abelian_group.sumsetp.sumsetI
thf(fact_72_additive__abelian__group_Osumset__iterated_Ocong,axiom,
pluenn1960970773371692859ated_a = pluenn1960970773371692859ated_a ).
% additive_abelian_group.sumset_iterated.cong
thf(fact_73_additive__abelian__group_Ofinite__sumset__iterated,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,R: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ R ) ) ) ) ).
% additive_abelian_group.finite_sumset_iterated
thf(fact_74_additive__abelian__group_Osumset__iterated__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,K: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ K ) @ G ) ) ).
% additive_abelian_group.sumset_iterated_subset_carrier
thf(fact_75_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_76_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_77_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_78_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_79_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_80_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_81_additive__abelian__group__def,axiom,
pluenn1164192988769422572roup_a = group_201663378560352916roup_a ).
% additive_abelian_group_def
thf(fact_82_additive__abelian__group_Ocard__le__sumset,axiom,
! [G: set_real,Addition: real > real > real,Zero: real,A2: set_real,A: real,B: set_real] :
( ( pluenn1014277435162747966p_real @ G @ Addition @ Zero )
=> ( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ( ( member_real @ A @ G )
=> ( ( finite_finite_real @ B )
=> ( ( ord_less_eq_set_real @ B @ G )
=> ( ord_less_eq_nat @ ( finite_card_real @ B ) @ ( finite_card_real @ ( pluenn7361685508355272389t_real @ G @ Addition @ A2 @ B ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.card_le_sumset
thf(fact_83_additive__abelian__group_Ocard__le__sumset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ G )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.card_le_sumset
thf(fact_84_additive__abelian__group_Ofinite__sumset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset
thf(fact_85_additive__abelian__group_Osumsetp_Ocong,axiom,
pluenn895083305082786853setp_a = pluenn895083305082786853setp_a ).
% additive_abelian_group.sumsetp.cong
thf(fact_86_additive__abelian__group_Osumset__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ G ) ) ).
% additive_abelian_group.sumset_subset_carrier
thf(fact_87_additive__abelian__group_Osumset__mono,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A5: set_a,A2: set_a,B5: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ B5 @ B )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A5 @ B5 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.sumset_mono
thf(fact_88_additive__abelian__group_Osumset__empty_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ) ).
% additive_abelian_group.sumset_empty(1)
thf(fact_89_additive__abelian__group_Osumset__empty_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ) ).
% additive_abelian_group.sumset_empty(2)
thf(fact_90_additive__abelian__group_Ofinite__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) ) ) ) ).
% additive_abelian_group.finite_minusset
thf(fact_91_additive__abelian__group_Ominusset__subset__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) @ G ) ) ).
% additive_abelian_group.minusset_subset_carrier
thf(fact_92_additive__abelian__group_Ominusset__iterated__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,K: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) @ K )
= ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ K ) ) ) ) ).
% additive_abelian_group.minusset_iterated_minusset
thf(fact_93_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_94_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_95_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_96_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_97_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_98_additive__abelian__group_ORuzsa__distance_Ocong,axiom,
pluenn5761198478017115492ance_a = pluenn5761198478017115492ance_a ).
% additive_abelian_group.Ruzsa_distance.cong
thf(fact_99_additive__abelian__group_ORuzsa__triangle__ineq1,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,U2: set_a,V3: set_a,W: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ U2 )
=> ( ( ord_less_eq_set_a @ U2 @ G )
=> ( ( finite_finite_a @ V3 )
=> ( ( ord_less_eq_set_a @ V3 @ G )
=> ( ( finite_finite_a @ W )
=> ( ( ord_less_eq_set_a @ W @ G )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( finite_card_a @ U2 ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ V3 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ W ) ) ) ) @ ( times_times_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ U2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ V3 ) ) ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ U2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ W ) ) ) ) ) ) ) ) ) ) ) ) ).
% additive_abelian_group.Ruzsa_triangle_ineq1
thf(fact_100_additive__abelian__group_Ocard__minusset_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A2 @ G )
=> ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% additive_abelian_group.card_minusset'
thf(fact_101_additive__abelian__group_Ofinite__differenceset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ) ) ).
% additive_abelian_group.finite_differenceset
thf(fact_102_additive__abelian__group_Osumset__commute,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ A2 ) ) ) ).
% additive_abelian_group.sumset_commute
thf(fact_103_additive__abelian__group_Osumset_OsumsetI,axiom,
! [G: set_real,Addition: real > real > real,Zero: real,A: real,A2: set_real,B4: real,B: set_real] :
( ( pluenn1014277435162747966p_real @ G @ Addition @ Zero )
=> ( ( member_real @ A @ A2 )
=> ( ( member_real @ A @ G )
=> ( ( member_real @ B4 @ B )
=> ( ( member_real @ B4 @ G )
=> ( member_real @ ( Addition @ A @ B4 ) @ ( pluenn7361685508355272389t_real @ G @ Addition @ A2 @ B ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.sumsetI
thf(fact_104_additive__abelian__group_Osumset_OsumsetI,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B4: a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ A2 )
=> ( ( member_a @ A @ G )
=> ( ( member_a @ B4 @ B )
=> ( ( member_a @ B4 @ G )
=> ( member_a @ ( Addition @ A @ B4 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.sumsetI
thf(fact_105_additive__abelian__group_Osumset__assoc,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,C: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ C )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ C ) ) ) ) ).
% additive_abelian_group.sumset_assoc
thf(fact_106_additive__abelian__group_Osumset_Osimps,axiom,
! [G: set_real,Addition: real > real > real,Zero: real,A: real,A2: set_real,B: set_real] :
( ( pluenn1014277435162747966p_real @ G @ Addition @ Zero )
=> ( ( member_real @ A @ ( pluenn7361685508355272389t_real @ G @ Addition @ A2 @ B ) )
= ( ? [A4: real,B3: real] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( member_real @ A4 @ A2 )
& ( member_real @ A4 @ G )
& ( member_real @ B3 @ B )
& ( member_real @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset.simps
thf(fact_107_additive__abelian__group_Osumset_Osimps,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= ( ? [A4: a,B3: a] :
( ( A
= ( Addition @ A4 @ B3 ) )
& ( member_a @ A4 @ A2 )
& ( member_a @ A4 @ G )
& ( member_a @ B3 @ B )
& ( member_a @ B3 @ G ) ) ) ) ) ).
% additive_abelian_group.sumset.simps
thf(fact_108_additive__abelian__group_Osumset_Ocases,axiom,
! [G: set_real,Addition: real > real > real,Zero: real,A: real,A2: set_real,B: set_real] :
( ( pluenn1014277435162747966p_real @ G @ Addition @ Zero )
=> ( ( member_real @ A @ ( pluenn7361685508355272389t_real @ G @ Addition @ A2 @ B ) )
=> ~ ! [A3: real,B2: real] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( member_real @ A3 @ A2 )
=> ( ( member_real @ A3 @ G )
=> ( ( member_real @ B2 @ B )
=> ~ ( member_real @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.cases
thf(fact_109_additive__abelian__group_Osumset_Ocases,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( member_a @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
=> ~ ! [A3: a,B2: a] :
( ( A
= ( Addition @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ A2 )
=> ( ( member_a @ A3 @ G )
=> ( ( member_a @ B2 @ B )
=> ~ ( member_a @ B2 @ G ) ) ) ) ) ) ) ).
% additive_abelian_group.sumset.cases
thf(fact_110_additive__abelian__group_Ocard__sumset__iterated__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,K: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_card_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) @ K ) )
= ( finite_card_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ K ) ) ) ) ).
% additive_abelian_group.card_sumset_iterated_minusset
thf(fact_111_additive__abelian__group_Odifferenceset__commute,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ).
% additive_abelian_group.differenceset_commute
thf(fact_112_additive__abelian__group_Ominusset__distrib__sum,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ).
% additive_abelian_group.minusset_distrib_sum
thf(fact_113_additive__abelian__group_Odiff__minus__set,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) )
= ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ).
% additive_abelian_group.diff_minus_set
thf(fact_114_additive__abelian__group_Ocard__differenceset__commute,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) ) )
= ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ) ).
% additive_abelian_group.card_differenceset_commute
thf(fact_115_additive__abelian__group_ORuzsa__distance__def,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn5761198478017115492ance_a @ G @ Addition @ Zero @ A2 @ B )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) @ ( times_times_real @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ A2 ) ) ) @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ B ) ) ) ) ) ) ) ).
% additive_abelian_group.Ruzsa_distance_def
thf(fact_116_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_117_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_118_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_119_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_120_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_121_real__sqrt__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% real_sqrt_le_iff
thf(fact_122_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_123_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_124_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_125_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_126_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_127_times__divide__eq__right,axiom,
! [A: real,B4: real,C2: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B4 @ C2 ) )
= ( divide_divide_real @ ( times_times_real @ A @ B4 ) @ C2 ) ) ).
% times_divide_eq_right
thf(fact_128_divide__divide__eq__right,axiom,
! [A: real,B4: real,C2: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B4 @ C2 ) )
= ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ B4 ) ) ).
% divide_divide_eq_right
thf(fact_129_divide__divide__eq__left,axiom,
! [A: real,B4: real,C2: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B4 ) @ C2 )
= ( divide_divide_real @ A @ ( times_times_real @ B4 @ C2 ) ) ) ).
% divide_divide_eq_left
thf(fact_130_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_131_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_132_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_133_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_134_empty__iff,axiom,
! [C2: real] :
~ ( member_real @ C2 @ bot_bot_set_real ) ).
% empty_iff
thf(fact_135_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_136_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_137_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_138_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_139_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_140_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_141_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_142_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_143_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_144_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_145_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_146_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_147_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_148_real__sqrt__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ X )
= ( sqrt @ Y ) )
= ( X = Y ) ) ).
% real_sqrt_eq_iff
thf(fact_149_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_150_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_151_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_152_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_153_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_154_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_155_divide__eq__0__iff,axiom,
! [A: real,B4: real] :
( ( ( divide_divide_real @ A @ B4 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B4 = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_156_divide__cancel__left,axiom,
! [C2: real,A: real,B4: real] :
( ( ( divide_divide_real @ C2 @ A )
= ( divide_divide_real @ C2 @ B4 ) )
= ( ( C2 = zero_zero_real )
| ( A = B4 ) ) ) ).
% divide_cancel_left
thf(fact_157_divide__cancel__right,axiom,
! [A: real,C2: real,B4: real] :
( ( ( divide_divide_real @ A @ C2 )
= ( divide_divide_real @ B4 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A = B4 ) ) ) ).
% divide_cancel_right
thf(fact_158_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_159_times__divide__eq__left,axiom,
! [B4: real,C2: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B4 @ C2 ) @ A )
= ( divide_divide_real @ ( times_times_real @ B4 @ A ) @ C2 ) ) ).
% times_divide_eq_left
thf(fact_160_singletonI,axiom,
! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).
% singletonI
thf(fact_161_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_162_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_163_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_164_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_165_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_166_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_167_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_168_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_169_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_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_178_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_179_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_180_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_181_mult__divide__mult__cancel__left__if,axiom,
! [C2: real,A: real,B4: real] :
( ( ( C2 = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) )
= zero_zero_real ) )
& ( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) )
= ( divide_divide_real @ A @ B4 ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_182_nonzero__mult__divide__mult__cancel__left,axiom,
! [C2: real,A: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) )
= ( divide_divide_real @ A @ B4 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_183_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C2: real,A: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ B4 @ C2 ) )
= ( divide_divide_real @ A @ B4 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_184_nonzero__mult__divide__mult__cancel__right,axiom,
! [C2: real,A: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) )
= ( divide_divide_real @ A @ B4 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_185_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C2: real,A: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ C2 @ B4 ) )
= ( divide_divide_real @ A @ B4 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_186_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_187_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_188_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_189_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_190_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_191_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_192_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_193_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_194_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_195_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_196_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_197_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_198_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_199_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_200_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_201_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_202_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_203_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_204_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_205_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_206_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_207_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_208_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_209_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_210_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_211_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_212_minus__minusset,axiom,
! [A2: set_a] :
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
= ( inf_inf_set_a @ A2 @ g ) ) ).
% minus_minusset
thf(fact_213_card__minusset,axiom,
! [A2: set_a] :
( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ g ) ) ) ).
% card_minusset
thf(fact_214_minusset__is__empty__iff,axiom,
! [A2: set_a] :
( ( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A2 )
= bot_bot_set_a )
= ( ( inf_inf_set_a @ A2 @ g )
= bot_bot_set_a ) ) ).
% minusset_is_empty_iff
thf(fact_215_minusset__triv,axiom,
( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( insert_a @ zero @ bot_bot_set_a ) )
= ( insert_a @ zero @ bot_bot_set_a ) ) ).
% minusset_triv
thf(fact_216_sumset__iterated__0,axiom,
! [A2: set_a] :
( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ zero_zero_nat )
= ( insert_a @ zero @ bot_bot_set_a ) ) ).
% sumset_iterated_0
thf(fact_217_sumset__D_I2_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ zero @ bot_bot_set_a ) @ A2 )
= ( inf_inf_set_a @ A2 @ g ) ) ).
% sumset_D(2)
thf(fact_218_sumset__D_I1_J,axiom,
! [A2: set_a] :
( ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( insert_a @ zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A2 @ g ) ) ).
% sumset_D(1)
thf(fact_219_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B6: set_a] :
( ( A2
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_220_mk__disjoint__insert,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ? [B6: set_real] :
( ( A2
= ( insert_real @ A @ B6 ) )
& ~ ( member_real @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_221_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_222_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_223_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_224_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_225_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_226_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_227_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_228_insert__eq__iff,axiom,
! [A: a,A2: set_a,B4: a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B4 @ B )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B4 @ B ) )
= ( ( ( A = B4 )
=> ( A2 = B ) )
& ( ( A != B4 )
=> ? [C3: set_a] :
( ( A2
= ( insert_a @ B4 @ C3 ) )
& ~ ( member_a @ B4 @ C3 )
& ( B
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_229_insert__eq__iff,axiom,
! [A: real,A2: set_real,B4: real,B: set_real] :
( ~ ( member_real @ A @ A2 )
=> ( ~ ( member_real @ B4 @ B )
=> ( ( ( insert_real @ A @ A2 )
= ( insert_real @ B4 @ B ) )
= ( ( ( A = B4 )
=> ( A2 = B ) )
& ( ( A != B4 )
=> ? [C3: set_real] :
( ( A2
= ( insert_real @ B4 @ C3 ) )
& ~ ( member_real @ B4 @ C3 )
& ( B
= ( insert_real @ A @ C3 ) )
& ~ ( member_real @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_230_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_231_insert__absorb,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ( ( insert_real @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_232_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_233_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_234_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B7: set_a] : ( inf_inf_set_a @ B7 @ A6 ) ) ) ).
% Int_commute
thf(fact_235_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B6: set_a] :
( ( A2
= ( insert_a @ X @ B6 ) )
=> ( member_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_236_Set_Oset__insert,axiom,
! [X: real,A2: set_real] :
( ( member_real @ X @ A2 )
=> ~ ! [B6: set_real] :
( ( A2
= ( insert_real @ X @ B6 ) )
=> ( member_real @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_237_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_238_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_239_insertI2,axiom,
! [A: a,B: set_a,B4: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ B4 @ B ) ) ) ).
% insertI2
thf(fact_240_insertI2,axiom,
! [A: real,B: set_real,B4: real] :
( ( member_real @ A @ B )
=> ( member_real @ A @ ( insert_real @ B4 @ B ) ) ) ).
% insertI2
thf(fact_241_insertI1,axiom,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).
% insertI1
thf(fact_242_insertI1,axiom,
! [A: real,B: set_real] : ( member_real @ A @ ( insert_real @ A @ B ) ) ).
% insertI1
thf(fact_243_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_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_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_252_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_253_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_254_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_255_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_256_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_257_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_258_singleton__inject,axiom,
! [A: a,B4: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B4 @ bot_bot_set_a ) )
=> ( A = B4 ) ) ).
% singleton_inject
thf(fact_259_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_260_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_261_singleton__iff,axiom,
! [B4: real,A: real] :
( ( member_real @ B4 @ ( insert_real @ A @ bot_bot_set_real ) )
= ( B4 = A ) ) ).
% singleton_iff
thf(fact_262_singleton__iff,axiom,
! [B4: a,A: a] :
( ( member_a @ B4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B4 = A ) ) ).
% singleton_iff
thf(fact_263_singletonD,axiom,
! [B4: real,A: real] :
( ( member_real @ B4 @ ( insert_real @ A @ bot_bot_set_real ) )
=> ( B4 = A ) ) ).
% singletonD
thf(fact_264_singletonD,axiom,
! [B4: a,A: a] :
( ( member_a @ B4 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B4 = A ) ) ).
% singletonD
thf(fact_265_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_266_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_267_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_268_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_269_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_270_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_271_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_272_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_273_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_274_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).
% subset_insertI
thf(fact_275_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_276_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_277_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_278_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_279_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_280_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_281_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_282_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_283_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_284_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_285_divide__right__mono__neg,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B4 @ C2 ) @ ( divide_divide_real @ A @ C2 ) ) ) ) ).
% divide_right_mono_neg
thf(fact_286_divide__nonpos__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_287_divide__nonpos__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_288_divide__nonneg__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_289_divide__nonneg__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_290_zero__le__divide__iff,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B4 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_291_divide__right__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B4 @ C2 ) ) ) ) ).
% divide_right_mono
thf(fact_292_divide__le__0__iff,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B4 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) ) ) ) ).
% divide_le_0_iff
thf(fact_293_frac__eq__eq,axiom,
! [Y: real,Z: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W2 @ Z ) )
= ( ( times_times_real @ X @ Z )
= ( times_times_real @ W2 @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_294_divide__eq__eq,axiom,
! [B4: real,C2: real,A: real] :
( ( ( divide_divide_real @ B4 @ C2 )
= A )
= ( ( ( C2 != zero_zero_real )
=> ( B4
= ( times_times_real @ A @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_295_eq__divide__eq,axiom,
! [A: real,B4: real,C2: real] :
( ( A
= ( divide_divide_real @ B4 @ C2 ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ A @ C2 )
= B4 ) )
& ( ( C2 = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_296_divide__eq__imp,axiom,
! [C2: real,B4: real,A: real] :
( ( C2 != zero_zero_real )
=> ( ( B4
= ( times_times_real @ A @ C2 ) )
=> ( ( divide_divide_real @ B4 @ C2 )
= A ) ) ) ).
% divide_eq_imp
thf(fact_297_eq__divide__imp,axiom,
! [C2: real,A: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ C2 )
= B4 )
=> ( A
= ( divide_divide_real @ B4 @ C2 ) ) ) ) ).
% eq_divide_imp
thf(fact_298_nonzero__divide__eq__eq,axiom,
! [C2: real,B4: real,A: real] :
( ( C2 != zero_zero_real )
=> ( ( ( divide_divide_real @ B4 @ C2 )
= A )
= ( B4
= ( times_times_real @ A @ C2 ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_299_nonzero__eq__divide__eq,axiom,
! [C2: real,A: real,B4: real] :
( ( C2 != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B4 @ C2 ) )
= ( ( times_times_real @ A @ C2 )
= B4 ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_300_additive__abelian__group_Osumset__D_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ Zero @ bot_bot_set_a ) )
= ( inf_inf_set_a @ A2 @ G ) ) ) ).
% additive_abelian_group.sumset_D(1)
thf(fact_301_additive__abelian__group_Osumset__D_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ Zero @ bot_bot_set_a ) @ A2 )
= ( inf_inf_set_a @ A2 @ G ) ) ) ).
% additive_abelian_group.sumset_D(2)
thf(fact_302_additive__abelian__group_Ocard__sumset__0__iff_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ ( inf_inf_set_a @ A2 @ G ) )
= zero_zero_nat )
| ( ( finite_card_a @ ( inf_inf_set_a @ B @ G ) )
= zero_zero_nat ) ) ) ) ).
% additive_abelian_group.card_sumset_0_iff'
thf(fact_303_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
! [G: set_real,Addition: real > real > real,Zero: real,A2: set_real,A: real] :
( ( pluenn1014277435162747966p_real @ G @ Addition @ Zero )
=> ( ( finite_finite_real @ A2 )
=> ( ( ( member_real @ A @ G )
=> ( ( finite_card_real @ ( pluenn7361685508355272389t_real @ G @ Addition @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= ( finite_card_real @ ( inf_inf_set_real @ A2 @ G ) ) ) )
& ( ~ ( member_real @ A @ G )
=> ( ( finite_card_real @ ( pluenn7361685508355272389t_real @ G @ Addition @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
= zero_zero_nat ) ) ) ) ) ).
% additive_abelian_group.card_sumset_singleton_eq
thf(fact_304_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ A @ G )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ G ) ) ) )
& ( ~ ( member_a @ A @ G )
=> ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= zero_zero_nat ) ) ) ) ) ).
% additive_abelian_group.card_sumset_singleton_eq
thf(fact_305_additive__abelian__group_Osumset__iterated__0,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A2 @ zero_zero_nat )
= ( insert_a @ Zero @ bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_iterated_0
thf(fact_306_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_307_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_308_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_309_additive__abelian__group_Osumset__Int__carrier__eq_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( inf_inf_set_a @ A2 @ G ) @ B )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier_eq(2)
thf(fact_310_additive__abelian__group_Osumset__Int__carrier__eq_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( inf_inf_set_a @ B @ G ) )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier_eq(1)
thf(fact_311_additive__abelian__group_Osumset__Int__carrier,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ G )
= ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ).
% additive_abelian_group.sumset_Int_carrier
thf(fact_312_additive__abelian__group_Ominus__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) )
= ( inf_inf_set_a @ A2 @ G ) ) ) ).
% additive_abelian_group.minus_minusset
thf(fact_313_additive__abelian__group_Osumset__is__empty__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= bot_bot_set_a )
= ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ G )
= bot_bot_set_a ) ) ) ) ).
% additive_abelian_group.sumset_is_empty_iff
thf(fact_314_additive__abelian__group_Osumset__empty_H_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_empty'(1)
thf(fact_315_additive__abelian__group_Osumset__empty_H_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_empty'(2)
thf(fact_316_additive__abelian__group_Ofinite__sumset_H,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ G ) )
=> ( ( finite_finite_a @ ( inf_inf_set_a @ B @ G ) )
=> ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) ) ) ) ).
% additive_abelian_group.finite_sumset'
thf(fact_317_additive__abelian__group_Osumset__subset__insert_I1_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ B ) ) ) ) ).
% additive_abelian_group.sumset_subset_insert(1)
thf(fact_318_additive__abelian__group_Osumset__subset__insert_I2_J,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A2 ) @ B ) ) ) ).
% additive_abelian_group.sumset_subset_insert(2)
thf(fact_319_ex__in__conv,axiom,
! [A2: set_real] :
( ( ? [X2: real] : ( member_real @ X2 @ A2 ) )
= ( A2 != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_320_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_321_equals0I,axiom,
! [A2: set_real] :
( ! [Y2: real] :
~ ( member_real @ Y2 @ A2 )
=> ( A2 = bot_bot_set_real ) ) ).
% equals0I
thf(fact_322_equals0I,axiom,
! [A2: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_323_equals0D,axiom,
! [A2: set_real,A: real] :
( ( A2 = bot_bot_set_real )
=> ~ ( member_real @ A @ A2 ) ) ).
% equals0D
thf(fact_324_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_325_emptyE,axiom,
! [A: real] :
~ ( member_real @ A @ bot_bot_set_real ) ).
% emptyE
thf(fact_326_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_327_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_328_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A6: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ A6 @ B7 )
& ( ord_less_eq_set_a @ B7 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_329_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_330_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_331_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_332_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B7: set_real] :
! [T: real] :
( ( member_real @ T @ A6 )
=> ( member_real @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_333_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B7: set_a] :
! [T: a] :
( ( member_a @ T @ A6 )
=> ( member_a @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_334_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_335_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_336_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B7: set_real] :
! [X2: real] :
( ( member_real @ X2 @ A6 )
=> ( member_real @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_337_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B7: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( member_a @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_338_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_339_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_340_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_341_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_342_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_343_additive__abelian__group_Ominusset__is__empty__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 )
= bot_bot_set_a )
= ( ( inf_inf_set_a @ A2 @ G )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.minusset_is_empty_iff
thf(fact_344_additive__abelian__group_Ominusset__triv,axiom,
! [G: set_a,Addition: a > a > a,Zero: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( insert_a @ Zero @ bot_bot_set_a ) )
= ( insert_a @ Zero @ bot_bot_set_a ) ) ) ).
% additive_abelian_group.minusset_triv
thf(fact_345_additive__abelian__group_Ocard__minusset,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A2 ) )
= ( finite_card_a @ ( inf_inf_set_a @ A2 @ G ) ) ) ) ).
% additive_abelian_group.card_minusset
thf(fact_346_additive__abelian__group_Oinfinite__sumset__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) )
= ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ G ) )
& ( ( inf_inf_set_a @ B @ G )
!= bot_bot_set_a ) )
| ( ( ( inf_inf_set_a @ A2 @ G )
!= bot_bot_set_a )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ B @ G ) ) ) ) ) ) ).
% additive_abelian_group.infinite_sumset_iff
thf(fact_347_additive__abelian__group_Oinfinite__sumset__aux,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A2 @ G ) )
=> ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) ) )
= ( ( inf_inf_set_a @ B @ G )
!= bot_bot_set_a ) ) ) ) ).
% additive_abelian_group.infinite_sumset_aux
thf(fact_348_additive__abelian__group_Ocard__sumset__0__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_eq_set_a @ A2 @ G )
=> ( ( ord_less_eq_set_a @ B @ G )
=> ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ B ) )
= zero_zero_nat )
= ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
| ( ( finite_card_a @ B )
= zero_zero_nat ) ) ) ) ) ) ).
% additive_abelian_group.card_sumset_0_iff
thf(fact_349_additive__abelian__group_Ocard__sumset__le,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,A: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( finite_finite_a @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ).
% additive_abelian_group.card_sumset_le
thf(fact_350_times__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W2: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W2 ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W2 ) ) ) ).
% times_divide_times_eq
thf(fact_351_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W2: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W2 ) )
= ( divide_divide_real @ ( times_times_real @ X @ W2 ) @ ( times_times_real @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_352_divide__divide__eq__left_H,axiom,
! [A: real,B4: real,C2: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B4 ) @ C2 )
= ( divide_divide_real @ A @ ( times_times_real @ C2 @ B4 ) ) ) ).
% divide_divide_eq_left'
thf(fact_353_real__sqrt__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_le_mono
thf(fact_354_real__sqrt__divide,axiom,
! [X: real,Y: real] :
( ( sqrt @ ( divide_divide_real @ X @ Y ) )
= ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_divide
thf(fact_355_real__sqrt__mult,axiom,
! [X: real,Y: real] :
( ( sqrt @ ( times_times_real @ X @ Y ) )
= ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_mult
thf(fact_356_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_357_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_358_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_359_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_360_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_361_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_362_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_363_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_364_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_365_div__mult__mult1,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) )
= ( divide_divide_nat @ A @ B4 ) ) ) ).
% div_mult_mult1
thf(fact_366_div__mult__mult1,axiom,
! [C2: int,A: int,B4: int] :
( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) )
= ( divide_divide_int @ A @ B4 ) ) ) ).
% div_mult_mult1
thf(fact_367_div__mult__mult2,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) )
= ( divide_divide_nat @ A @ B4 ) ) ) ).
% div_mult_mult2
thf(fact_368_div__mult__mult2,axiom,
! [C2: int,A: int,B4: int] :
( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ C2 ) )
= ( divide_divide_int @ A @ B4 ) ) ) ).
% div_mult_mult2
thf(fact_369_div__mult__mult1__if,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ( C2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) )
= zero_zero_nat ) )
& ( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) )
= ( divide_divide_nat @ A @ B4 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_370_div__mult__mult1__if,axiom,
! [C2: int,A: int,B4: int] :
( ( ( C2 = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) )
= zero_zero_int ) )
& ( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) )
= ( divide_divide_int @ A @ B4 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_371_sumset__iterated__empty,axiom,
! [R: nat] :
( ( ord_less_nat @ zero_zero_nat @ R )
=> ( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ bot_bot_set_a @ R )
= bot_bot_set_a ) ) ).
% sumset_iterated_empty
thf(fact_372_real__sqrt__eq__zero__cancel__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% real_sqrt_eq_zero_cancel_iff
thf(fact_373_real__sqrt__zero,axiom,
( ( sqrt @ zero_zero_real )
= zero_zero_real ) ).
% real_sqrt_zero
thf(fact_374_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_375_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_376_mult__cancel__right,axiom,
! [A: int,C2: int,B4: int] :
( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B4 @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A = B4 ) ) ) ).
% mult_cancel_right
thf(fact_377_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_378_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_379_mult__cancel__left,axiom,
! [C2: int,A: int,B4: int] :
( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B4 ) )
= ( ( C2 = zero_zero_int )
| ( A = B4 ) ) ) ).
% mult_cancel_left
thf(fact_380_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_381_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_382_mult__eq__0__iff,axiom,
! [A: int,B4: int] :
( ( ( times_times_int @ A @ B4 )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B4 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_383_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_384_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_385_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_386_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_387_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_388_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_389_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_390_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_391_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_392_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_393_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_394_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_395_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_396_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_397_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_398_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_399_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_400_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_401_real__sqrt__ge__0__iff,axiom,
! [Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
= ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).
% real_sqrt_ge_0_iff
thf(fact_402_real__sqrt__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% real_sqrt_le_0_iff
thf(fact_403_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_404_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_405_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_406_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_407_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_408_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_409_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_410_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_411_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_412_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_413_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_414_linordered__field__no__ub,axiom,
! [X5: real] :
? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_415_linordered__field__no__lb,axiom,
! [X5: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X5 ) ).
% linordered_field_no_lb
thf(fact_416_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_417_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_418_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_419_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_420_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_421_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_422_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_423_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_424_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_425_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_426_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_427_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_428_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_429_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_430_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_431_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_432_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_433_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_434_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_435_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_436_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_437_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_438_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_439_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_440_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_441_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_442_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_443_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_444_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_445_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_446_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_447_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_448_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_449_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_int @ A @ B4 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_450_mult__less__cancel__right__disj,axiom,
! [A: real,C2: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A @ B4 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B4 @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_451_mult__less__cancel__right__disj,axiom,
! [A: int,C2: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A @ B4 ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B4 @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_452_mult__strict__right__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_453_mult__strict__right__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_454_mult__strict__right__mono,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_int @ A @ B4 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_455_mult__strict__right__mono__neg,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_real @ B4 @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_456_mult__strict__right__mono__neg,axiom,
! [B4: int,A: int,C2: int] :
( ( ord_less_int @ B4 @ A )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_457_mult__less__cancel__left__disj,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A @ B4 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B4 @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_458_mult__less__cancel__left__disj,axiom,
! [C2: int,A: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A @ B4 ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B4 @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_459_mult__strict__left__mono,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_460_mult__strict__left__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_461_mult__strict__left__mono,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_int @ A @ B4 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_462_mult__strict__left__mono__neg,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_real @ B4 @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_463_mult__strict__left__mono__neg,axiom,
! [B4: int,A: int,C2: int] :
( ( ord_less_int @ B4 @ A )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_464_mult__less__cancel__left__pos,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) )
= ( ord_less_real @ A @ B4 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_465_mult__less__cancel__left__pos,axiom,
! [C2: int,A: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) )
= ( ord_less_int @ A @ B4 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_466_mult__less__cancel__left__neg,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) )
= ( ord_less_real @ B4 @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_467_mult__less__cancel__left__neg,axiom,
! [C2: int,A: int,B4: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) )
= ( ord_less_int @ B4 @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_468_zero__less__mult__pos2,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B4 @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B4 ) ) ) ).
% zero_less_mult_pos2
thf(fact_469_zero__less__mult__pos2,axiom,
! [B4: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B4 @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B4 ) ) ) ).
% zero_less_mult_pos2
thf(fact_470_zero__less__mult__pos2,axiom,
! [B4: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B4 @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B4 ) ) ) ).
% zero_less_mult_pos2
thf(fact_471_zero__less__mult__pos,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B4 ) ) ) ).
% zero_less_mult_pos
thf(fact_472_zero__less__mult__pos,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B4 ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B4 ) ) ) ).
% zero_less_mult_pos
thf(fact_473_zero__less__mult__pos,axiom,
! [A: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B4 ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B4 ) ) ) ).
% zero_less_mult_pos
thf(fact_474_zero__less__mult__iff,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B4 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B4 @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_475_zero__less__mult__iff,axiom,
! [A: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B4 ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B4 ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B4 @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_476_mult__pos__neg2,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B4 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B4 @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_477_mult__pos__neg2,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B4 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B4 @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_478_mult__pos__neg2,axiom,
! [A: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B4 @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_479_mult__pos__pos,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B4 ) ) ) ) ).
% mult_pos_pos
thf(fact_480_mult__pos__pos,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B4 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B4 ) ) ) ) ).
% mult_pos_pos
thf(fact_481_mult__pos__pos,axiom,
! [A: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B4 ) ) ) ) ).
% mult_pos_pos
thf(fact_482_mult__pos__neg,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B4 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_483_mult__pos__neg,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B4 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B4 ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_484_mult__pos__neg,axiom,
! [A: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B4 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_485_mult__neg__pos,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ord_less_nat @ ( times_times_nat @ A @ B4 ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_486_mult__neg__pos,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B4 )
=> ( ord_less_real @ ( times_times_real @ A @ B4 ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_487_mult__neg__pos,axiom,
! [A: int,B4: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ord_less_int @ ( times_times_int @ A @ B4 ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_488_mult__less__0__iff,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ A @ B4 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B4 @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B4 ) ) ) ) ).
% mult_less_0_iff
thf(fact_489_mult__less__0__iff,axiom,
! [A: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ A @ B4 ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B4 @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B4 ) ) ) ) ).
% mult_less_0_iff
thf(fact_490_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_491_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_492_mult__neg__neg,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B4 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B4 ) ) ) ) ).
% mult_neg_neg
thf(fact_493_mult__neg__neg,axiom,
! [A: int,B4: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B4 ) ) ) ) ).
% mult_neg_neg
thf(fact_494_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_495_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_496_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_497_mult__le__cancel__left,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B4 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B4 @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_498_mult__le__cancel__left,axiom,
! [C2: int,A: int,B4: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B4 ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B4 @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_499_mult__le__cancel__right,axiom,
! [A: real,C2: real,B4: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B4 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B4 @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_500_mult__le__cancel__right,axiom,
! [A: int,C2: int,B4: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B4 ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B4 @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_501_mult__left__less__imp__less,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B4 ) ) ) ).
% mult_left_less_imp_less
thf(fact_502_mult__left__less__imp__less,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A @ B4 ) ) ) ).
% mult_left_less_imp_less
thf(fact_503_mult__left__less__imp__less,axiom,
! [C2: int,A: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B4 ) ) ) ).
% mult_left_less_imp_less
thf(fact_504_mult__strict__mono,axiom,
! [A: real,B4: real,C2: real,D: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_real @ zero_zero_real @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_505_mult__strict__mono,axiom,
! [A: nat,B4: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_506_mult__strict__mono,axiom,
! [A: int,B4: int,C2: int,D: int] :
( ( ord_less_int @ A @ B4 )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_507_mult__less__cancel__left,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B4 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B4 @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_508_mult__less__cancel__left,axiom,
! [C2: int,A: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B4 ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B4 @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_509_mult__right__less__imp__less,axiom,
! [A: real,C2: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B4 ) ) ) ).
% mult_right_less_imp_less
thf(fact_510_mult__right__less__imp__less,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A @ B4 ) ) ) ).
% mult_right_less_imp_less
thf(fact_511_mult__right__less__imp__less,axiom,
! [A: int,C2: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ C2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B4 ) ) ) ).
% mult_right_less_imp_less
thf(fact_512_mult__strict__mono_H,axiom,
! [A: real,B4: real,C2: real,D: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_513_mult__strict__mono_H,axiom,
! [A: nat,B4: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_514_mult__strict__mono_H,axiom,
! [A: int,B4: int,C2: int,D: int] :
( ( ord_less_int @ A @ B4 )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_515_mult__less__cancel__right,axiom,
! [A: real,C2: real,B4: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B4 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B4 @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_516_mult__less__cancel__right,axiom,
! [A: int,C2: int,B4: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ C2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B4 ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B4 @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_517_mult__le__cancel__left__neg,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) )
= ( ord_less_eq_real @ B4 @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_518_mult__le__cancel__left__neg,axiom,
! [C2: int,A: int,B4: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) )
= ( ord_less_eq_int @ B4 @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_519_mult__le__cancel__left__pos,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) )
= ( ord_less_eq_real @ A @ B4 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_520_mult__le__cancel__left__pos,axiom,
! [C2: int,A: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) )
= ( ord_less_eq_int @ A @ B4 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_521_mult__left__le__imp__le,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B4 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B4 ) ) ) ).
% mult_left_le_imp_le
thf(fact_522_mult__left__le__imp__le,axiom,
! [C2: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A @ B4 ) ) ) ).
% mult_left_le_imp_le
thf(fact_523_mult__left__le__imp__le,axiom,
! [C2: int,A: int,B4: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B4 ) ) ) ).
% mult_left_le_imp_le
thf(fact_524_mult__right__le__imp__le,axiom,
! [A: real,C2: real,B4: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ C2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B4 ) ) ) ).
% mult_right_le_imp_le
thf(fact_525_mult__right__le__imp__le,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A @ B4 ) ) ) ).
% mult_right_le_imp_le
thf(fact_526_mult__right__le__imp__le,axiom,
! [A: int,C2: int,B4: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ C2 ) )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B4 ) ) ) ).
% mult_right_le_imp_le
thf(fact_527_mult__le__less__imp__less,axiom,
! [A: real,B4: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_528_mult__le__less__imp__less,axiom,
! [A: nat,B4: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_529_mult__le__less__imp__less,axiom,
! [A: int,B4: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_530_mult__less__le__imp__less,axiom,
! [A: real,B4: real,C2: real,D: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B4 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_531_mult__less__le__imp__less,axiom,
! [A: nat,B4: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B4 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_532_mult__less__le__imp__less,axiom,
! [A: int,B4: int,C2: int,D: int] :
( ( ord_less_int @ A @ B4 )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_533_divide__neg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_534_divide__neg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_535_divide__pos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_536_divide__pos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_537_divide__less__0__iff,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B4 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B4 @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B4 ) ) ) ) ).
% divide_less_0_iff
thf(fact_538_divide__less__cancel,axiom,
! [A: real,C2: real,B4: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B4 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B4 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B4 @ A ) )
& ( C2 != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_539_zero__less__divide__iff,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B4 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B4 @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_540_divide__strict__right__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B4 @ C2 ) ) ) ) ).
% divide_strict_right_mono
thf(fact_541_divide__strict__right__mono__neg,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_real @ B4 @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B4 @ C2 ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_542_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_543_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_544_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_545_card__ge__0__finite,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ( finite_finite_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_546_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_547_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_548_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_549_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_550_real__sqrt__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_zero
thf(fact_551_real__sqrt__eq__zero__cancel,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( sqrt @ X )
= zero_zero_real )
=> ( X = zero_zero_real ) ) ) ).
% real_sqrt_eq_zero_cancel
thf(fact_552_card__gt__0__iff,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
= ( ( A2 != bot_bot_set_a )
& ( finite_finite_a @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_553_frac__le,axiom,
! [Y: real,X: real,W2: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_le
thf(fact_554_frac__less,axiom,
! [X: real,Y: real,W2: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_less
thf(fact_555_frac__less2,axiom,
! [X: real,Y: real,W2: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_real @ W2 @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_less2
thf(fact_556_divide__le__cancel,axiom,
! [A: real,C2: real,B4: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B4 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B4 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B4 @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_557_divide__nonneg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_558_divide__nonneg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_pos
thf(fact_559_divide__nonpos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_neg
thf(fact_560_divide__nonpos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_561_divide__strict__left__mono__neg,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B4 ) )
=> ( ord_less_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B4 ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_562_divide__strict__left__mono,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_real @ B4 @ A )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B4 ) )
=> ( ord_less_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B4 ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_563_mult__imp__less__div__pos,axiom,
! [Y: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
=> ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_564_mult__imp__div__pos__less,axiom,
! [Y: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_565_pos__less__divide__eq,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B4 @ C2 ) )
= ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B4 ) ) ) ).
% pos_less_divide_eq
thf(fact_566_pos__divide__less__eq,axiom,
! [C2: real,B4: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( divide_divide_real @ B4 @ C2 ) @ A )
= ( ord_less_real @ B4 @ ( times_times_real @ A @ C2 ) ) ) ) ).
% pos_divide_less_eq
thf(fact_567_neg__less__divide__eq,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B4 @ C2 ) )
= ( ord_less_real @ B4 @ ( times_times_real @ A @ C2 ) ) ) ) ).
% neg_less_divide_eq
thf(fact_568_neg__divide__less__eq,axiom,
! [C2: real,B4: real,A: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B4 @ C2 ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B4 ) ) ) ).
% neg_divide_less_eq
thf(fact_569_less__divide__eq,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_real @ A @ ( divide_divide_real @ B4 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B4 ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B4 @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_570_divide__less__eq,axiom,
! [B4: real,C2: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B4 @ C2 ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ B4 @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B4 ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_571_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_572_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_573_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_574_real__div__sqrt,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( divide_divide_real @ X @ ( sqrt @ X ) )
= ( sqrt @ X ) ) ) ).
% real_div_sqrt
thf(fact_575_divide__left__mono__neg,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B4 ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B4 ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_576_mult__imp__le__div__pos,axiom,
! [Y: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X )
=> ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_577_mult__imp__div__pos__le,axiom,
! [Y: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_le
thf(fact_578_pos__le__divide__eq,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B4 @ C2 ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B4 ) ) ) ).
% pos_le_divide_eq
thf(fact_579_pos__divide__le__eq,axiom,
! [C2: real,B4: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B4 @ C2 ) @ A )
= ( ord_less_eq_real @ B4 @ ( times_times_real @ A @ C2 ) ) ) ) ).
% pos_divide_le_eq
thf(fact_580_neg__le__divide__eq,axiom,
! [C2: real,A: real,B4: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B4 @ C2 ) )
= ( ord_less_eq_real @ B4 @ ( times_times_real @ A @ C2 ) ) ) ) ).
% neg_le_divide_eq
thf(fact_581_neg__divide__le__eq,axiom,
! [C2: real,B4: real,A: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B4 @ C2 ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B4 ) ) ) ).
% neg_divide_le_eq
thf(fact_582_divide__left__mono,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B4 ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B4 ) ) ) ) ) ).
% divide_left_mono
thf(fact_583_le__divide__eq,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B4 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B4 ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B4 @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_584_divide__le__eq,axiom,
! [B4: real,C2: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B4 @ C2 ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ B4 @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B4 ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_585_additive__abelian__group_Osumset__iterated__empty,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,R: nat] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( ord_less_nat @ zero_zero_nat @ R )
=> ( ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ bot_bot_set_a @ R )
= bot_bot_set_a ) ) ) ).
% additive_abelian_group.sumset_iterated_empty
thf(fact_586_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_587_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_588_mult__right__cancel,axiom,
! [C2: int,A: int,B4: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B4 @ C2 ) )
= ( A = B4 ) ) ) ).
% mult_right_cancel
thf(fact_589_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_590_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_591_mult__left__cancel,axiom,
! [C2: int,A: int,B4: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B4 ) )
= ( A = B4 ) ) ) ).
% mult_left_cancel
thf(fact_592_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_593_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_594_no__zero__divisors,axiom,
! [A: int,B4: int] :
( ( A != zero_zero_int )
=> ( ( B4 != zero_zero_int )
=> ( ( times_times_int @ A @ B4 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_595_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_596_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_597_divisors__zero,axiom,
! [A: int,B4: int] :
( ( ( times_times_int @ A @ B4 )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B4 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_598_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_599_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_600_mult__not__zero,axiom,
! [A: int,B4: int] :
( ( ( times_times_int @ A @ B4 )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B4 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_601_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_602_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_603_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_604_finite__has__maximal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ord_less_eq_int @ A @ X3 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_605_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_606_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_607_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_608_finite__has__minimal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ord_less_eq_int @ X3 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_609_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_610_infinite__imp__nonempty,axiom,
! [S2: set_a] :
( ~ ( finite_finite_a @ S2 )
=> ( S2 != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_611_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_612_infinite__super,axiom,
! [S2: set_a,T3: set_a] :
( ( ord_less_eq_set_a @ S2 @ T3 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ T3 ) ) ) ).
% infinite_super
thf(fact_613_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_614_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_615_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_616_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_617_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_618_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_619_mult__mono,axiom,
! [A: int,B4: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_620_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_621_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_622_mult__mono_H,axiom,
! [A: int,B4: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_623_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_624_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_625_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_626_split__mult__pos__le,axiom,
! [A: int,B4: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B4 ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B4 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B4 ) ) ) ).
% split_mult_pos_le
thf(fact_627_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_628_mult__left__mono__neg,axiom,
! [B4: int,A: int,C2: int] :
( ( ord_less_eq_int @ B4 @ A )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_629_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_630_mult__nonpos__nonpos,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B4 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B4 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_631_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_632_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_633_mult__left__mono,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) ) ) ) ).
% mult_left_mono
thf(fact_634_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_635_mult__right__mono__neg,axiom,
! [B4: int,A: int,C2: int] :
( ( ord_less_eq_int @ B4 @ A )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_636_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_637_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_638_mult__right__mono,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_639_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_640_mult__le__0__iff,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B4 ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B4 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B4 ) ) ) ) ).
% mult_le_0_iff
thf(fact_641_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_642_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_643_split__mult__neg__le,axiom,
! [A: int,B4: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B4 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B4 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B4 ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_644_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_645_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_646_mult__nonneg__nonneg,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B4 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_647_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_648_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_649_mult__nonneg__nonpos,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B4 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B4 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_650_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_651_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_652_mult__nonpos__nonneg,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B4 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B4 ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_653_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_654_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_655_mult__nonneg__nonpos2,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B4 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B4 @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_656_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_657_zero__le__mult__iff,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B4 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B4 ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B4 @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_658_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_659_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_660_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B4 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_661_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_662_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_663_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_664_finite__has__maximal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_665_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_666_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_667_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_668_finite__has__minimal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_669_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_670_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_671_finite__induct,axiom,
! [F: set_real,P: set_real > $o] :
( ( finite_finite_real @ F )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ~ ( member_real @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X3 @ F3 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_672_finite__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_673_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,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( F3 != bot_bot_set_real )
=> ( ~ ( member_real @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_674_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,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_675_infinite__finite__induct,axiom,
! [P: set_real > $o,A2: set_real] :
( ! [A7: set_real] :
( ~ ( finite_finite_real @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X3: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ~ ( member_real @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_676_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A7: set_a] :
( ~ ( finite_finite_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_677_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_678_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ( finite_card_a @ B6 )
= N )
& ( ord_less_eq_set_a @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_679_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_680_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,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( member_real @ A3 @ A2 )
=> ( ~ ( member_real @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_681_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,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A2 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_682_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,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( member_real @ A3 @ A2 )
=> ( ( ord_less_eq_set_real @ F3 @ A2 )
=> ( ~ ( member_real @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_683_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,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_684_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_685_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_686_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_687_exists__subset__between,axiom,
! [A2: set_a,N: nat,C: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C ) )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( finite_finite_a @ C )
=> ? [B6: set_a] :
( ( ord_less_eq_set_a @ A2 @ B6 )
& ( ord_less_eq_set_a @ B6 @ C )
& ( ( finite_card_a @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_688_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S2 ) )
=> ~ ! [T4: set_a] :
( ( ord_less_eq_set_a @ T4 @ S2 )
=> ( ( ( finite_card_a @ T4 )
= N )
=> ~ ( finite_finite_a @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_689_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_690_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_691_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_692_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_693_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C2 ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_694_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_695_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_696_psubsetI,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_a @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_697_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_698_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_699_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_700_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_701_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_702_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_703_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_704_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_705_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_706_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_707_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_708_real__sqrt__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% real_sqrt_less_iff
thf(fact_709_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_710_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_711_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_712_inf_Obounded__iff,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B4 @ C2 ) )
= ( ( ord_less_eq_int @ A @ B4 )
& ( ord_less_eq_int @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_713_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_714_le__inf__iff,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z ) )
= ( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_715_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_716_le__inf__iff,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) )
= ( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_717_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_718_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_719_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_720_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_721_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_722_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_723_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_724_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_725_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_726_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_727_real__sqrt__lt__0__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% real_sqrt_lt_0_iff
thf(fact_728_real__sqrt__gt__0__iff,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
= ( ord_less_real @ zero_zero_real @ Y ) ) ).
% real_sqrt_gt_0_iff
thf(fact_729_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_730_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_731_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_732_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_733_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_734_psubset__imp__ex__mem,axiom,
! [A2: set_real,B: set_real] :
( ( ord_less_set_real @ A2 @ B )
=> ? [B2: real] : ( member_real @ B2 @ ( minus_minus_set_real @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_735_psubset__imp__ex__mem,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ? [B2: a] : ( member_a @ B2 @ ( minus_minus_set_a @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_736_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_737_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_738_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_739_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_740_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_741_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_742_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_743_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_744_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_745_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_746_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_747_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_748_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_749_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_750_right__diff__distrib_H,axiom,
! [A: int,B4: int,C2: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B4 @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ A @ B4 ) @ ( times_times_int @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_751_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_752_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_753_left__diff__distrib_H,axiom,
! [B4: int,C2: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B4 @ C2 ) @ A )
= ( minus_minus_int @ ( times_times_int @ B4 @ A ) @ ( times_times_int @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_754_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_755_right__diff__distrib,axiom,
! [A: int,B4: int,C2: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B4 @ C2 ) )
= ( minus_minus_int @ ( times_times_int @ A @ B4 ) @ ( times_times_int @ A @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_756_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_757_left__diff__distrib,axiom,
! [A: int,B4: int,C2: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B4 ) @ C2 )
= ( minus_minus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B4 @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_758_diff__divide__distrib,axiom,
! [A: real,B4: real,C2: real] :
( ( divide_divide_real @ ( minus_minus_real @ A @ B4 ) @ C2 )
= ( minus_minus_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B4 @ C2 ) ) ) ).
% diff_divide_distrib
thf(fact_759_Diff__infinite__finite,axiom,
! [T3: set_a,S2: set_a] :
( ( finite_finite_a @ T3 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_760_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_761_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_762_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_763_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_764_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_765_not__psubset__empty,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_766_finite__psubset__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ! [B8: set_a] :
( ( ord_less_set_a @ B8 @ A7 )
=> ( P @ B8 ) )
=> ( P @ A7 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_767_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B7: set_a] :
( ( ord_less_set_a @ A6 @ B7 )
| ( A6 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_768_subset__psubset__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_769_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ A6 @ B7 )
& ~ ( ord_less_eq_set_a @ B7 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_770_psubset__subset__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_771_psubset__imp__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% psubset_imp_subset
thf(fact_772_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ A6 @ B7 )
& ( A6 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_773_psubsetE,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% psubsetE
thf(fact_774_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_775_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_776_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_777_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_778_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_779_real__sqrt__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_less_mono
thf(fact_780_real__sqrt__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% real_sqrt_gt_zero
thf(fact_781_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y3: real] :
? [N2: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_782_finite__induct__select,axiom,
! [S2: set_a,P: set_a > $o] :
( ( finite_finite_a @ S2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [T4: set_a] :
( ( ord_less_set_a @ T4 @ S2 )
=> ( ( P @ T4 )
=> ? [X5: a] :
( ( member_a @ X5 @ ( minus_minus_set_a @ S2 @ T4 ) )
& ( P @ ( insert_a @ X5 @ T4 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_783_psubset__insert__iff,axiom,
! [A2: set_real,X: real,B: set_real] :
( ( ord_less_set_real @ A2 @ ( insert_real @ X @ B ) )
= ( ( ( member_real @ X @ B )
=> ( ord_less_set_real @ A2 @ B ) )
& ( ~ ( member_real @ X @ B )
=> ( ( ( member_real @ X @ A2 )
=> ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B ) )
& ( ~ ( member_real @ X @ A2 )
=> ( ord_less_eq_set_real @ A2 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_784_psubset__insert__iff,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ B )
=> ( ord_less_set_a @ A2 @ B ) )
& ( ~ ( member_a @ X @ B )
=> ( ( ( member_a @ X @ A2 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_785_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_786_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_787_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_788_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_789_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_790_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_791_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_792_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_793_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_794_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_795_add__divide__eq__if__simps_I4_J,axiom,
! [Z: real,A: real,B4: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B4 @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B4 @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B4 ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_796_diff__frac__eq,axiom,
! [Y: real,Z: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_797_diff__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_798_divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_799_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_800_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_801_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_802_infinite__remove,axiom,
! [S2: set_a,A: a] :
( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_803_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_804_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_805_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_806_card__less__sym__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_807_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_808_psubset__card__mono,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_set_a @ A2 @ B )
=> ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_809_complete__real,axiom,
! [S2: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S2 )
=> ( ? [Z3: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ? [Y2: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S2 )
=> ( ord_less_eq_real @ X5 @ Y2 ) )
& ! [Z3: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ) ) ).
% complete_real
thf(fact_810_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_811_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_812_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X2 ) ) ) ).
% inf_commute
thf(fact_813_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_814_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_815_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_816_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_817_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_818_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_819_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_820_frac__le__eq,axiom,
! [Y: real,Z: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
= ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% frac_le_eq
thf(fact_821_frac__less__eq,axiom,
! [Y: real,Z: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_822_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_823_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_824_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_825_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_826_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_827_card__psubset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) )
=> ( ord_less_set_a @ A2 @ B ) ) ) ) ).
% card_psubset
thf(fact_828_additive__abelian__group_Osumsetdiff__sing,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A2: set_a,B: set_a,X: a] :
( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
=> ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( minus_minus_set_a @ A2 @ B ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% additive_abelian_group.sumsetdiff_sing
thf(fact_829_card__Diff1__less__iff,axiom,
! [A2: set_real,X: real] :
( ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) )
= ( ( finite_finite_real @ A2 )
& ( member_real @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_830_card__Diff1__less__iff,axiom,
! [A2: set_a,X: a] :
( ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) )
= ( ( finite_finite_a @ A2 )
& ( member_a @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_831_card__Diff2__less,axiom,
! [A2: set_real,X: real,Y: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ X @ A2 )
=> ( ( member_real @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ ( insert_real @ Y @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_832_card__Diff2__less,axiom,
! [A2: set_a,X: a,Y: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( insert_a @ Y @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_833_card__Diff1__less,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_834_card__Diff1__less,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_835_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_836_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_837_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_838_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_839_inf_OcoboundedI2,axiom,
! [B4: int,C2: int,A: int] :
( ( ord_less_eq_int @ B4 @ C2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_840_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_841_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_842_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_843_inf_OcoboundedI1,axiom,
! [A: int,C2: int,B4: int] :
( ( ord_less_eq_int @ A @ C2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B4 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_844_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_845_inf_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( inf_inf_real @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_846_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_847_inf_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( inf_inf_int @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_848_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_849_inf_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( inf_inf_real @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_850_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_851_inf_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( inf_inf_int @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_852_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_853_inf_Ocobounded2,axiom,
! [A: real,B4: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_854_inf_Ocobounded2,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_855_inf_Ocobounded2,axiom,
! [A: int,B4: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_856_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_857_inf_Ocobounded1,axiom,
! [A: real,B4: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_858_inf_Ocobounded1,axiom,
! [A: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_859_inf_Ocobounded1,axiom,
! [A: int,B4: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B4 ) @ A ) ).
% inf.cobounded1
thf(fact_860_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_861_inf_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( A4
= ( inf_inf_real @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_862_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( A4
= ( inf_inf_nat @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_863_inf_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( A4
= ( inf_inf_int @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_864_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_865_inf__greatest,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Z )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_866_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_867_inf__greatest,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Z )
=> ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_868_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_869_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_870_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_871_inf_OboundedI,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( ord_less_eq_int @ A @ C2 )
=> ( ord_less_eq_int @ A @ ( inf_inf_int @ B4 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_872_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_873_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_874_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_875_inf_OboundedE,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B4 @ C2 ) )
=> ~ ( ( ord_less_eq_int @ A @ B4 )
=> ~ ( ord_less_eq_int @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_876_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_877_inf__absorb2,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( inf_inf_real @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_878_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_879_inf__absorb2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( inf_inf_int @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_880_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_881_inf__absorb1,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( inf_inf_real @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_882_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_883_inf__absorb1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( inf_inf_int @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_884_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_885_inf_Oabsorb2,axiom,
! [B4: real,A: real] :
( ( ord_less_eq_real @ B4 @ A )
=> ( ( inf_inf_real @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_886_inf_Oabsorb2,axiom,
! [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
=> ( ( inf_inf_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_887_inf_Oabsorb2,axiom,
! [B4: int,A: int] :
( ( ord_less_eq_int @ B4 @ A )
=> ( ( inf_inf_int @ A @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_888_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_889_inf_Oabsorb1,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ( inf_inf_real @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_890_inf_Oabsorb1,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( ( inf_inf_nat @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_891_inf_Oabsorb1,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( inf_inf_int @ A @ B4 )
= A ) ) ).
% inf.absorb1
thf(fact_892_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_893_le__iff__inf,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y4: real] :
( ( inf_inf_real @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_894_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( inf_inf_nat @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_895_le__iff__inf,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y4: int] :
( ( inf_inf_int @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_896_inf__unique,axiom,
! [F2: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F2 @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F2 @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_a,Y2: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a @ X3 @ Z4 )
=> ( ord_less_eq_set_a @ X3 @ ( F2 @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_897_inf__unique,axiom,
! [F2: real > real > real,X: real,Y: real] :
( ! [X3: real,Y2: real] : ( ord_less_eq_real @ ( F2 @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: real,Y2: real] : ( ord_less_eq_real @ ( F2 @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: real,Y2: real,Z4: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ord_less_eq_real @ X3 @ Z4 )
=> ( ord_less_eq_real @ X3 @ ( F2 @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf_real @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_898_inf__unique,axiom,
! [F2: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F2 @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F2 @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: nat,Y2: nat,Z4: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Z4 )
=> ( ord_less_eq_nat @ X3 @ ( F2 @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_899_inf__unique,axiom,
! [F2: int > int > int,X: int,Y: int] :
( ! [X3: int,Y2: int] : ( ord_less_eq_int @ ( F2 @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: int,Y2: int] : ( ord_less_eq_int @ ( F2 @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: int,Y2: int,Z4: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ( ord_less_eq_int @ X3 @ Z4 )
=> ( ord_less_eq_int @ X3 @ ( F2 @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf_int @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_900_inf_OorderI,axiom,
! [A: set_a,B4: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B4 ) )
=> ( ord_less_eq_set_a @ A @ B4 ) ) ).
% inf.orderI
thf(fact_901_inf_OorderI,axiom,
! [A: real,B4: real] :
( ( A
= ( inf_inf_real @ A @ B4 ) )
=> ( ord_less_eq_real @ A @ B4 ) ) ).
% inf.orderI
thf(fact_902_inf_OorderI,axiom,
! [A: nat,B4: nat] :
( ( A
= ( inf_inf_nat @ A @ B4 ) )
=> ( ord_less_eq_nat @ A @ B4 ) ) ).
% inf.orderI
thf(fact_903_inf_OorderI,axiom,
! [A: int,B4: int] :
( ( A
= ( inf_inf_int @ A @ B4 ) )
=> ( ord_less_eq_int @ A @ B4 ) ) ).
% inf.orderI
thf(fact_904_inf_OorderE,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( A
= ( inf_inf_set_a @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_905_inf_OorderE,axiom,
! [A: real,B4: real] :
( ( ord_less_eq_real @ A @ B4 )
=> ( A
= ( inf_inf_real @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_906_inf_OorderE,axiom,
! [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
=> ( A
= ( inf_inf_nat @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_907_inf_OorderE,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( A
= ( inf_inf_int @ A @ B4 ) ) ) ).
% inf.orderE
thf(fact_908_le__infI2,axiom,
! [B4: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_909_le__infI2,axiom,
! [B4: real,X: real,A: real] :
( ( ord_less_eq_real @ B4 @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_910_le__infI2,axiom,
! [B4: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_911_le__infI2,axiom,
! [B4: int,X: int,A: int] :
( ( ord_less_eq_int @ B4 @ X )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B4 ) @ X ) ) ).
% le_infI2
thf(fact_912_le__infI1,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_913_le__infI1,axiom,
! [A: real,X: real,B4: real] :
( ( ord_less_eq_real @ A @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_914_le__infI1,axiom,
! [A: nat,X: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_915_le__infI1,axiom,
! [A: int,X: int,B4: int] :
( ( ord_less_eq_int @ A @ X )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B4 ) @ X ) ) ).
% le_infI1
thf(fact_916_inf__mono,axiom,
! [A: set_a,C2: set_a,B4: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B4 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_917_inf__mono,axiom,
! [A: real,C2: real,B4: real,D: real] :
( ( ord_less_eq_real @ A @ C2 )
=> ( ( ord_less_eq_real @ B4 @ D )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B4 ) @ ( inf_inf_real @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_918_inf__mono,axiom,
! [A: nat,C2: nat,B4: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B4 ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_919_inf__mono,axiom,
! [A: int,C2: int,B4: int,D: int] :
( ( ord_less_eq_int @ A @ C2 )
=> ( ( ord_less_eq_int @ B4 @ D )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B4 ) @ ( inf_inf_int @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_920_le__infI,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B4 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_921_le__infI,axiom,
! [X: real,A: real,B4: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ X @ B4 )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_922_le__infI,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B4 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_923_le__infI,axiom,
! [X: int,A: int,B4: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ X @ B4 )
=> ( ord_less_eq_int @ X @ ( inf_inf_int @ A @ B4 ) ) ) ) ).
% le_infI
thf(fact_924_le__infE,axiom,
! [X: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B4 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B4 ) ) ) ).
% le_infE
thf(fact_925_le__infE,axiom,
! [X: real,A: real,B4: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ A @ B4 ) )
=> ~ ( ( ord_less_eq_real @ X @ A )
=> ~ ( ord_less_eq_real @ X @ B4 ) ) ) ).
% le_infE
thf(fact_926_le__infE,axiom,
! [X: nat,A: nat,B4: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B4 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B4 ) ) ) ).
% le_infE
thf(fact_927_le__infE,axiom,
! [X: int,A: int,B4: int] :
( ( ord_less_eq_int @ X @ ( inf_inf_int @ A @ B4 ) )
=> ~ ( ( ord_less_eq_int @ X @ A )
=> ~ ( ord_less_eq_int @ X @ B4 ) ) ) ).
% le_infE
thf(fact_928_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_929_inf__le2,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_930_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_931_inf__le2,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_932_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_933_inf__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_934_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_935_inf__le1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_936_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_937_inf__sup__ord_I1_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_938_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_939_inf__sup__ord_I1_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_940_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_941_inf__sup__ord_I2_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_942_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_943_inf__sup__ord_I2_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_944_inf_Ostrict__coboundedI2,axiom,
! [B4: set_a,C2: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_945_inf_Ostrict__coboundedI2,axiom,
! [B4: nat,C2: nat,A: nat] :
( ( ord_less_nat @ B4 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_946_inf_Ostrict__coboundedI2,axiom,
! [B4: real,C2: real,A: real] :
( ( ord_less_real @ B4 @ C2 )
=> ( ord_less_real @ ( inf_inf_real @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_947_inf_Ostrict__coboundedI2,axiom,
! [B4: int,C2: int,A: int] :
( ( ord_less_int @ B4 @ C2 )
=> ( ord_less_int @ ( inf_inf_int @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_948_inf_Ostrict__coboundedI1,axiom,
! [A: set_a,C2: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ C2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_949_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C2: nat,B4: nat] :
( ( ord_less_nat @ A @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_950_inf_Ostrict__coboundedI1,axiom,
! [A: real,C2: real,B4: real] :
( ( ord_less_real @ A @ C2 )
=> ( ord_less_real @ ( inf_inf_real @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_951_inf_Ostrict__coboundedI1,axiom,
! [A: int,C2: int,B4: int] :
( ( ord_less_int @ A @ C2 )
=> ( ord_less_int @ ( inf_inf_int @ A @ B4 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_952_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( A4
= ( inf_inf_set_a @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_953_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( A4
= ( inf_inf_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_954_inf_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( A4
= ( inf_inf_real @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_955_inf_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( A4
= ( inf_inf_int @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_956_inf_Ostrict__boundedE,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) )
=> ~ ( ( ord_less_set_a @ A @ B4 )
=> ~ ( ord_less_set_a @ A @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_957_inf_Ostrict__boundedE,axiom,
! [A: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B4 @ C2 ) )
=> ~ ( ( ord_less_nat @ A @ B4 )
=> ~ ( ord_less_nat @ A @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_958_inf_Ostrict__boundedE,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_real @ A @ ( inf_inf_real @ B4 @ C2 ) )
=> ~ ( ( ord_less_real @ A @ B4 )
=> ~ ( ord_less_real @ A @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_959_inf_Ostrict__boundedE,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_int @ A @ ( inf_inf_int @ B4 @ C2 ) )
=> ~ ( ( ord_less_int @ A @ B4 )
=> ~ ( ord_less_int @ A @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_960_inf_Oabsorb4,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ A )
=> ( ( inf_inf_set_a @ A @ B4 )
= B4 ) ) ).
% inf.absorb4
thf(fact_961_inf_Oabsorb4,axiom,
! [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
=> ( ( inf_inf_nat @ A @ B4 )
= B4 ) ) ).
% inf.absorb4
thf(fact_962_inf_Oabsorb4,axiom,
! [B4: real,A: real] :
( ( ord_less_real @ B4 @ A )
=> ( ( inf_inf_real @ A @ B4 )
= B4 ) ) ).
% inf.absorb4
thf(fact_963_inf_Oabsorb4,axiom,
! [B4: int,A: int] :
( ( ord_less_int @ B4 @ A )
=> ( ( inf_inf_int @ A @ B4 )
= B4 ) ) ).
% inf.absorb4
thf(fact_964_inf_Oabsorb3,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ B4 )
=> ( ( inf_inf_set_a @ A @ B4 )
= A ) ) ).
% inf.absorb3
thf(fact_965_inf_Oabsorb3,axiom,
! [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
=> ( ( inf_inf_nat @ A @ B4 )
= A ) ) ).
% inf.absorb3
thf(fact_966_inf_Oabsorb3,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( inf_inf_real @ A @ B4 )
= A ) ) ).
% inf.absorb3
thf(fact_967_inf_Oabsorb3,axiom,
! [A: int,B4: int] :
( ( ord_less_int @ A @ B4 )
=> ( ( inf_inf_int @ A @ B4 )
= A ) ) ).
% inf.absorb3
thf(fact_968_less__infI2,axiom,
! [B4: set_a,X: set_a,A: set_a] :
( ( ord_less_set_a @ B4 @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% less_infI2
thf(fact_969_less__infI2,axiom,
! [B4: nat,X: nat,A: nat] :
( ( ord_less_nat @ B4 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% less_infI2
thf(fact_970_less__infI2,axiom,
! [B4: real,X: real,A: real] :
( ( ord_less_real @ B4 @ X )
=> ( ord_less_real @ ( inf_inf_real @ A @ B4 ) @ X ) ) ).
% less_infI2
thf(fact_971_less__infI2,axiom,
! [B4: int,X: int,A: int] :
( ( ord_less_int @ B4 @ X )
=> ( ord_less_int @ ( inf_inf_int @ A @ B4 ) @ X ) ) ).
% less_infI2
thf(fact_972_less__infI1,axiom,
! [A: set_a,X: set_a,B4: set_a] :
( ( ord_less_set_a @ A @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B4 ) @ X ) ) ).
% less_infI1
thf(fact_973_less__infI1,axiom,
! [A: nat,X: nat,B4: nat] :
( ( ord_less_nat @ A @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B4 ) @ X ) ) ).
% less_infI1
thf(fact_974_less__infI1,axiom,
! [A: real,X: real,B4: real] :
( ( ord_less_real @ A @ X )
=> ( ord_less_real @ ( inf_inf_real @ A @ B4 ) @ X ) ) ).
% less_infI1
thf(fact_975_less__infI1,axiom,
! [A: int,X: int,B4: int] :
( ( ord_less_int @ A @ X )
=> ( ord_less_int @ ( inf_inf_int @ A @ B4 ) @ X ) ) ).
% less_infI1
thf(fact_976_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_977_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_978_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_979_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_980_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_981_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_982_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_983_diff__gt__0__iff__gt,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B4 ) )
= ( ord_less_real @ B4 @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_984_diff__gt__0__iff__gt,axiom,
! [A: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B4 ) )
= ( ord_less_int @ B4 @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_985_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_986_diff__ge__0__iff__ge,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B4 ) )
= ( ord_less_eq_int @ B4 @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_987_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_988_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_989_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_990_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_991_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_992_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_993_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_994_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_995_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_996_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_997_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_998_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_999_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1000_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_1001_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_1002_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_1003_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_1004_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1005_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1006_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_1007_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_1008_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_1009_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_1010_zdiv__mono1,axiom,
! [A: int,A8: int,B4: int] :
( ( ord_less_eq_int @ A @ A8 )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A8 @ B4 ) ) ) ) ).
% zdiv_mono1
thf(fact_1011_zdiv__mono2,axiom,
! [A: int,B9: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B9 )
=> ( ( ord_less_eq_int @ B9 @ B4 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B9 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1012_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1013_zdiv__mono1__neg,axiom,
! [A: int,A8: int,B4: int] :
( ( ord_less_eq_int @ A @ A8 )
=> ( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A8 @ B4 ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1014_zdiv__mono2__neg,axiom,
! [A: int,B9: int,B4: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B9 )
=> ( ( ord_less_eq_int @ B9 @ B4 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B9 ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1015_zdiv__zmult2__eq,axiom,
! [C2: int,A: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B4 @ C2 ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B4 ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1016_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1017_div__nonneg__neg__le0,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1018_div__nonpos__pos__le0,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1019_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1020_neg__imp__zdiv__nonneg__iff,axiom,
! [B4: int,A: int] :
( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B4 ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1021_pos__imp__zdiv__nonneg__iff,axiom,
! [B4: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B4 ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1022_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B4 ) )
= ( ( ord_less_eq_int @ B4 @ A )
& ( ord_less_int @ zero_zero_int @ B4 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1023_pos__imp__zdiv__neg__iff,axiom,
! [B4: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B4 ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1024_neg__imp__zdiv__neg__iff,axiom,
! [B4: int,A: int] :
( ( ord_less_int @ B4 @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B4 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1025_div__neg__pos__less0,axiom,
! [A: int,B4: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ord_less_int @ ( divide_divide_int @ A @ B4 ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1026_psubsetD,axiom,
! [A2: set_a,B: set_a,C2: a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_1027_psubsetD,axiom,
! [A2: set_real,B: set_real,C2: real] :
( ( ord_less_set_real @ A2 @ B )
=> ( ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_1028_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_1029_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1030_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_1031_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_1032_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_1033_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_1034_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_1035_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1036_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_1037_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_1038_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_1039_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_1040_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_1041_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_1042_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_1043_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_1044_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_1045_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_1046_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_1047_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_1048_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_1049_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_1050_mult_Oleft__commute,axiom,
! [B4: int,A: int,C2: int] :
( ( times_times_int @ B4 @ ( times_times_int @ A @ C2 ) )
= ( times_times_int @ A @ ( times_times_int @ B4 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_1051_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_1052_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B3: real] : ( times_times_real @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_1053_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B3: int] : ( times_times_int @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_1054_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_1055_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_1056_mult_Oassoc,axiom,
! [A: int,B4: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A @ B4 ) @ C2 )
= ( times_times_int @ A @ ( times_times_int @ B4 @ C2 ) ) ) ).
% mult.assoc
thf(fact_1057_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_1058_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_1059_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B4: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A @ B4 ) @ C2 )
= ( times_times_int @ A @ ( times_times_int @ B4 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1060_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_1061_diff__eq__diff__eq,axiom,
! [A: int,B4: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B4 )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( A = B4 )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_1062_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_1063_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_1064_diff__right__commute,axiom,
! [A: int,C2: int,B4: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B4 )
= ( minus_minus_int @ ( minus_minus_int @ A @ B4 ) @ C2 ) ) ).
% diff_right_commute
thf(fact_1065_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_1066_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_1067_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_1068_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_1069_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_1070_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_1071_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_1072_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1073_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1074_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1075_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_1076_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_1077_diff__eq__diff__less__eq,axiom,
! [A: int,B4: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B4 )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_eq_int @ A @ B4 )
= ( ord_less_eq_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1078_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_1079_diff__right__mono,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B4 @ C2 ) ) ) ).
% diff_right_mono
thf(fact_1080_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_1081_diff__left__mono,axiom,
! [B4: int,A: int,C2: int] :
( ( ord_less_eq_int @ B4 @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B4 ) ) ) ).
% diff_left_mono
thf(fact_1082_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_1083_diff__mono,axiom,
! [A: int,B4: int,D: int,C2: int] :
( ( ord_less_eq_int @ A @ B4 )
=> ( ( ord_less_eq_int @ D @ C2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B4 @ D ) ) ) ) ).
% diff_mono
thf(fact_1084_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [A4: real,B3: real] :
( ( minus_minus_real @ A4 @ B3 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1085_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A4: int,B3: int] :
( ( minus_minus_int @ A4 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1086_diff__strict__right__mono,axiom,
! [A: real,B4: real,C2: real] :
( ( ord_less_real @ A @ B4 )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B4 @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_1087_diff__strict__right__mono,axiom,
! [A: int,B4: int,C2: int] :
( ( ord_less_int @ A @ B4 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B4 @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_1088_diff__strict__left__mono,axiom,
! [B4: real,A: real,C2: real] :
( ( ord_less_real @ B4 @ A )
=> ( ord_less_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B4 ) ) ) ).
% diff_strict_left_mono
thf(fact_1089_diff__strict__left__mono,axiom,
! [B4: int,A: int,C2: int] :
( ( ord_less_int @ B4 @ A )
=> ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B4 ) ) ) ).
% diff_strict_left_mono
thf(fact_1090_diff__eq__diff__less,axiom,
! [A: real,B4: real,C2: real,D: real] :
( ( ( minus_minus_real @ A @ B4 )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( ord_less_real @ A @ B4 )
= ( ord_less_real @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1091_diff__eq__diff__less,axiom,
! [A: int,B4: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B4 )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_int @ A @ B4 )
= ( ord_less_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1092_diff__strict__mono,axiom,
! [A: real,B4: real,D: real,C2: real] :
( ( ord_less_real @ A @ B4 )
=> ( ( ord_less_real @ D @ C2 )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B4 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1093_diff__strict__mono,axiom,
! [A: int,B4: int,D: int,C2: int] :
( ( ord_less_int @ A @ B4 )
=> ( ( ord_less_int @ D @ C2 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B4 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1094_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_1095_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1096_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B3 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_1097_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_1098_diff__shunt__var,axiom,
! [X: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1099_sumset__iterated__r,axiom,
! [R: nat,A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ R )
=> ( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ R )
= ( pluenn3038260743871226533mset_a @ g @ addition @ A2 @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A2 @ ( minus_minus_nat @ R @ one_one_nat ) ) ) ) ) ).
% sumset_iterated_r
thf(fact_1100_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_1101_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_1102_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_1103_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_1104_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_1105_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_1106_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_1107_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_1108_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_1109_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_1110_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_1111_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_1112_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1113_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_1114_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_1115_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1116_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_1117_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_1118_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_1119_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_1120_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_1121_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_1122_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_1123_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1124_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1125_mult__cancel__left1,axiom,
! [C2: real,B4: real] :
( ( C2
= ( times_times_real @ C2 @ B4 ) )
= ( ( C2 = zero_zero_real )
| ( B4 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_1126_mult__cancel__left1,axiom,
! [C2: int,B4: int] :
( ( C2
= ( times_times_int @ C2 @ B4 ) )
= ( ( C2 = zero_zero_int )
| ( B4 = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_1127_mult__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ( times_times_real @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_1128_mult__cancel__left2,axiom,
! [C2: int,A: int] :
( ( ( times_times_int @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_1129_mult__cancel__right1,axiom,
! [C2: real,B4: real] :
( ( C2
= ( times_times_real @ B4 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B4 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_1130_mult__cancel__right1,axiom,
! [C2: int,B4: int] :
( ( C2
= ( times_times_int @ B4 @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( B4 = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1131_mult__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ( times_times_real @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_1132_mult__cancel__right2,axiom,
! [A: int,C2: int] :
( ( ( times_times_int @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1133_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_1134_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_1135_eq__divide__eq__1,axiom,
! [B4: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B4 @ A ) )
= ( ( A != zero_zero_real )
& ( A = B4 ) ) ) ).
% eq_divide_eq_1
thf(fact_1136_divide__eq__eq__1,axiom,
! [B4: real,A: real] :
( ( ( divide_divide_real @ B4 @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B4 ) ) ) ).
% divide_eq_eq_1
thf(fact_1137_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_1138_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_1139_one__eq__divide__iff,axiom,
! [A: real,B4: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B4 ) )
= ( ( B4 != zero_zero_real )
& ( A = B4 ) ) ) ).
% one_eq_divide_iff
thf(fact_1140_divide__eq__1__iff,axiom,
! [A: real,B4: real] :
( ( ( divide_divide_real @ A @ B4 )
= one_one_real )
= ( ( B4 != zero_zero_real )
& ( A = B4 ) ) ) ).
% divide_eq_1_iff
thf(fact_1141_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_1142_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_1143_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_1144_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1145_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1146_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1147_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1148_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1149_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1150_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_1151_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_1152_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_1153_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1154_zero__le__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_1155_divide__le__0__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_1156_zero__less__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_1157_less__divide__eq__1__pos,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B4 @ A ) )
= ( ord_less_real @ A @ B4 ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1158_less__divide__eq__1__neg,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B4 @ A ) )
= ( ord_less_real @ B4 @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1159_divide__less__eq__1__pos,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ ( divide_divide_real @ B4 @ A ) @ one_one_real )
= ( ord_less_real @ B4 @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1160_divide__less__eq__1__neg,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B4 @ A ) @ one_one_real )
= ( ord_less_real @ A @ B4 ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1161_divide__less__0__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_1162_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B4: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B4 ) )
= ( divide_divide_real @ one_one_real @ B4 ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1163_nonzero__divide__mult__cancel__right,axiom,
! [B4: real,A: real] :
( ( B4 != zero_zero_real )
=> ( ( divide_divide_real @ B4 @ ( times_times_real @ A @ B4 ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1164_le__divide__eq__1__pos,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B4 @ A ) )
= ( ord_less_eq_real @ A @ B4 ) ) ) ).
% le_divide_eq_1_pos
thf(fact_1165_le__divide__eq__1__neg,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B4 @ A ) )
= ( ord_less_eq_real @ B4 @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_1166_divide__le__eq__1__pos,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B4 @ A ) @ one_one_real )
= ( ord_less_eq_real @ B4 @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1167_divide__le__eq__1__neg,axiom,
! [A: real,B4: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B4 @ A ) @ one_one_real )
= ( ord_less_eq_real @ A @ B4 ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1168_card__Diff__insert,axiom,
! [A: real,A2: set_real,B: set_real] :
( ( member_real @ A @ A2 )
=> ( ~ ( member_real @ A @ B )
=> ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B ) ) )
= ( minus_minus_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1169_card__Diff__insert,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ~ ( member_a @ A @ B )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1170_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_1171_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_1172_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_1173_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1174_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1175_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1176_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_1177_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_1178_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_1179_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1180_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1181_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1182_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_1183_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_1184_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_1185_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1186_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_1187_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1188_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1189_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1190_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_1191_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1192_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1193_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1194_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_1195_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_1196_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1197_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_1198_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_1199_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1200_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1201_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1202_right__inverse__eq,axiom,
! [B4: real,A: real] :
( ( B4 != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B4 )
= one_one_real )
= ( A = B4 ) ) ) ).
% right_inverse_eq
thf(fact_1203_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1204_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_1205_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_1206_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_1207_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_1208_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_1209_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_1210_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1211_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1212_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1213_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1214_real__sqrt__eq__1__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% real_sqrt_eq_1_iff
thf(fact_1215_real__sqrt__one,axiom,
( ( sqrt @ one_one_real )
= one_one_real ) ).
% real_sqrt_one
thf(fact_1216_zle__diff1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% zle_diff1_eq
thf(fact_1217_real__sqrt__lt__1__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
= ( ord_less_real @ X @ one_one_real ) ) ).
% real_sqrt_lt_1_iff
thf(fact_1218_real__sqrt__gt__1__iff,axiom,
! [Y: real] :
( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
= ( ord_less_real @ one_one_real @ Y ) ) ).
% real_sqrt_gt_1_iff
thf(fact_1219_real__sqrt__ge__1__iff,axiom,
! [Y: real] :
( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
= ( ord_less_eq_real @ one_one_real @ Y ) ) ).
% real_sqrt_ge_1_iff
thf(fact_1220_real__sqrt__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ).
% real_sqrt_le_1_iff
thf(fact_1221_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1222_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1223_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1224_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1225_real__sqrt__ge__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_one
thf(fact_1226_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1227_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1228_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_1229_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1230_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N2: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1231_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_1232_real__of__nat__div3,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).
% real_of_nat_div3
thf(fact_1233_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1234_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1235_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_1236_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1237_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1238_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_1239_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1240_group__axioms,axiom,
group_group_a @ g @ addition @ zero ).
% group_axioms
thf(fact_1241_plusinfinity,axiom,
! [D: int,P2: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [X_12: int] : ( P2 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1242_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1243_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1244_int__ops_I6_J,axiom,
! [A: nat,B4: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B4 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B4 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B4 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B4 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1245_group__of__Units,axiom,
group_group_a @ ( group_Units_a @ g @ addition @ zero ) @ addition @ zero ).
% group_of_Units
thf(fact_1246_int__if,axiom,
! [P: $o,A: nat,B4: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B4 ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B4 ) )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% int_if
thf(fact_1247_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1248_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1249_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1250_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1251_int__ops_I7_J,axiom,
! [A: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B4 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ).
% int_ops(7)
thf(fact_1252_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1253_int__ops_I8_J,axiom,
! [A: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B4 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ).
% int_ops(8)
thf(fact_1254_mem__UnitsI,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ( member_a @ U @ g )
=> ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) ) ) ) ).
% mem_UnitsI
thf(fact_1255_mem__UnitsD,axiom,
! [U: a] :
( ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
& ( member_a @ U @ g ) ) ) ).
% mem_UnitsD
thf(fact_1256_invertibleE,axiom,
! [U: a] :
( ( group_invertible_a @ g @ addition @ zero @ U )
=> ( ! [V4: a] :
( ( ( ( addition @ U @ V4 )
= zero )
& ( ( addition @ V4 @ U )
= zero ) )
=> ~ ( member_a @ V4 @ g ) )
=> ~ ( member_a @ U @ g ) ) ) ).
% invertibleE
thf(fact_1257_invertible__def,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( ( group_invertible_a @ g @ addition @ zero @ U )
= ( ? [X2: a] :
( ( member_a @ X2 @ g )
& ( ( addition @ U @ X2 )
= zero )
& ( ( addition @ X2 @ U )
= zero ) ) ) ) ) ).
% invertible_def
thf(fact_1258_unit__invertible,axiom,
group_invertible_a @ g @ addition @ zero @ zero ).
% unit_invertible
thf(fact_1259_composition__invertible,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( group_invertible_a @ g @ addition @ zero @ Y )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( group_invertible_a @ g @ addition @ zero @ ( addition @ X @ Y ) ) ) ) ) ) ).
% composition_invertible
thf(fact_1260_invertible,axiom,
! [U: a] :
( ( member_a @ U @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ).
% invertible
thf(fact_1261_invertibleI,axiom,
! [U: a,V2: a] :
( ( ( addition @ U @ V2 )
= zero )
=> ( ( ( addition @ V2 @ U )
= zero )
=> ( ( member_a @ U @ g )
=> ( ( member_a @ V2 @ g )
=> ( group_invertible_a @ g @ addition @ zero @ U ) ) ) ) ) ).
% invertibleI
thf(fact_1262_invertible__left__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z @ g )
=> ( ( ( addition @ X @ Y )
= ( addition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ).
% invertible_left_cancel
thf(fact_1263_invertible__right__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( group_invertible_a @ g @ addition @ zero @ X )
=> ( ( member_a @ X @ g )
=> ( ( member_a @ Y @ g )
=> ( ( member_a @ Z @ g )
=> ( ( ( addition @ Y @ X )
= ( addition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% invertible_right_cancel
thf(fact_1264_Bolzano,axiom,
! [A: real,B4: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B4 )
=> ( ! [A3: real,B2: real,C4: real] :
( ( P @ A3 @ B2 )
=> ( ( P @ B2 @ C4 )
=> ( ( ord_less_eq_real @ A3 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C4 )
=> ( P @ A3 @ C4 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B4 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A3: real,B2: real] :
( ( ( ord_less_eq_real @ A3 @ X3 )
& ( ord_less_eq_real @ X3 @ B2 )
& ( ord_less_real @ ( minus_minus_real @ B2 @ A3 ) @ D3 ) )
=> ( P @ A3 @ B2 ) ) ) ) )
=> ( P @ A @ B4 ) ) ) ) ).
% Bolzano
thf(fact_1265_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_1266_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_1267_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_1268_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
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ v ) ) ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ u ) ) @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ v ) ) ) ) @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ w ) ) ) ) )
= ( divide_divide_real @ ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ u ) ) ) ) @ ( times_times_real @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ u ) ) ) @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ v ) ) ) ) ) @ ( semiri5074537144036343181t_real @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ) ) ) @ ( times_times_real @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ u ) ) ) @ ( sqrt @ ( semiri5074537144036343181t_real @ ( finite_card_a @ w ) ) ) ) ) ) ).
%------------------------------------------------------------------------------