TPTP Problem File: SLH0010^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Dedekind_Real/0000_Dedekind_Real/prob_00728_022731__5669108_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1345 ( 551 unt; 75 typ; 0 def)
% Number of atoms : 3499 (1279 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9948 ( 326 ~; 92 |; 143 &;7919 @)
% ( 0 <=>;1468 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 270 ( 270 >; 0 *; 0 +; 0 <<)
% Number of symbols : 69 ( 68 usr; 11 con; 0-2 aty)
% Number of variables : 3278 ( 126 ^;3120 !; 32 ?;3278 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:25:03.291
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $tType ).
thf(ty_n_t__Dedekind____Real__Opreal,type,
dedekind_preal: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (68)
thf(sy_c_Dedekind__Real_Opreal_ORep__preal,type,
dedekind_Rep_preal: dedekind_preal > set_rat ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Dedekind____Real__Opreal,type,
invers3090987106763476162_preal: dedekind_preal > dedekind_preal ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
inverse_inverse_rat: rat > rat ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Groups_Ocomm__monoid_001t__Dedekind____Real__Opreal,type,
comm_m4700367977445107531_preal: ( dedekind_preal > dedekind_preal > dedekind_preal ) > dedekind_preal > $o ).
thf(sy_c_Groups_Ocomm__monoid_001t__Int__Oint,type,
comm_monoid_int: ( int > int > int ) > int > $o ).
thf(sy_c_Groups_Ocomm__monoid_001t__Nat__Onat,type,
comm_monoid_nat: ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Groups_Ocomm__monoid_001t__Rat__Orat,type,
comm_monoid_rat: ( rat > rat > rat ) > rat > $o ).
thf(sy_c_Groups_Ocomm__monoid_001t__Real__Oreal,type,
comm_monoid_real: ( real > real > real ) > real > $o ).
thf(sy_c_Groups_Omonoid_001t__Dedekind____Real__Opreal,type,
monoid4946265034669045783_preal: ( dedekind_preal > dedekind_preal > dedekind_preal ) > dedekind_preal > $o ).
thf(sy_c_Groups_Omonoid_001t__Int__Oint,type,
monoid_int: ( int > int > int ) > int > $o ).
thf(sy_c_Groups_Omonoid_001t__Nat__Onat,type,
monoid_nat: ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Groups_Omonoid_001t__Rat__Orat,type,
monoid_rat: ( rat > rat > rat ) > rat > $o ).
thf(sy_c_Groups_Omonoid_001t__Real__Oreal,type,
monoid_real: ( real > real > real ) > real > $o ).
thf(sy_c_Groups_Oone__class_Oone_001t__Dedekind____Real__Opreal,type,
one_on9143529541772854033_preal: dedekind_preal ).
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__Rat__Orat,type,
one_one_rat: rat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Dedekind____Real__Opreal,type,
plus_p3173629198307831117_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
plus_plus_rat: rat > rat > rat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Dedekind____Real__Opreal,type,
times_3000655703912201937_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
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__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
times_times_rat: rat > rat > rat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
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__Rat__Orat,type,
zero_zero_rat: rat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
neg_nu5219082963157363817nc_rat: rat > rat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Dedekind____Real__Opreal,type,
ord_le5708704896291381698_preal: dedekind_preal > dedekind_preal > $o ).
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__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
ord_less_rat: rat > rat > $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__Rat__Orat_J,type,
ord_less_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Dedekind____Real__Opreal,type,
ord_le5604041210740703414_preal: dedekind_preal > dedekind_preal > $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__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
ord_less_eq_rat: rat > rat > $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__Rat__Orat_J,type,
ord_less_eq_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Dedekind____Real__Opreal,type,
divide4190755330972744004_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
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__Rat__Orat,type,
divide_divide_rat: rat > rat > rat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
collect_rat: ( rat > $o ) > set_rat ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > set_rat > $o ).
thf(sy_v_r,type,
r: dedekind_preal ).
% Relevant facts (1269)
thf(fact_0_preal__mult__commute,axiom,
( times_3000655703912201937_preal
= ( ^ [X: dedekind_preal,Y: dedekind_preal] : ( times_3000655703912201937_preal @ Y @ X ) ) ) ).
% preal_mult_commute
thf(fact_1_preal__mult__inverse,axiom,
! [R: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( invers3090987106763476162_preal @ R ) @ R )
= one_on9143529541772854033_preal ) ).
% preal_mult_inverse
thf(fact_2_preal__mult__1,axiom,
! [Z: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ Z )
= Z ) ).
% preal_mult_1
thf(fact_3_preal__mult__assoc,axiom,
! [X2: dedekind_preal,Y2: dedekind_preal,Z: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ X2 @ Y2 ) @ Z )
= ( times_3000655703912201937_preal @ X2 @ ( times_3000655703912201937_preal @ Y2 @ Z ) ) ) ).
% preal_mult_assoc
thf(fact_4_inverse__eq__1__iff,axiom,
! [X2: rat] :
( ( ( inverse_inverse_rat @ X2 )
= one_one_rat )
= ( X2 = one_one_rat ) ) ).
% inverse_eq_1_iff
thf(fact_5_inverse__eq__1__iff,axiom,
! [X2: real] :
( ( ( inverse_inverse_real @ X2 )
= one_one_real )
= ( X2 = one_one_real ) ) ).
% inverse_eq_1_iff
thf(fact_6_inverse__1,axiom,
( ( inverse_inverse_rat @ one_one_rat )
= one_one_rat ) ).
% inverse_1
thf(fact_7_inverse__1,axiom,
( ( inverse_inverse_real @ one_one_real )
= one_one_real ) ).
% inverse_1
thf(fact_8_inverse__mult__distrib,axiom,
! [A: rat,B: rat] :
( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
= ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) ) ) ).
% inverse_mult_distrib
thf(fact_9_inverse__mult__distrib,axiom,
! [A: real,B: real] :
( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) ) ) ).
% inverse_mult_distrib
thf(fact_10_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_11_mult_Oright__neutral,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ A @ one_on9143529541772854033_preal )
= A ) ).
% mult.right_neutral
thf(fact_12_mult_Oright__neutral,axiom,
! [A: rat] :
( ( times_times_rat @ A @ one_one_rat )
= A ) ).
% mult.right_neutral
thf(fact_13_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_14_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_15_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_16_mult__1,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ A )
= A ) ).
% mult_1
thf(fact_17_mult__1,axiom,
! [A: rat] :
( ( times_times_rat @ one_one_rat @ A )
= A ) ).
% mult_1
thf(fact_18_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_19_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_20_inverse__unique,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= one_one_rat )
=> ( ( inverse_inverse_rat @ A )
= B ) ) ).
% inverse_unique
thf(fact_21_inverse__unique,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= one_one_real )
=> ( ( inverse_inverse_real @ A )
= B ) ) ).
% inverse_unique
thf(fact_22_inverse__eq__iff__eq,axiom,
! [A: rat,B: rat] :
( ( ( inverse_inverse_rat @ A )
= ( inverse_inverse_rat @ B ) )
= ( A = B ) ) ).
% inverse_eq_iff_eq
thf(fact_23_inverse__eq__iff__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
= ( A = B ) ) ).
% inverse_eq_iff_eq
thf(fact_24_inverse__inverse__eq,axiom,
! [A: rat] :
( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_25_inverse__inverse__eq,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_26_mult__commute__imp__mult__inverse__commute,axiom,
! [Y2: rat,X2: rat] :
( ( ( times_times_rat @ Y2 @ X2 )
= ( times_times_rat @ X2 @ Y2 ) )
=> ( ( times_times_rat @ ( inverse_inverse_rat @ Y2 ) @ X2 )
= ( times_times_rat @ X2 @ ( inverse_inverse_rat @ Y2 ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_27_mult__commute__imp__mult__inverse__commute,axiom,
! [Y2: real,X2: real] :
( ( ( times_times_real @ Y2 @ X2 )
= ( times_times_real @ X2 @ Y2 ) )
=> ( ( times_times_real @ ( inverse_inverse_real @ Y2 ) @ X2 )
= ( times_times_real @ X2 @ ( inverse_inverse_real @ Y2 ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_28_comm__monoid__mult__class_Omult__1,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_29_comm__monoid__mult__class_Omult__1,axiom,
! [A: rat] :
( ( times_times_rat @ one_one_rat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_30_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_31_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_32_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_33_mult_Ocomm__neutral,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ A @ one_on9143529541772854033_preal )
= A ) ).
% mult.comm_neutral
thf(fact_34_mult_Ocomm__neutral,axiom,
! [A: rat] :
( ( times_times_rat @ A @ one_one_rat )
= A ) ).
% mult.comm_neutral
thf(fact_35_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_36_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_37_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_38_mult_Oleft__commute,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ B @ ( times_3000655703912201937_preal @ A @ C ) )
= ( times_3000655703912201937_preal @ A @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_39_mult_Oleft__commute,axiom,
! [B: rat,A: rat,C: rat] :
( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_40_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_41_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_42_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_43_mult_Ocommute,axiom,
( times_3000655703912201937_preal
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] : ( times_3000655703912201937_preal @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_44_mult_Ocommute,axiom,
( times_times_rat
= ( ^ [A2: rat,B2: rat] : ( times_times_rat @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_45_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_46_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A2: int,B2: int] : ( times_times_int @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_47_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A2: real,B2: real] : ( times_times_real @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_48_mult_Oassoc,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ A @ B ) @ C )
= ( times_3000655703912201937_preal @ A @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% mult.assoc
thf(fact_49_mult_Oassoc,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_50_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_51_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_52_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_53_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ A @ B ) @ C )
= ( times_3000655703912201937_preal @ A @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_54_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_55_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_56_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_57_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_58_one__reorient,axiom,
! [X2: dedekind_preal] :
( ( one_on9143529541772854033_preal = X2 )
= ( X2 = one_on9143529541772854033_preal ) ) ).
% one_reorient
thf(fact_59_one__reorient,axiom,
! [X2: rat] :
( ( one_one_rat = X2 )
= ( X2 = one_one_rat ) ) ).
% one_reorient
thf(fact_60_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_61_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_62_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_63_inverse__eq__imp__eq,axiom,
! [A: rat,B: rat] :
( ( ( inverse_inverse_rat @ A )
= ( inverse_inverse_rat @ B ) )
=> ( A = B ) ) ).
% inverse_eq_imp_eq
thf(fact_64_inverse__eq__imp__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
=> ( A = B ) ) ).
% inverse_eq_imp_eq
thf(fact_65_right__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
= one_one_rat ) ) ).
% right_inverse
thf(fact_66_right__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
= one_one_real ) ) ).
% right_inverse
thf(fact_67_left__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
= one_one_rat ) ) ).
% left_inverse
thf(fact_68_left__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
= one_one_real ) ) ).
% left_inverse
thf(fact_69_field__class_Ofield__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
= one_one_rat ) ) ).
% field_class.field_inverse
thf(fact_70_field__class_Ofield__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
= one_one_real ) ) ).
% field_class.field_inverse
thf(fact_71_mult_Omonoid__axioms,axiom,
monoid4946265034669045783_preal @ times_3000655703912201937_preal @ one_on9143529541772854033_preal ).
% mult.monoid_axioms
thf(fact_72_mult_Omonoid__axioms,axiom,
monoid_rat @ times_times_rat @ one_one_rat ).
% mult.monoid_axioms
thf(fact_73_mult_Omonoid__axioms,axiom,
monoid_nat @ times_times_nat @ one_one_nat ).
% mult.monoid_axioms
thf(fact_74_mult_Omonoid__axioms,axiom,
monoid_int @ times_times_int @ one_one_int ).
% mult.monoid_axioms
thf(fact_75_mult_Omonoid__axioms,axiom,
monoid_real @ times_times_real @ one_one_real ).
% mult.monoid_axioms
thf(fact_76_divide__preal__def,axiom,
( divide4190755330972744004_preal
= ( ^ [R2: dedekind_preal,S: dedekind_preal] : ( times_3000655703912201937_preal @ R2 @ ( invers3090987106763476162_preal @ S ) ) ) ) ).
% divide_preal_def
thf(fact_77_mult_Ocomm__monoid__axioms,axiom,
comm_m4700367977445107531_preal @ times_3000655703912201937_preal @ one_on9143529541772854033_preal ).
% mult.comm_monoid_axioms
thf(fact_78_mult_Ocomm__monoid__axioms,axiom,
comm_monoid_rat @ times_times_rat @ one_one_rat ).
% mult.comm_monoid_axioms
thf(fact_79_mult_Ocomm__monoid__axioms,axiom,
comm_monoid_nat @ times_times_nat @ one_one_nat ).
% mult.comm_monoid_axioms
thf(fact_80_mult_Ocomm__monoid__axioms,axiom,
comm_monoid_int @ times_times_int @ one_one_int ).
% mult.comm_monoid_axioms
thf(fact_81_mult_Ocomm__monoid__axioms,axiom,
comm_monoid_real @ times_times_real @ one_one_real ).
% mult.comm_monoid_axioms
thf(fact_82_nonzero__inverse__mult__distrib,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
= ( times_times_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_83_nonzero__inverse__mult__distrib,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_84_subset__inverse__mult,axiom,
! [R: dedekind_preal] : ( ord_less_eq_set_rat @ ( dedekind_Rep_preal @ one_on9143529541772854033_preal ) @ ( dedekind_Rep_preal @ ( times_3000655703912201937_preal @ ( invers3090987106763476162_preal @ R ) @ R ) ) ) ).
% subset_inverse_mult
thf(fact_85_inverse__mult__subset,axiom,
! [R: dedekind_preal] : ( ord_less_eq_set_rat @ ( dedekind_Rep_preal @ ( times_3000655703912201937_preal @ ( invers3090987106763476162_preal @ R ) @ R ) ) @ ( dedekind_Rep_preal @ one_on9143529541772854033_preal ) ) ).
% inverse_mult_subset
thf(fact_86_inverse__eq__divide,axiom,
( inverse_inverse_rat
= ( divide_divide_rat @ one_one_rat ) ) ).
% inverse_eq_divide
thf(fact_87_inverse__eq__divide,axiom,
( inverse_inverse_real
= ( divide_divide_real @ one_one_real ) ) ).
% inverse_eq_divide
thf(fact_88_divide__inverse__commute,axiom,
( divide_divide_rat
= ( ^ [A2: rat,B2: rat] : ( times_times_rat @ ( inverse_inverse_rat @ B2 ) @ A2 ) ) ) ).
% divide_inverse_commute
thf(fact_89_divide__inverse__commute,axiom,
( divide_divide_real
= ( ^ [A2: real,B2: real] : ( times_times_real @ ( inverse_inverse_real @ B2 ) @ A2 ) ) ) ).
% divide_inverse_commute
thf(fact_90_divide__inverse,axiom,
( divide_divide_rat
= ( ^ [A2: rat,B2: rat] : ( times_times_rat @ A2 @ ( inverse_inverse_rat @ B2 ) ) ) ) ).
% divide_inverse
thf(fact_91_divide__inverse,axiom,
( divide_divide_real
= ( ^ [A2: real,B2: real] : ( times_times_real @ A2 @ ( inverse_inverse_real @ B2 ) ) ) ) ).
% divide_inverse
thf(fact_92_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_93_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_94_divide__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divide_eq_0_iff
thf(fact_95_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_96_divide__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( divide_divide_rat @ C @ A )
= ( divide_divide_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_97_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_98_divide__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ( divide_divide_rat @ A @ C )
= ( divide_divide_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_99_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_100_division__ring__divide__zero,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% division_ring_divide_zero
thf(fact_101_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_102_times__divide__eq__left,axiom,
! [B: rat,C: rat,A: rat] :
( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_103_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_104_divide__divide__eq__left,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
= ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_105_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_106_divide__divide__eq__right,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_107_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_108_times__divide__eq__right,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_109_inverse__zero,axiom,
( ( inverse_inverse_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% inverse_zero
thf(fact_110_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_111_mem__Collect__eq,axiom,
! [A: rat,P: rat > $o] :
( ( member_rat @ A @ ( collect_rat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_112_Collect__mem__eq,axiom,
! [A3: set_rat] :
( ( collect_rat
@ ^ [X: rat] : ( member_rat @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_113_inverse__nonzero__iff__nonzero,axiom,
! [A: rat] :
( ( ( inverse_inverse_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_114_inverse__nonzero__iff__nonzero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_115_inverse__divide,axiom,
! [A: rat,B: rat] :
( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ B @ A ) ) ).
% inverse_divide
thf(fact_116_inverse__divide,axiom,
! [A: real,B: real] :
( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ B @ A ) ) ).
% inverse_divide
thf(fact_117_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_118_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_119_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_120_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_121_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_122_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_123_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_124_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_125_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_126_mult__divide__mult__cancel__left__if,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( C = zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= zero_zero_rat ) )
& ( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_127_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_128_zero__eq__1__divide__iff,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( divide_divide_rat @ one_one_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% zero_eq_1_divide_iff
thf(fact_129_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_130_one__divide__eq__0__iff,axiom,
! [A: rat] :
( ( ( divide_divide_rat @ one_one_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% one_divide_eq_0_iff
thf(fact_131_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_132_eq__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( one_one_rat
= ( divide_divide_rat @ B @ A ) )
= ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_133_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_134_divide__eq__eq__1,axiom,
! [B: rat,A: rat] :
( ( ( divide_divide_rat @ B @ A )
= one_one_rat )
= ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_135_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_136_divide__self__if,axiom,
! [A: rat] :
( ( ( A = zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= zero_zero_rat ) )
& ( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ) ).
% divide_self_if
thf(fact_137_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_138_divide__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% divide_self
thf(fact_139_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_140_one__eq__divide__iff,axiom,
! [A: rat,B: rat] :
( ( one_one_rat
= ( divide_divide_rat @ A @ B ) )
= ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_141_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_142_divide__eq__1__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= one_one_rat )
= ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_143_inverse__nonpositive__iff__nonpositive,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_144_inverse__nonpositive__iff__nonpositive,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_145_inverse__nonnegative__iff__nonnegative,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_146_inverse__nonnegative__iff__nonnegative,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_147_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_148_divide__le__0__1__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% divide_le_0_1_iff
thf(fact_149_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_150_zero__le__divide__1__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_151_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_152_nonzero__divide__mult__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
= ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_153_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_154_nonzero__divide__mult__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
= ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_155_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_156_divide__le__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_157_divide__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_158_divide__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_159_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_160_zero__le__divide__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_le_divide_iff
thf(fact_161_divide__nonneg__nonneg,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_162_divide__nonneg__nonneg,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_163_divide__nonneg__nonpos,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_164_divide__nonneg__nonpos,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_eq_rat @ Y2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ zero_zero_rat ) ) ) ).
% divide_nonneg_nonpos
thf(fact_165_divide__nonpos__nonneg,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_166_divide__nonpos__nonneg,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ zero_zero_rat ) ) ) ).
% divide_nonpos_nonneg
thf(fact_167_divide__nonpos__nonpos,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_168_divide__nonpos__nonpos,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_169_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_170_divide__right__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_171_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_172_zero__reorient,axiom,
! [X2: rat] :
( ( zero_zero_rat = X2 )
= ( X2 = zero_zero_rat ) ) ).
% zero_reorient
thf(fact_173_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_174_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_175_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_176_mem__Rep__preal__Ex,axiom,
! [X3: dedekind_preal] :
? [X4: rat] : ( member_rat @ X4 @ ( dedekind_Rep_preal @ X3 ) ) ).
% mem_Rep_preal_Ex
thf(fact_177_Rep__preal__inject,axiom,
! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ( dedekind_Rep_preal @ X2 )
= ( dedekind_Rep_preal @ Y2 ) )
= ( X2 = Y2 ) ) ).
% Rep_preal_inject
thf(fact_178_nonzero__eq__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_179_nonzero__eq__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( A
= ( divide_divide_rat @ B @ C ) )
= ( ( times_times_rat @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_180_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A )
= ( B
= ( times_times_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_181_nonzero__divide__eq__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( ( divide_divide_rat @ B @ C )
= A )
= ( B
= ( times_times_rat @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_182_eq__divide__imp,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= B )
=> ( A
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_183_eq__divide__imp,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ C )
= B )
=> ( A
= ( divide_divide_rat @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_184_divide__eq__imp,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_185_divide__eq__imp,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( B
= ( times_times_rat @ A @ C ) )
=> ( ( divide_divide_rat @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_186_eq__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_187_eq__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( A
= ( divide_divide_rat @ B @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ A @ C )
= B ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq
thf(fact_188_divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( divide_divide_real @ B @ C )
= A )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_189_divide__eq__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ( divide_divide_rat @ B @ C )
= A )
= ( ( ( C != zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq
thf(fact_190_frac__eq__eq,axiom,
! [Y2: real,Z: real,X2: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X2 @ Y2 )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X2 @ Z )
= ( times_times_real @ W @ Y2 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_191_frac__eq__eq,axiom,
! [Y2: rat,Z: rat,X2: rat,W: rat] :
( ( Y2 != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ( divide_divide_rat @ X2 @ Y2 )
= ( divide_divide_rat @ W @ Z ) )
= ( ( times_times_rat @ X2 @ Z )
= ( times_times_rat @ W @ Y2 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_192_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_193_right__inverse__eq,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( ( divide_divide_rat @ A @ B )
= one_one_rat )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_194_nonzero__inverse__eq__divide,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( inverse_inverse_rat @ A )
= ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_195_nonzero__inverse__eq__divide,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_196_inverse__le__1__iff,axiom,
! [X2: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X2 ) @ one_one_rat )
= ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
| ( ord_less_eq_rat @ one_one_rat @ X2 ) ) ) ).
% inverse_le_1_iff
thf(fact_197_inverse__le__1__iff,axiom,
! [X2: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ X2 ) @ one_one_real )
= ( ( ord_less_eq_real @ X2 @ zero_zero_real )
| ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% inverse_le_1_iff
thf(fact_198_times__divide__times__eq,axiom,
! [X2: real,Y2: real,Z: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y2 @ W ) ) ) ).
% times_divide_times_eq
thf(fact_199_times__divide__times__eq,axiom,
! [X2: rat,Y2: rat,Z: rat,W: rat] :
( ( times_times_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ ( divide_divide_rat @ Z @ W ) )
= ( divide_divide_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y2 @ W ) ) ) ).
% times_divide_times_eq
thf(fact_200_divide__divide__times__eq,axiom,
! [X2: real,Y2: real,Z: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X2 @ W ) @ ( times_times_real @ Y2 @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_201_divide__divide__times__eq,axiom,
! [X2: rat,Y2: rat,Z: rat,W: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ ( divide_divide_rat @ Z @ W ) )
= ( divide_divide_rat @ ( times_times_rat @ X2 @ W ) @ ( times_times_rat @ Y2 @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_202_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_203_divide__divide__eq__left_H,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
= ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_204_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% field_class.field_inverse_zero
thf(fact_205_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_206_inverse__zero__imp__zero,axiom,
! [A: rat] :
( ( ( inverse_inverse_rat @ A )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ).
% inverse_zero_imp_zero
thf(fact_207_inverse__zero__imp__zero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_208_nonzero__inverse__eq__imp__eq,axiom,
! [A: rat,B: rat] :
( ( ( inverse_inverse_rat @ A )
= ( inverse_inverse_rat @ B ) )
=> ( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_209_nonzero__inverse__eq__imp__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
=> ( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_210_nonzero__inverse__inverse__eq,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_211_nonzero__inverse__inverse__eq,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_212_nonzero__imp__inverse__nonzero,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( inverse_inverse_rat @ A )
!= zero_zero_rat ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_213_nonzero__imp__inverse__nonzero,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_214_mem__Rep__preal__mult__iff,axiom,
! [Z: rat,R: dedekind_preal,S2: dedekind_preal] :
( ( member_rat @ Z @ ( dedekind_Rep_preal @ ( times_3000655703912201937_preal @ R @ S2 ) ) )
= ( ? [X: rat] :
( ( member_rat @ X @ ( dedekind_Rep_preal @ R ) )
& ? [Y: rat] :
( ( member_rat @ Y @ ( dedekind_Rep_preal @ S2 ) )
& ( Z
= ( times_times_rat @ X @ Y ) ) ) ) ) ) ).
% mem_Rep_preal_mult_iff
thf(fact_215_field__class_Ofield__divide__inverse,axiom,
( divide_divide_rat
= ( ^ [A2: rat,B2: rat] : ( times_times_rat @ A2 @ ( inverse_inverse_rat @ B2 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_216_field__class_Ofield__divide__inverse,axiom,
( divide_divide_real
= ( ^ [A2: real,B2: real] : ( times_times_real @ A2 @ ( inverse_inverse_real @ B2 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_217_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_218_div__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% div_self
thf(fact_219_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_220_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_221_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_222_nonzero__mult__div__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_223_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_224_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_225_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_226_nonzero__mult__div__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_227_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_228_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_229_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_230_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_231_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_232_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_233_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_234_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_235_mult__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( C
= ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_left1
thf(fact_236_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_237_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_238_mult__cancel__left2,axiom,
! [C: rat,A: rat] :
( ( ( times_times_rat @ C @ A )
= C )
= ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_left2
thf(fact_239_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_240_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_241_mult__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( C
= ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_right1
thf(fact_242_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_243_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_244_mult__cancel__right2,axiom,
! [A: rat,C: rat] :
( ( ( times_times_rat @ A @ C )
= C )
= ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_right2
thf(fact_245_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_246_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_247_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_248_div__by__1,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ one_one_rat )
= A ) ).
% div_by_1
thf(fact_249_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_250_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_251_mult__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ( times_times_rat @ A @ C )
= ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_252_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_253_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_254_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_255_mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( times_times_rat @ C @ A )
= ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_256_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_257_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_258_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_259_mult__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% mult_eq_0_iff
thf(fact_260_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_261_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_262_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_263_mult__zero__right,axiom,
! [A: rat] :
( ( times_times_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% mult_zero_right
thf(fact_264_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_265_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_266_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_267_mult__zero__left,axiom,
! [A: rat] :
( ( times_times_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% mult_zero_left
thf(fact_268_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_269_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_270_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_271_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_272_div__0,axiom,
! [A: rat] :
( ( divide_divide_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% div_0
thf(fact_273_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_274_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_275_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_276_div__by__0,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% div_by_0
thf(fact_277_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_278_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_279_preal__le__def,axiom,
( ord_le5604041210740703414_preal
= ( ^ [R2: dedekind_preal,S: dedekind_preal] : ( ord_less_eq_set_rat @ ( dedekind_Rep_preal @ R2 ) @ ( dedekind_Rep_preal @ S ) ) ) ) ).
% preal_le_def
thf(fact_280_mult__right__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ C )
= ( times_times_rat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_281_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_282_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_283_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_284_mult__left__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ C @ A )
= ( times_times_rat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_285_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_286_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_287_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_288_no__zero__divisors,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( times_times_rat @ A @ B )
!= zero_zero_rat ) ) ) ).
% no_zero_divisors
thf(fact_289_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_290_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_291_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_292_divisors__zero,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= zero_zero_rat )
=> ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divisors_zero
thf(fact_293_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_294_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_295_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_296_mult__not__zero,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
!= zero_zero_rat )
=> ( ( A != zero_zero_rat )
& ( B != zero_zero_rat ) ) ) ).
% mult_not_zero
thf(fact_297_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_298_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_299_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_300_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
% zero_neq_one
thf(fact_301_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_302_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_303_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_304_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_305_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_306_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_307_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_308_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_309_zero__le__mult__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_le_mult_iff
thf(fact_310_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_311_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_312_mult__nonneg__nonpos2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_313_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_314_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_315_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_316_mult__nonpos__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_317_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_318_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_319_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_320_mult__nonneg__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_321_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_322_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_323_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_324_mult__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_325_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_326_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_327_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_328_split__mult__neg__le,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% split_mult_neg_le
thf(fact_329_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_330_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_331_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_332_mult__le__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_333_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_334_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_335_mult__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_336_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_337_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_338_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_339_mult__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_340_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_341_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_342_mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_343_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_344_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_345_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_346_mult__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_347_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_348_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_349_mult__left__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_350_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_351_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_352_split__mult__pos__le,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_353_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_354_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_355_zero__le__square,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% zero_le_square
thf(fact_356_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_357_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_358_mult__mono_H,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_359_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_360_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_361_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_362_mult__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_363_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_364_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_365_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_366_not__one__le__zero,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_le_zero
thf(fact_367_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_368_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_369_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_370_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_371_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_372_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_373_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_374_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one_class.zero_le_one
thf(fact_375_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_376_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_377_mult__left__le,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_378_mult__left__le,axiom,
! [C: rat,A: rat] :
( ( ord_less_eq_rat @ C @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_379_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_380_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_381_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_382_mult__le__one,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ B @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% mult_le_one
thf(fact_383_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_384_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_385_mult__right__le__one__le,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X2 @ Y2 ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_386_mult__right__le__one__le,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ X2 @ Y2 ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_387_mult__right__le__one__le,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X2 @ Y2 ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_388_mult__left__le__one__le,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y2 @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_389_mult__left__le__one__le,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Y2 @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_390_mult__left__le__one__le,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y2 @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_391_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_392_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_393_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_394_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_395_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_396_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_397_subsetI,axiom,
! [A3: set_rat,B3: set_rat] :
( ! [X4: rat] :
( ( member_rat @ X4 @ A3 )
=> ( member_rat @ X4 @ B3 ) )
=> ( ord_less_eq_set_rat @ A3 @ B3 ) ) ).
% subsetI
thf(fact_398_subset__antisym,axiom,
! [A3: set_rat,B3: set_rat] :
( ( ord_less_eq_set_rat @ A3 @ B3 )
=> ( ( ord_less_eq_set_rat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_399_order__refl,axiom,
! [X2: set_rat] : ( ord_less_eq_set_rat @ X2 @ X2 ) ).
% order_refl
thf(fact_400_order__refl,axiom,
! [X2: rat] : ( ord_less_eq_rat @ X2 @ X2 ) ).
% order_refl
thf(fact_401_order__refl,axiom,
! [X2: num] : ( ord_less_eq_num @ X2 @ X2 ) ).
% order_refl
thf(fact_402_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_403_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_404_dual__order_Orefl,axiom,
! [A: set_rat] : ( ord_less_eq_set_rat @ A @ A ) ).
% dual_order.refl
thf(fact_405_dual__order_Orefl,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% dual_order.refl
thf(fact_406_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_407_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_408_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_409_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_410_le__numeral__extra_I4_J,axiom,
ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% le_numeral_extra(4)
thf(fact_411_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_412_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_413_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_414_le__numeral__extra_I3_J,axiom,
ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% le_numeral_extra(3)
thf(fact_415_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_416_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_417_order__antisym__conv,axiom,
! [Y2: set_rat,X2: set_rat] :
( ( ord_less_eq_set_rat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_rat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_418_order__antisym__conv,axiom,
! [Y2: rat,X2: rat] :
( ( ord_less_eq_rat @ Y2 @ X2 )
=> ( ( ord_less_eq_rat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_419_order__antisym__conv,axiom,
! [Y2: num,X2: num] :
( ( ord_less_eq_num @ Y2 @ X2 )
=> ( ( ord_less_eq_num @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_420_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_421_order__antisym__conv,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_422_linorder__le__cases,axiom,
! [X2: rat,Y2: rat] :
( ~ ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_423_linorder__le__cases,axiom,
! [X2: num,Y2: num] :
( ~ ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_num @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_424_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_425_linorder__le__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_426_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_427_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_428_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_429_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_430_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_431_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_432_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_433_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_434_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_435_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_436_ord__eq__le__subst,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_437_ord__eq__le__subst,axiom,
! [A: num,F: rat > num,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_438_ord__eq__le__subst,axiom,
! [A: nat,F: rat > nat,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_439_ord__eq__le__subst,axiom,
! [A: int,F: rat > int,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_440_ord__eq__le__subst,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_441_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_442_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_443_ord__eq__le__subst,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_444_ord__eq__le__subst,axiom,
! [A: rat,F: nat > rat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_445_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_446_linorder__linear,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
| ( ord_less_eq_rat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_447_linorder__linear,axiom,
! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
| ( ord_less_eq_num @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_448_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_449_linorder__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_450_order__eq__refl,axiom,
! [X2: set_rat,Y2: set_rat] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_rat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_451_order__eq__refl,axiom,
! [X2: rat,Y2: rat] :
( ( X2 = Y2 )
=> ( ord_less_eq_rat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_452_order__eq__refl,axiom,
! [X2: num,Y2: num] :
( ( X2 = Y2 )
=> ( ord_less_eq_num @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_453_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_454_order__eq__refl,axiom,
! [X2: int,Y2: int] :
( ( X2 = Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_455_order__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_456_order__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_457_order__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_458_order__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_459_order__subst2,axiom,
! [A: num,B: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_460_order__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_461_order__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_462_order__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_463_order__subst2,axiom,
! [A: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_464_order__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_465_order__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_466_order__subst1,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_467_order__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_468_order__subst1,axiom,
! [A: rat,F: int > rat,B: int,C: int] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_469_order__subst1,axiom,
! [A: num,F: rat > num,B: rat,C: rat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_470_order__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_471_order__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_472_order__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_473_order__subst1,axiom,
! [A: nat,F: rat > nat,B: rat,C: rat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_474_order__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_475_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_rat,Z2: set_rat] : ( Y4 = Z2 ) )
= ( ^ [A2: set_rat,B2: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
& ( ord_less_eq_set_rat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_476_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: rat,Z2: rat] : ( Y4 = Z2 ) )
= ( ^ [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
& ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_477_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: num,Z2: num] : ( Y4 = Z2 ) )
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
& ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_478_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_479_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_480_antisym,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( ord_less_eq_set_rat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_481_antisym,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_482_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_483_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_484_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_485_dual__order_Otrans,axiom,
! [B: set_rat,A: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ B @ A )
=> ( ( ord_less_eq_set_rat @ C @ B )
=> ( ord_less_eq_set_rat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_486_dual__order_Otrans,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_eq_rat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_487_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_488_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_489_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_490_dual__order_Oantisym,axiom,
! [B: set_rat,A: set_rat] :
( ( ord_less_eq_set_rat @ B @ A )
=> ( ( ord_less_eq_set_rat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_491_dual__order_Oantisym,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_492_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_493_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_494_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_495_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_rat,Z2: set_rat] : ( Y4 = Z2 ) )
= ( ^ [A2: set_rat,B2: set_rat] :
( ( ord_less_eq_set_rat @ B2 @ A2 )
& ( ord_less_eq_set_rat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_496_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: rat,Z2: rat] : ( Y4 = Z2 ) )
= ( ^ [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
& ( ord_less_eq_rat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_497_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: num,Z2: num] : ( Y4 = Z2 ) )
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
& ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_498_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_499_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_500_linorder__wlog,axiom,
! [P: rat > rat > $o,A: rat,B: rat] :
( ! [A4: rat,B4: rat] :
( ( ord_less_eq_rat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: rat,B4: rat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_501_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B4: num] :
( ( ord_less_eq_num @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: num,B4: num] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_502_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_503_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_504_order__trans,axiom,
! [X2: set_rat,Y2: set_rat,Z: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_rat @ Y2 @ Z )
=> ( ord_less_eq_set_rat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_505_order__trans,axiom,
! [X2: rat,Y2: rat,Z: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ Z )
=> ( ord_less_eq_rat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_506_order__trans,axiom,
! [X2: num,Y2: num,Z: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ( ord_less_eq_num @ Y2 @ Z )
=> ( ord_less_eq_num @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_507_order__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_508_order__trans,axiom,
! [X2: int,Y2: int,Z: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z )
=> ( ord_less_eq_int @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_509_order_Otrans,axiom,
! [A: set_rat,B: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( ord_less_eq_set_rat @ B @ C )
=> ( ord_less_eq_set_rat @ A @ C ) ) ) ).
% order.trans
thf(fact_510_order_Otrans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ A @ C ) ) ) ).
% order.trans
thf(fact_511_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_512_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_513_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_514_order__antisym,axiom,
! [X2: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_rat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_515_order__antisym,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_516_order__antisym,axiom,
! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ( ord_less_eq_num @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_517_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_518_order__antisym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_519_ord__le__eq__trans,axiom,
! [A: set_rat,B: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_rat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_520_ord__le__eq__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_rat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_521_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_522_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_523_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_524_ord__eq__le__trans,axiom,
! [A: set_rat,B: set_rat,C: set_rat] :
( ( A = B )
=> ( ( ord_less_eq_set_rat @ B @ C )
=> ( ord_less_eq_set_rat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_525_ord__eq__le__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_526_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_527_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_528_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_529_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_rat,Z2: set_rat] : ( Y4 = Z2 ) )
= ( ^ [X: set_rat,Y: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y )
& ( ord_less_eq_set_rat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_530_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: rat,Z2: rat] : ( Y4 = Z2 ) )
= ( ^ [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
& ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_531_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: num,Z2: num] : ( Y4 = Z2 ) )
= ( ^ [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
& ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_532_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_533_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_534_le__cases3,axiom,
! [X2: rat,Y2: rat,Z: rat] :
( ( ( ord_less_eq_rat @ X2 @ Y2 )
=> ~ ( ord_less_eq_rat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_rat @ Y2 @ X2 )
=> ~ ( ord_less_eq_rat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_rat @ X2 @ Z )
=> ~ ( ord_less_eq_rat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_rat @ Z @ Y2 )
=> ~ ( ord_less_eq_rat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_rat @ Y2 @ Z )
=> ~ ( ord_less_eq_rat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_rat @ Z @ X2 )
=> ~ ( ord_less_eq_rat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_535_le__cases3,axiom,
! [X2: num,Y2: num,Z: num] :
( ( ( ord_less_eq_num @ X2 @ Y2 )
=> ~ ( ord_less_eq_num @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_num @ Y2 @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ Z ) )
=> ( ( ( ord_less_eq_num @ X2 @ Z )
=> ~ ( ord_less_eq_num @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_num @ Z @ Y2 )
=> ~ ( ord_less_eq_num @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_num @ Y2 @ Z )
=> ~ ( ord_less_eq_num @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_num @ Z @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_536_le__cases3,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_537_le__cases3,axiom,
! [X2: int,Y2: int,Z: int] :
( ( ( ord_less_eq_int @ X2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z )
=> ~ ( ord_less_eq_int @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_538_nle__le,axiom,
! [A: rat,B: rat] :
( ( ~ ( ord_less_eq_rat @ A @ B ) )
= ( ( ord_less_eq_rat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_539_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_540_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_541_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_542_Collect__mono__iff,axiom,
! [P: rat > $o,Q: rat > $o] :
( ( ord_less_eq_set_rat @ ( collect_rat @ P ) @ ( collect_rat @ Q ) )
= ( ! [X: rat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_543_set__eq__subset,axiom,
( ( ^ [Y4: set_rat,Z2: set_rat] : ( Y4 = Z2 ) )
= ( ^ [A5: set_rat,B5: set_rat] :
( ( ord_less_eq_set_rat @ A5 @ B5 )
& ( ord_less_eq_set_rat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_544_subset__trans,axiom,
! [A3: set_rat,B3: set_rat,C2: set_rat] :
( ( ord_less_eq_set_rat @ A3 @ B3 )
=> ( ( ord_less_eq_set_rat @ B3 @ C2 )
=> ( ord_less_eq_set_rat @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_545_Collect__mono,axiom,
! [P: rat > $o,Q: rat > $o] :
( ! [X4: rat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_rat @ ( collect_rat @ P ) @ ( collect_rat @ Q ) ) ) ).
% Collect_mono
thf(fact_546_subset__refl,axiom,
! [A3: set_rat] : ( ord_less_eq_set_rat @ A3 @ A3 ) ).
% subset_refl
thf(fact_547_subset__iff,axiom,
( ord_less_eq_set_rat
= ( ^ [A5: set_rat,B5: set_rat] :
! [T: rat] :
( ( member_rat @ T @ A5 )
=> ( member_rat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_548_equalityD2,axiom,
! [A3: set_rat,B3: set_rat] :
( ( A3 = B3 )
=> ( ord_less_eq_set_rat @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_549_equalityD1,axiom,
! [A3: set_rat,B3: set_rat] :
( ( A3 = B3 )
=> ( ord_less_eq_set_rat @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_550_subset__eq,axiom,
( ord_less_eq_set_rat
= ( ^ [A5: set_rat,B5: set_rat] :
! [X: rat] :
( ( member_rat @ X @ A5 )
=> ( member_rat @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_551_equalityE,axiom,
! [A3: set_rat,B3: set_rat] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq_set_rat @ A3 @ B3 )
=> ~ ( ord_less_eq_set_rat @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_552_subsetD,axiom,
! [A3: set_rat,B3: set_rat,C: rat] :
( ( ord_less_eq_set_rat @ A3 @ B3 )
=> ( ( member_rat @ C @ A3 )
=> ( member_rat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_553_in__mono,axiom,
! [A3: set_rat,B3: set_rat,X2: rat] :
( ( ord_less_eq_set_rat @ A3 @ B3 )
=> ( ( member_rat @ X2 @ A3 )
=> ( member_rat @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_554_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
= one_one_rat ) ).
% dbl_inc_simps(2)
thf(fact_555_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_556_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_557_divide__le__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_558_divide__le__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_559_divide__le__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_560_divide__le__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_561_le__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_562_le__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_563_le__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_564_le__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_565_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_566_divide__rat__def,axiom,
( divide_divide_rat
= ( ^ [Q2: rat,R2: rat] : ( times_times_rat @ Q2 @ ( inverse_inverse_rat @ R2 ) ) ) ) ).
% divide_rat_def
thf(fact_567_distrib__subset1,axiom,
! [W: dedekind_preal,X2: dedekind_preal,Y2: dedekind_preal] : ( ord_less_eq_set_rat @ ( dedekind_Rep_preal @ ( times_3000655703912201937_preal @ W @ ( plus_p3173629198307831117_preal @ X2 @ Y2 ) ) ) @ ( dedekind_Rep_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ W @ X2 ) @ ( times_3000655703912201937_preal @ W @ Y2 ) ) ) ) ).
% distrib_subset1
thf(fact_568_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_569_add__right__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_570_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_571_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_572_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_573_add__left__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_574_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_575_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_576_psubsetI,axiom,
! [A3: set_rat,B3: set_rat] :
( ( ord_less_eq_set_rat @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_set_rat @ A3 @ B3 ) ) ) ).
% psubsetI
thf(fact_577_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_578_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_579_add__le__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_580_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_581_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_582_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_583_add__le__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_584_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_585_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_586_add__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add_0
thf(fact_587_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_588_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_589_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_590_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y2 ) )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_591_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_592_add__cancel__right__right,axiom,
! [A: rat,B: rat] :
( ( A
= ( plus_plus_rat @ A @ B ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_right
thf(fact_593_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_594_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_595_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_596_add__cancel__right__left,axiom,
! [A: rat,B: rat] :
( ( A
= ( plus_plus_rat @ B @ A ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_left
thf(fact_597_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_598_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_599_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_600_add__cancel__left__right,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= A )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_right
thf(fact_601_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_602_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_603_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_604_add__cancel__left__left,axiom,
! [B: rat,A: rat] :
( ( ( plus_plus_rat @ B @ A )
= A )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_left
thf(fact_605_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_606_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_607_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_608_double__zero__sym,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A @ A ) )
= ( A = zero_zero_rat ) ) ).
% double_zero_sym
thf(fact_609_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_610_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_611_add_Oright__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.right_neutral
thf(fact_612_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_613_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_614_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_615_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_616_add__less__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
= ( ord_less_rat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_617_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_618_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_619_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_620_add__less__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
= ( ord_less_rat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_621_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_622_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_623_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_624_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_625_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_626_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_627_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_628_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_629_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_630_le__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel2
thf(fact_631_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_632_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_633_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_634_le__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel1
thf(fact_635_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_636_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_637_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_638_add__le__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% add_le_same_cancel2
thf(fact_639_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_640_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_641_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_642_add__le__same__cancel1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% add_le_same_cancel1
thf(fact_643_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_644_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_645_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_646_add__less__same__cancel1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% add_less_same_cancel1
thf(fact_647_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_648_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_649_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_650_add__less__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% add_less_same_cancel2
thf(fact_651_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_652_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_653_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_654_less__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel1
thf(fact_655_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_656_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_657_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_658_less__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel2
thf(fact_659_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_660_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_661_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_662_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_663_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_664_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_665_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_666_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_667_inverse__less__iff__less,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( ord_less_rat @ B @ A ) ) ) ) ).
% inverse_less_iff_less
thf(fact_668_inverse__less__iff__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A ) ) ) ) ).
% inverse_less_iff_less
thf(fact_669_inverse__less__iff__less__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( ord_less_rat @ B @ A ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_670_inverse__less__iff__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_671_inverse__negative__iff__negative,axiom,
! [A: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% inverse_negative_iff_negative
thf(fact_672_inverse__negative__iff__negative,axiom,
! [A: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% inverse_negative_iff_negative
thf(fact_673_inverse__positive__iff__positive,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% inverse_positive_iff_positive
thf(fact_674_inverse__positive__iff__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% inverse_positive_iff_positive
thf(fact_675_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_676_zero__less__divide__1__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_677_less__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_678_less__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_rat @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_679_less__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_680_less__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_rat @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_681_divide__less__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_682_divide__less__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_rat @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_683_divide__less__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_684_divide__less__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_rat @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_685_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_686_divide__less__0__1__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% divide_less_0_1_iff
thf(fact_687_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_688_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_689_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_690_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_691_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_692_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_693_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_694_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_695_inverse__le__iff__le__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_696_inverse__le__iff__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_697_inverse__le__iff__le,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% inverse_le_iff_le
thf(fact_698_inverse__le__iff__le,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ) ).
% inverse_le_iff_le
thf(fact_699_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_700_add__less__le__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_701_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_702_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_703_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_704_add__le__less__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_705_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_706_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_707_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_708_add__mono__thms__linordered__field_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( ord_less_eq_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_709_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_710_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_711_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_712_add__mono__thms__linordered__field_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( ord_less_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_713_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_714_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_715_order__less__imp__not__less,axiom,
! [X2: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X2 @ Y2 )
=> ~ ( ord_less_set_rat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_716_order__less__imp__not__less,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ~ ( ord_less_rat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_717_order__less__imp__not__less,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ~ ( ord_less_num @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_718_order__less__imp__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_719_order__less__imp__not__less,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_720_order__less__imp__not__eq2,axiom,
! [X2: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_721_order__less__imp__not__eq2,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_722_order__less__imp__not__eq2,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_723_order__less__imp__not__eq2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_724_order__less__imp__not__eq2,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_725_order__less__imp__not__eq,axiom,
! [X2: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_726_order__less__imp__not__eq,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_727_order__less__imp__not__eq,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_728_order__less__imp__not__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_729_order__less__imp__not__eq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_730_linorder__less__linear,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_rat @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_731_linorder__less__linear,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_num @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_732_linorder__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_733_linorder__less__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_734_order__less__imp__triv,axiom,
! [X2: set_rat,Y2: set_rat,P: $o] :
( ( ord_less_set_rat @ X2 @ Y2 )
=> ( ( ord_less_set_rat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_735_order__less__imp__triv,axiom,
! [X2: rat,Y2: rat,P: $o] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ( ord_less_rat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_736_order__less__imp__triv,axiom,
! [X2: num,Y2: num,P: $o] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ( ord_less_num @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_737_order__less__imp__triv,axiom,
! [X2: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_738_order__less__imp__triv,axiom,
! [X2: int,Y2: int,P: $o] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_739_order__less__not__sym,axiom,
! [X2: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X2 @ Y2 )
=> ~ ( ord_less_set_rat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_740_order__less__not__sym,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ~ ( ord_less_rat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_741_order__less__not__sym,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ~ ( ord_less_num @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_742_order__less__not__sym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_743_order__less__not__sym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_744_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_745_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_746_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_747_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_748_order__less__subst2,axiom,
! [A: num,B: num,F: num > rat,C: rat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_749_order__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_750_order__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_751_order__less__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_752_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_753_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_754_order__less__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_755_order__less__subst1,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_756_order__less__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_757_order__less__subst1,axiom,
! [A: rat,F: int > rat,B: int,C: int] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_758_order__less__subst1,axiom,
! [A: num,F: rat > num,B: rat,C: rat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_759_order__less__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_760_order__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_761_order__less__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_762_order__less__subst1,axiom,
! [A: nat,F: rat > nat,B: rat,C: rat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_763_order__less__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_764_order__less__irrefl,axiom,
! [X2: set_rat] :
~ ( ord_less_set_rat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_765_order__less__irrefl,axiom,
! [X2: rat] :
~ ( ord_less_rat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_766_order__less__irrefl,axiom,
! [X2: num] :
~ ( ord_less_num @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_767_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_768_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_769_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_770_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_771_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_772_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_773_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > rat,C: rat] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_774_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_775_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_776_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_777_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_778_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_779_ord__eq__less__subst,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_780_ord__eq__less__subst,axiom,
! [A: num,F: rat > num,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_781_ord__eq__less__subst,axiom,
! [A: nat,F: rat > nat,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_782_ord__eq__less__subst,axiom,
! [A: int,F: rat > int,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_783_ord__eq__less__subst,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_784_ord__eq__less__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_785_ord__eq__less__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_786_ord__eq__less__subst,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_787_ord__eq__less__subst,axiom,
! [A: rat,F: nat > rat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_788_ord__eq__less__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_789_order__less__trans,axiom,
! [X2: set_rat,Y2: set_rat,Z: set_rat] :
( ( ord_less_set_rat @ X2 @ Y2 )
=> ( ( ord_less_set_rat @ Y2 @ Z )
=> ( ord_less_set_rat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_790_order__less__trans,axiom,
! [X2: rat,Y2: rat,Z: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ( ord_less_rat @ Y2 @ Z )
=> ( ord_less_rat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_791_order__less__trans,axiom,
! [X2: num,Y2: num,Z: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ( ord_less_num @ Y2 @ Z )
=> ( ord_less_num @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_792_order__less__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_793_order__less__trans,axiom,
! [X2: int,Y2: int,Z: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z )
=> ( ord_less_int @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_794_order__less__asym_H,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ~ ( ord_less_set_rat @ B @ A ) ) ).
% order_less_asym'
thf(fact_795_order__less__asym_H,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order_less_asym'
thf(fact_796_order__less__asym_H,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order_less_asym'
thf(fact_797_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_798_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_799_linorder__neq__iff,axiom,
! [X2: rat,Y2: rat] :
( ( X2 != Y2 )
= ( ( ord_less_rat @ X2 @ Y2 )
| ( ord_less_rat @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_800_linorder__neq__iff,axiom,
! [X2: num,Y2: num] :
( ( X2 != Y2 )
= ( ( ord_less_num @ X2 @ Y2 )
| ( ord_less_num @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_801_linorder__neq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_802_linorder__neq__iff,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
= ( ( ord_less_int @ X2 @ Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_803_order__less__asym,axiom,
! [X2: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X2 @ Y2 )
=> ~ ( ord_less_set_rat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_804_order__less__asym,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ~ ( ord_less_rat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_805_order__less__asym,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ~ ( ord_less_num @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_806_order__less__asym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_807_order__less__asym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_808_linorder__neqE,axiom,
! [X2: rat,Y2: rat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_rat @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_809_linorder__neqE,axiom,
! [X2: num,Y2: num] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_num @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_810_linorder__neqE,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_811_linorder__neqE,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_812_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_rat,A: set_rat] :
( ( ord_less_set_rat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_813_dual__order_Ostrict__implies__not__eq,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_814_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_815_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_816_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_817_order_Ostrict__implies__not__eq,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_818_order_Ostrict__implies__not__eq,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_819_order_Ostrict__implies__not__eq,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_820_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_821_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_822_dual__order_Ostrict__trans,axiom,
! [B: set_rat,A: set_rat,C: set_rat] :
( ( ord_less_set_rat @ B @ A )
=> ( ( ord_less_set_rat @ C @ B )
=> ( ord_less_set_rat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_823_dual__order_Ostrict__trans,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C @ B )
=> ( ord_less_rat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_824_dual__order_Ostrict__trans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_825_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_826_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_827_not__less__iff__gr__or__eq,axiom,
! [X2: rat,Y2: rat] :
( ( ~ ( ord_less_rat @ X2 @ Y2 ) )
= ( ( ord_less_rat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_828_not__less__iff__gr__or__eq,axiom,
! [X2: num,Y2: num] :
( ( ~ ( ord_less_num @ X2 @ Y2 ) )
= ( ( ord_less_num @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_829_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_830_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_831_order_Ostrict__trans,axiom,
! [A: set_rat,B: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ( ord_less_set_rat @ B @ C )
=> ( ord_less_set_rat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_832_order_Ostrict__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_833_order_Ostrict__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_834_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_835_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_836_linorder__less__wlog,axiom,
! [P: rat > rat > $o,A: rat,B: rat] :
( ! [A4: rat,B4: rat] :
( ( ord_less_rat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: rat] : ( P @ A4 @ A4 )
=> ( ! [A4: rat,B4: rat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_837_linorder__less__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B4: num] :
( ( ord_less_num @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: num] : ( P @ A4 @ A4 )
=> ( ! [A4: num,B4: num] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_838_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_839_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_840_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_841_dual__order_Oirrefl,axiom,
! [A: set_rat] :
~ ( ord_less_set_rat @ A @ A ) ).
% dual_order.irrefl
thf(fact_842_dual__order_Oirrefl,axiom,
! [A: rat] :
~ ( ord_less_rat @ A @ A ) ).
% dual_order.irrefl
thf(fact_843_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_844_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_845_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_846_dual__order_Oasym,axiom,
! [B: set_rat,A: set_rat] :
( ( ord_less_set_rat @ B @ A )
=> ~ ( ord_less_set_rat @ A @ B ) ) ).
% dual_order.asym
thf(fact_847_dual__order_Oasym,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ~ ( ord_less_rat @ A @ B ) ) ).
% dual_order.asym
thf(fact_848_dual__order_Oasym,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ A @ B ) ) ).
% dual_order.asym
thf(fact_849_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_850_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_851_linorder__cases,axiom,
! [X2: rat,Y2: rat] :
( ~ ( ord_less_rat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_rat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_852_linorder__cases,axiom,
! [X2: num,Y2: num] :
( ~ ( ord_less_num @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_num @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_853_linorder__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_854_linorder__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_855_antisym__conv3,axiom,
! [Y2: rat,X2: rat] :
( ~ ( ord_less_rat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_rat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_856_antisym__conv3,axiom,
! [Y2: num,X2: num] :
( ~ ( ord_less_num @ Y2 @ X2 )
=> ( ( ~ ( ord_less_num @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_857_antisym__conv3,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_858_antisym__conv3,axiom,
! [Y2: int,X2: int] :
( ~ ( ord_less_int @ Y2 @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_859_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_860_ord__less__eq__trans,axiom,
! [A: set_rat,B: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ( B = C )
=> ( ord_less_set_rat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_861_ord__less__eq__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( B = C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_862_ord__less__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_863_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_864_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_865_ord__eq__less__trans,axiom,
! [A: set_rat,B: set_rat,C: set_rat] :
( ( A = B )
=> ( ( ord_less_set_rat @ B @ C )
=> ( ord_less_set_rat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_866_ord__eq__less__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_867_ord__eq__less__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_868_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_869_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_870_order_Oasym,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ~ ( ord_less_set_rat @ B @ A ) ) ).
% order.asym
thf(fact_871_order_Oasym,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order.asym
thf(fact_872_order_Oasym,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order.asym
thf(fact_873_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_874_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_875_less__imp__neq,axiom,
! [X2: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_876_less__imp__neq,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_877_less__imp__neq,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_878_less__imp__neq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_879_less__imp__neq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_880_dense,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ? [Z3: rat] :
( ( ord_less_rat @ X2 @ Z3 )
& ( ord_less_rat @ Z3 @ Y2 ) ) ) ).
% dense
thf(fact_881_gt__ex,axiom,
! [X2: rat] :
? [X_1: rat] : ( ord_less_rat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_882_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_883_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_884_lt__ex,axiom,
! [X2: rat] :
? [Y3: rat] : ( ord_less_rat @ Y3 @ X2 ) ).
% lt_ex
thf(fact_885_lt__ex,axiom,
! [X2: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X2 ) ).
% lt_ex
thf(fact_886_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_887_is__num__normalize_I1_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_888_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_889_preal__less__def,axiom,
( ord_le5708704896291381698_preal
= ( ^ [R2: dedekind_preal,S: dedekind_preal] : ( ord_less_set_rat @ ( dedekind_Rep_preal @ R2 ) @ ( dedekind_Rep_preal @ S ) ) ) ) ).
% preal_less_def
thf(fact_890_add__eq__exists,axiom,
! [A: real,B: real] :
? [X4: real] :
( ( plus_plus_real @ A @ X4 )
= B ) ).
% add_eq_exists
thf(fact_891_add__eq__exists,axiom,
! [A: rat,B: rat] :
? [X4: rat] :
( ( plus_plus_rat @ A @ X4 )
= B ) ).
% add_eq_exists
thf(fact_892_add__eq__exists,axiom,
! [A: int,B: int] :
? [X4: int] :
( ( plus_plus_int @ A @ X4 )
= B ) ).
% add_eq_exists
thf(fact_893_preal__add__assoc,axiom,
! [X2: dedekind_preal,Y2: dedekind_preal,Z: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ X2 @ Y2 ) @ Z )
= ( plus_p3173629198307831117_preal @ X2 @ ( plus_p3173629198307831117_preal @ Y2 @ Z ) ) ) ).
% preal_add_assoc
thf(fact_894_preal__add__commute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [X: dedekind_preal,Y: dedekind_preal] : ( plus_p3173629198307831117_preal @ Y @ X ) ) ) ).
% preal_add_commute
thf(fact_895_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_896_add__less__imp__less__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
=> ( ord_less_rat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_897_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_898_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_899_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_900_add__less__imp__less__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
=> ( ord_less_rat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_901_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_902_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_903_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_904_add__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_905_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_906_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_907_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_908_add__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_909_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_910_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_911_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_912_add__strict__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_913_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_914_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_915_linordered__field__no__ub,axiom,
! [X6: rat] :
? [X_1: rat] : ( ord_less_rat @ X6 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_916_linordered__field__no__lb,axiom,
! [X6: rat] :
? [Y3: rat] : ( ord_less_rat @ Y3 @ X6 ) ).
% linordered_field_no_lb
thf(fact_917_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_918_add__right__imp__eq,axiom,
! [B: rat,A: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_919_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_920_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_921_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_922_add__left__imp__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_923_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_924_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_925_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_926_add_Oleft__commute,axiom,
! [B: rat,A: rat,C: rat] :
( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_927_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_928_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_929_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A2: real,B2: real] : ( plus_plus_real @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_930_add_Ocommute,axiom,
( plus_plus_rat
= ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_931_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_932_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_933_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_934_add_Oright__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_935_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_936_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_937_add_Oleft__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_938_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_939_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_940_add_Oassoc,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% add.assoc
thf(fact_941_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_942_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_943_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_944_group__cancel_Oadd2,axiom,
! [B3: rat,K: rat,B: rat,A: rat] :
( ( B3
= ( plus_plus_rat @ K @ B ) )
=> ( ( plus_plus_rat @ A @ B3 )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_945_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_946_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_947_group__cancel_Oadd1,axiom,
! [A3: real,K: real,A: real,B: real] :
( ( A3
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A3 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_948_group__cancel_Oadd1,axiom,
! [A3: rat,K: rat,A: rat,B: rat] :
( ( A3
= ( plus_plus_rat @ K @ A ) )
=> ( ( plus_plus_rat @ A3 @ B )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_949_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_950_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_951_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_952_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_rat @ I @ K )
= ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_953_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_954_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_955_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_956_add__mono__thms__linordered__field_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( K = L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_957_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_958_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_959_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_960_add__mono__thms__linordered__field_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( ord_less_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_961_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_962_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_963_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_964_add__mono__thms__linordered__field_I5_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( ord_less_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_965_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_966_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_967_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_968_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_969_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_970_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_971_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_972_add__neg__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_neg
thf(fact_973_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_974_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_975_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_976_add__pos__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_977_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_978_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_979_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_980_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_981_pos__add__strict,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_982_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_983_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_984_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_985_less__add__one,axiom,
! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% less_add_one
thf(fact_986_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_987_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_988_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_989_add__mono1,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% add_mono1
thf(fact_990_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_991_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_992_linorder__neqE__linordered__idom,axiom,
! [X2: rat,Y2: rat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_rat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_993_linorder__neqE__linordered__idom,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_994_add__less__zeroD,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ ( plus_plus_real @ X2 @ Y2 ) @ zero_zero_real )
=> ( ( ord_less_real @ X2 @ zero_zero_real )
| ( ord_less_real @ Y2 @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_995_add__less__zeroD,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ X2 @ Y2 ) @ zero_zero_rat )
=> ( ( ord_less_rat @ X2 @ zero_zero_rat )
| ( ord_less_rat @ Y2 @ zero_zero_rat ) ) ) ).
% add_less_zeroD
thf(fact_996_add__less__zeroD,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ ( plus_plus_int @ X2 @ Y2 ) @ zero_zero_int )
=> ( ( ord_less_int @ X2 @ zero_zero_int )
| ( ord_less_int @ Y2 @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_997_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X: real] : ( plus_plus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_998_dbl__inc__def,axiom,
( neg_nu5219082963157363817nc_rat
= ( ^ [X: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).
% dbl_inc_def
thf(fact_999_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X: int] : ( plus_plus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_1000_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1001_add__strict__increasing2,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1002_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1003_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1004_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1005_add__strict__increasing,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1006_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1007_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1008_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1009_add__pos__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1010_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1011_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1012_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_1013_add__nonpos__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_nonpos_neg
thf(fact_1014_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1015_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_1016_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1017_add__nonneg__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1018_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1019_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1020_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_1021_add__neg__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_nonpos
thf(fact_1022_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1023_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_1024_field__le__epsilon,axiom,
! [X2: real,Y2: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ X2 @ ( plus_plus_real @ Y2 @ E ) ) )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% field_le_epsilon
thf(fact_1025_field__le__epsilon,axiom,
! [X2: rat,Y2: rat] :
( ! [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
=> ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ Y2 @ E ) ) )
=> ( ord_less_eq_rat @ X2 @ Y2 ) ) ).
% field_le_epsilon
thf(fact_1026_not__sum__squares__lt__zero,axiom,
! [X2: real,Y2: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_1027_not__sum__squares__lt__zero,axiom,
! [X2: rat,Y2: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y2 @ Y2 ) ) @ zero_zero_rat ) ).
% not_sum_squares_lt_zero
thf(fact_1028_not__sum__squares__lt__zero,axiom,
! [X2: int,Y2: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_1029_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_1030_zero__less__two,axiom,
ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% zero_less_two
thf(fact_1031_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1032_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1033_less__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% less_half_sum
thf(fact_1034_less__half__sum,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% less_half_sum
thf(fact_1035_gt__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% gt_half_sum
thf(fact_1036_gt__half__sum,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% gt_half_sum
thf(fact_1037_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1038_add__le__imp__le__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1039_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1040_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1041_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1042_add__le__imp__le__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1043_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1044_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1045_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C4: nat] :
( B2
= ( plus_plus_nat @ A2 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_1046_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1047_add__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1048_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1049_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1050_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_1051_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1052_add__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1053_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1054_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1055_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1056_add__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1057_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1058_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1059_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1060_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( ord_less_eq_rat @ K @ L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1061_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1062_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1063_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1064_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( ord_less_eq_rat @ K @ L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1065_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1066_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1067_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1068_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1069_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1070_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1071_add_Ogroup__left__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1072_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1073_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1074_add_Ocomm__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.comm_neutral
thf(fact_1075_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1076_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_1077_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_1078_comm__monoid__add__class_Oadd__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1079_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1080_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1081_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1082_order__le__imp__less__or__eq,axiom,
! [X2: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ Y2 )
=> ( ( ord_less_set_rat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1083_order__le__imp__less__or__eq,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ( ord_less_rat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1084_order__le__imp__less__or__eq,axiom,
! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ( ord_less_num @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1085_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1086_order__le__imp__less__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1087_linorder__le__less__linear,axiom,
! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
| ( ord_less_rat @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_1088_linorder__le__less__linear,axiom,
! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
| ( ord_less_num @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_1089_linorder__le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_1090_linorder__le__less__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_1091_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1092_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > rat,C: rat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1093_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1094_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > rat,C: rat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1095_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1096_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1097_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1098_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1099_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1100_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1101_order__less__le__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1102_order__less__le__subst1,axiom,
! [A: num,F: rat > num,B: rat,C: rat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1103_order__less__le__subst1,axiom,
! [A: nat,F: rat > nat,B: rat,C: rat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1104_order__less__le__subst1,axiom,
! [A: int,F: rat > int,B: rat,C: rat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1105_order__less__le__subst1,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1106_order__less__le__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1107_order__less__le__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1108_order__less__le__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1109_order__less__le__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1110_order__less__le__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1111_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1112_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1113_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1114_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1115_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1116_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1117_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1118_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1119_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1120_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1121_order__le__less__subst1,axiom,
! [A: int,F: rat > int,B: rat,C: rat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1122_order__le__less__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1123_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1124_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1125_not__in__Rep__preal__ub,axiom,
! [X2: rat,Xa: dedekind_preal,Y2: rat] :
( ~ ( member_rat @ X2 @ ( dedekind_Rep_preal @ Xa ) )
=> ( ( member_rat @ Y2 @ ( dedekind_Rep_preal @ Xa ) )
=> ( ( ord_less_rat @ zero_zero_rat @ X2 )
=> ( ord_less_rat @ Y2 @ X2 ) ) ) ) ).
% not_in_Rep_preal_ub
thf(fact_1126_Rep__preal__exists__bound,axiom,
! [X3: dedekind_preal] :
? [X4: rat] :
( ( ord_less_rat @ zero_zero_rat @ X4 )
& ~ ( member_rat @ X4 @ ( dedekind_Rep_preal @ X3 ) ) ) ).
% Rep_preal_exists_bound
thf(fact_1127_preal__add__mult__distrib2,axiom,
! [W: dedekind_preal,X2: dedekind_preal,Y2: dedekind_preal] :
( ( times_3000655703912201937_preal @ W @ ( plus_p3173629198307831117_preal @ X2 @ Y2 ) )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ W @ X2 ) @ ( times_3000655703912201937_preal @ W @ Y2 ) ) ) ).
% preal_add_mult_distrib2
thf(fact_1128_preal__add__mult__distrib,axiom,
! [X2: dedekind_preal,Y2: dedekind_preal,W: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ X2 @ Y2 ) @ W )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X2 @ W ) @ ( times_3000655703912201937_preal @ Y2 @ W ) ) ) ).
% preal_add_mult_distrib
thf(fact_1129_subset__inverse__mult__lemma,axiom,
! [X2: rat,R3: dedekind_preal] :
( ( ord_less_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_rat @ X2 @ one_one_rat )
=> ? [V: rat,U: rat,Y3: rat] :
( ( ord_less_rat @ zero_zero_rat @ V )
& ( ord_less_rat @ V @ Y3 )
& ~ ( member_rat @ ( inverse_inverse_rat @ Y3 ) @ ( dedekind_Rep_preal @ R3 ) )
& ( member_rat @ U @ ( dedekind_Rep_preal @ R3 ) )
& ( X2
= ( times_times_rat @ V @ U ) ) ) ) ) ).
% subset_inverse_mult_lemma
thf(fact_1130_mem__Rep__preal__inverse__iff,axiom,
! [Z: rat,R: dedekind_preal] :
( ( member_rat @ Z @ ( dedekind_Rep_preal @ ( invers3090987106763476162_preal @ R ) ) )
= ( ( ord_less_rat @ zero_zero_rat @ Z )
& ? [Y: rat] :
( ( ord_less_rat @ Z @ Y )
& ~ ( member_rat @ ( inverse_inverse_rat @ Y ) @ ( dedekind_Rep_preal @ R ) ) ) ) ) ).
% mem_Rep_preal_inverse_iff
thf(fact_1131_distrib__subset2,axiom,
! [W: dedekind_preal,X2: dedekind_preal,Y2: dedekind_preal] : ( ord_less_eq_set_rat @ ( dedekind_Rep_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ W @ X2 ) @ ( times_3000655703912201937_preal @ W @ Y2 ) ) ) @ ( dedekind_Rep_preal @ ( times_3000655703912201937_preal @ W @ ( plus_p3173629198307831117_preal @ X2 @ Y2 ) ) ) ) ).
% distrib_subset2
thf(fact_1132_less__eq__rat__def,axiom,
( ord_less_eq_rat
= ( ^ [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
| ( X = Y ) ) ) ) ).
% less_eq_rat_def
thf(fact_1133_preal__add__mult__distrib__mean,axiom,
! [A: rat,W: dedekind_preal,B: rat,D: rat,X2: dedekind_preal,E2: rat,Y2: dedekind_preal] :
( ( member_rat @ A @ ( dedekind_Rep_preal @ W ) )
=> ( ( member_rat @ B @ ( dedekind_Rep_preal @ W ) )
=> ( ( member_rat @ D @ ( dedekind_Rep_preal @ X2 ) )
=> ( ( member_rat @ E2 @ ( dedekind_Rep_preal @ Y2 ) )
=> ? [X4: rat] :
( ( member_rat @ X4 @ ( dedekind_Rep_preal @ W ) )
& ( ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ E2 ) )
= ( times_times_rat @ X4 @ ( plus_plus_rat @ D @ E2 ) ) ) ) ) ) ) ) ).
% preal_add_mult_distrib_mean
thf(fact_1134_mem__Rep__preal__add__iff,axiom,
! [Z: rat,R: dedekind_preal,S2: dedekind_preal] :
( ( member_rat @ Z @ ( dedekind_Rep_preal @ ( plus_p3173629198307831117_preal @ R @ S2 ) ) )
= ( ? [X: rat] :
( ( member_rat @ X @ ( dedekind_Rep_preal @ R ) )
& ? [Y: rat] :
( ( member_rat @ Y @ ( dedekind_Rep_preal @ S2 ) )
& ( Z
= ( plus_plus_rat @ X @ Y ) ) ) ) ) ) ).
% mem_Rep_preal_add_iff
thf(fact_1135_obtain__pos__sum,axiom,
! [R: rat] :
( ( ord_less_rat @ zero_zero_rat @ R )
=> ~ ! [S3: rat] :
( ( ord_less_rat @ zero_zero_rat @ S3 )
=> ! [T2: rat] :
( ( ord_less_rat @ zero_zero_rat @ T2 )
=> ( R
!= ( plus_plus_rat @ S3 @ T2 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_1136_add__inc,axiom,
! [X2: num,Y2: num] :
( ( plus_plus_num @ X2 @ ( inc @ Y2 ) )
= ( inc @ ( plus_plus_num @ X2 @ Y2 ) ) ) ).
% add_inc
thf(fact_1137_mult__inc,axiom,
! [X2: num,Y2: num] :
( ( times_times_num @ X2 @ ( inc @ Y2 ) )
= ( plus_plus_num @ ( times_times_num @ X2 @ Y2 ) @ X2 ) ) ).
% mult_inc
thf(fact_1138_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_1139_semiring__norm_I75_J,axiom,
! [M2: num] :
~ ( ord_less_num @ M2 @ one ) ).
% semiring_norm(75)
thf(fact_1140_num__induct,axiom,
! [P: num > $o,X2: num] :
( ( P @ one )
=> ( ! [X4: num] :
( ( P @ X4 )
=> ( P @ ( inc @ X4 ) ) )
=> ( P @ X2 ) ) ) ).
% num_induct
thf(fact_1141_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1142_le__num__One__iff,axiom,
! [X2: num] :
( ( ord_less_eq_num @ X2 @ one )
= ( X2 = one ) ) ).
% le_num_One_iff
thf(fact_1143_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1144_add__One,axiom,
! [X2: num] :
( ( plus_plus_num @ X2 @ one )
= ( inc @ X2 ) ) ).
% add_One
thf(fact_1145_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N )
= M2 )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1146_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1147_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1148_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1149_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1150_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1151_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1152_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1153_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1154_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1155_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1156_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1157_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1158_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1159_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_1160_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1161_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1162_div__mult__self__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N ) @ N )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1163_div__mult__self1__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M2 ) @ N )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1164_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1165_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1166_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1167_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ N @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1168_div__le__mono2,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1169_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1170_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1171_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1172_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1173_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1174_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1175_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_1176_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1177_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1178_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_1179_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_1180_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1181_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1182_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1183_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1184_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1185_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1186_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1187_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1188_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1189_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1190_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1191_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1192_dividend__less__times__div,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1193_dividend__less__div__times,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1194_split__div,axiom,
! [P: nat > $o,M2: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N )
& ( M2
= ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_div
thf(fact_1195_div__le__mono,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1196_div__le__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ).
% div_le_dividend
thf(fact_1197_times__div__less__eq__dividend,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_1198_div__times__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_1199_div__mult2__eq,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( divide_divide_nat @ M2 @ ( times_times_nat @ N @ Q3 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M2 @ N ) @ Q3 ) ) ).
% div_mult2_eq
thf(fact_1200_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_1201_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_1202_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1203_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1204_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1205_div__less__iff__less__mult,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q3 ) @ N )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1206_less__mult__imp__div__less,axiom,
! [M2: nat,I: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1207_nat__mult__div__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1208_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1209_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_1210_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1211_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1212_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1213_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_1214_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_1215_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1216_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_1217_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1218_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1219_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1220_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1221_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1222_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1223_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1224_zdiv__numeral__Bit0,axiom,
! [V2: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_1225_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_1226_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_1227_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1228_semiring__norm_I6_J,axiom,
! [M2: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M2 @ N ) ) ) ).
% semiring_norm(6)
thf(fact_1229_semiring__norm_I78_J,axiom,
! [M2: num,N: num] :
( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% semiring_norm(78)
thf(fact_1230_semiring__norm_I71_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% semiring_norm(71)
thf(fact_1231_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_1232_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1233_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_1234_semiring__norm_I69_J,axiom,
! [M2: num] :
~ ( ord_less_eq_num @ ( bit0 @ M2 ) @ one ) ).
% semiring_norm(69)
thf(fact_1235_add__self__div__2,axiom,
! [M2: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M2 ) ).
% add_self_div_2
thf(fact_1236_pos__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1237_neg__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1238_plus__inverse__ge__2,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_1239_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2 != zero_zero_int )
=> ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_1240_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1241_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1242_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1243_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1244_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1245_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1246_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1247_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_1248_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1249_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1250_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1251_add__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1252_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1253_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1254_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_1255_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_1256_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_1257_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_1258_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1259_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_1260_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_1261_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_1262_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1263_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1264_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_1265_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M5: nat] :
( ( P @ X2 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1266_left__add__mult__distrib,axiom,
! [I: nat,U2: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U2 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1267_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1268_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
% Conjectures (1)
thf(conj_0,conjecture,
( ( times_3000655703912201937_preal @ r @ ( invers3090987106763476162_preal @ r ) )
= one_on9143529541772854033_preal ) ).
%------------------------------------------------------------------------------