TPTP Problem File: SLH0011^1.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_01633_053304__5806592_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1337 ( 458 unt; 65 typ; 0 def)
% Number of atoms : 3892 (1026 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10756 ( 432 ~; 159 |; 198 &;8169 @)
% ( 0 <=>;1798 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 317 ( 317 >; 0 *; 0 +; 0 <<)
% Number of symbols : 59 ( 58 usr; 10 con; 0-2 aty)
% Number of variables : 3410 ( 193 ^;3063 !; 154 ?;3410 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:36:09.020
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Dedekind____Real__Oreal_J,type,
set_Dedekind_real: $tType ).
thf(ty_n_t__Dedekind____Real__Opreal,type,
dedekind_preal: $tType ).
thf(ty_n_t__Dedekind____Real__Oreal,type,
dedekind_real: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $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 (58)
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
archim6058952711729229775r_real: real > int ).
thf(sy_c_Dedekind__Real_Oreal__of__preal,type,
dedeki829075424511776921_preal: dedekind_preal > dedekind_real ).
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__Dedekind____Real__Oreal,type,
invers3762989301784728874d_real: dedekind_real > dedekind_real ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Dedekind____Real__Oreal,type,
one_on6069100329679821595d_real: dedekind_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_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__Dedekind____Real__Oreal,type,
plus_p4060926892116697567d_real: dedekind_real > dedekind_real > dedekind_real ).
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__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__Dedekind____Real__Oreal,type,
times_2157731159493324635d_real: dedekind_real > dedekind_real > dedekind_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
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__Dedekind____Real__Oreal,type,
uminus7714077491378687647d_real: dedekind_real > dedekind_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Dedekind____Real__Oreal,type,
zero_z580800474297136991d_real: dedekind_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
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__Dedekind____Real__Oreal,type,
ord_le2991122432403439658d_real: dedekind_real > dedekind_real > $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__Real__Oreal,type,
ord_less_real: real > real > $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__Dedekind____Real__Oreal,type,
ord_le2716243287969276982d_real: dedekind_real > dedekind_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $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__Dedekind____Real__Oreal,type,
divide9119111104558704680d_real: dedekind_real > dedekind_real > dedekind_real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Dedekind____Real__Oreal,type,
collec1873404778653292075d_real: ( dedekind_real > $o ) > set_Dedekind_real ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
cosh_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_member_001t__Dedekind____Real__Oreal,type,
member_Dedekind_real: dedekind_real > set_Dedekind_real > $o ).
thf(sy_v_x,type,
x: dedekind_real ).
% Relevant facts (1271)
thf(fact_0_x__greater__zero,axiom,
ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ x ).
% x_greater_zero
thf(fact_1__092_060open_062_092_060not_062_Ax_A_092_060le_062_A0_092_060close_062,axiom,
~ ( ord_le2716243287969276982d_real @ x @ zero_z580800474297136991d_real ) ).
% \<open>\<not> x \<le> 0\<close>
thf(fact_2_posreal__complete,axiom,
! [P: set_Dedekind_real] :
( ! [X: dedekind_real] :
( ( member_Dedekind_real @ X @ P )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ X ) )
=> ( ? [X2: dedekind_real] : ( member_Dedekind_real @ X2 @ P )
=> ( ? [Y: dedekind_real] :
! [X: dedekind_real] :
( ( member_Dedekind_real @ X @ P )
=> ( ord_le2991122432403439658d_real @ X @ Y ) )
=> ? [S: dedekind_real] :
! [Y: dedekind_real] :
( ( ? [X3: dedekind_real] :
( ( member_Dedekind_real @ X3 @ P )
& ( ord_le2991122432403439658d_real @ Y @ X3 ) ) )
= ( ord_le2991122432403439658d_real @ Y @ S ) ) ) ) ) ).
% posreal_complete
thf(fact_3_inverse__less__iff__less,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B )
=> ( ( ord_le2991122432403439658d_real @ ( invers3762989301784728874d_real @ A ) @ ( invers3762989301784728874d_real @ B ) )
= ( ord_le2991122432403439658d_real @ B @ A ) ) ) ) ).
% inverse_less_iff_less
thf(fact_4_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_5_inverse__less__iff__less__neg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ ( invers3762989301784728874d_real @ A ) @ ( invers3762989301784728874d_real @ B ) )
= ( ord_le2991122432403439658d_real @ B @ A ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_6_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_7_inverse__negative__iff__negative,axiom,
! [A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( invers3762989301784728874d_real @ A ) @ zero_z580800474297136991d_real )
= ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real ) ) ).
% inverse_negative_iff_negative
thf(fact_8_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_9_inverse__positive__iff__positive,axiom,
! [A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( invers3762989301784728874d_real @ A ) )
= ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A ) ) ).
% inverse_positive_iff_positive
thf(fact_10_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_11_inverse__zero,axiom,
( ( invers3762989301784728874d_real @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% inverse_zero
thf(fact_12_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_13_inverse__nonzero__iff__nonzero,axiom,
! [A: dedekind_real] :
( ( ( invers3762989301784728874d_real @ A )
= zero_z580800474297136991d_real )
= ( A = zero_z580800474297136991d_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_14_inverse__nonzero__iff__nonzero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_15_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_16_inverse__less__imp__less,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( invers3762989301784728874d_real @ A ) @ ( invers3762989301784728874d_real @ B ) )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ord_le2991122432403439658d_real @ B @ A ) ) ) ).
% inverse_less_imp_less
thf(fact_17_inverse__less__imp__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ B @ A ) ) ) ).
% inverse_less_imp_less
thf(fact_18_less__imp__inverse__less,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ord_le2991122432403439658d_real @ ( invers3762989301784728874d_real @ B ) @ ( invers3762989301784728874d_real @ A ) ) ) ) ).
% less_imp_inverse_less
thf(fact_19_less__imp__inverse__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% less_imp_inverse_less
thf(fact_20_inverse__less__imp__less__neg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( invers3762989301784728874d_real @ A ) @ ( invers3762989301784728874d_real @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ B @ A ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_21_inverse__less__imp__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_22_less__imp__inverse__less__neg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( invers3762989301784728874d_real @ B ) @ ( invers3762989301784728874d_real @ A ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_23_less__imp__inverse__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_24_inverse__negative__imp__negative,axiom,
! [A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( invers3762989301784728874d_real @ A ) @ zero_z580800474297136991d_real )
=> ( ( A != zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real ) ) ) ).
% inverse_negative_imp_negative
thf(fact_25_inverse__negative__imp__negative,axiom,
! [A: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
=> ( ( A != zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% inverse_negative_imp_negative
thf(fact_26_inverse__inverse__eq,axiom,
! [A: dedekind_real] :
( ( invers3762989301784728874d_real @ ( invers3762989301784728874d_real @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_27_inverse__inverse__eq,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_28_inverse__eq__iff__eq,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( invers3762989301784728874d_real @ A )
= ( invers3762989301784728874d_real @ B ) )
= ( A = B ) ) ).
% inverse_eq_iff_eq
thf(fact_29_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_30_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_31_inverse__nonpositive__iff__nonpositive,axiom,
! [A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( invers3762989301784728874d_real @ A ) @ zero_z580800474297136991d_real )
= ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_32_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_33_inverse__nonnegative__iff__nonnegative,axiom,
! [A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( invers3762989301784728874d_real @ A ) )
= ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_34_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_35_inverse__le__iff__le__neg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ ( invers3762989301784728874d_real @ A ) @ ( invers3762989301784728874d_real @ B ) )
= ( ord_le2716243287969276982d_real @ B @ A ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_36_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_37_inverse__le__iff__le,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B )
=> ( ( ord_le2716243287969276982d_real @ ( invers3762989301784728874d_real @ A ) @ ( invers3762989301784728874d_real @ B ) )
= ( ord_le2716243287969276982d_real @ B @ A ) ) ) ) ).
% inverse_le_iff_le
thf(fact_38_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_39_real__le__refl,axiom,
! [W: dedekind_real] : ( ord_le2716243287969276982d_real @ W @ W ) ).
% real_le_refl
thf(fact_40_real__le__trans,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real] :
( ( ord_le2716243287969276982d_real @ I @ J )
=> ( ( ord_le2716243287969276982d_real @ J @ K )
=> ( ord_le2716243287969276982d_real @ I @ K ) ) ) ).
% real_le_trans
thf(fact_41_real__le__antisym,axiom,
! [Z: dedekind_real,W: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Z @ W )
=> ( ( ord_le2716243287969276982d_real @ W @ Z )
=> ( Z = W ) ) ) ).
% real_le_antisym
thf(fact_42_zero__le,axiom,
! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).
% zero_le
thf(fact_43_real__less__def,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X3: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% real_less_def
thf(fact_44_le__imp__inverse__le__neg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ ( invers3762989301784728874d_real @ B ) @ ( invers3762989301784728874d_real @ A ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_45_le__imp__inverse__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_46_inverse__le__imp__le__neg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( invers3762989301784728874d_real @ A ) @ ( invers3762989301784728874d_real @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ B @ A ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_47_inverse__le__imp__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_48_le__imp__inverse__le,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ord_le2716243287969276982d_real @ ( invers3762989301784728874d_real @ B ) @ ( invers3762989301784728874d_real @ A ) ) ) ) ).
% le_imp_inverse_le
thf(fact_49_le__imp__inverse__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% le_imp_inverse_le
thf(fact_50_inverse__le__imp__le,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( invers3762989301784728874d_real @ A ) @ ( invers3762989301784728874d_real @ B ) )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ord_le2716243287969276982d_real @ B @ A ) ) ) ).
% inverse_le_imp_le
thf(fact_51_inverse__le__imp__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ B @ A ) ) ) ).
% inverse_le_imp_le
thf(fact_52_zero__reorient,axiom,
! [X4: dedekind_real] :
( ( zero_z580800474297136991d_real = X4 )
= ( X4 = zero_z580800474297136991d_real ) ) ).
% zero_reorient
thf(fact_53_zero__reorient,axiom,
! [X4: nat] :
( ( zero_zero_nat = X4 )
= ( X4 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_54_zero__reorient,axiom,
! [X4: int] :
( ( zero_zero_int = X4 )
= ( X4 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_55_zero__reorient,axiom,
! [X4: real] :
( ( zero_zero_real = X4 )
= ( X4 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_56_linordered__field__no__ub,axiom,
! [X2: dedekind_real] :
? [X_1: dedekind_real] : ( ord_le2991122432403439658d_real @ X2 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_57_linordered__field__no__ub,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_58_linordered__field__no__lb,axiom,
! [X2: dedekind_real] :
? [Y3: dedekind_real] : ( ord_le2991122432403439658d_real @ Y3 @ X2 ) ).
% linordered_field_no_lb
thf(fact_59_linordered__field__no__lb,axiom,
! [X2: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X2 ) ).
% linordered_field_no_lb
thf(fact_60_inverse__eq__imp__eq,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( invers3762989301784728874d_real @ A )
= ( invers3762989301784728874d_real @ B ) )
=> ( A = B ) ) ).
% inverse_eq_imp_eq
thf(fact_61_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_62_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_63_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_64_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_65_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_66_nonzero__imp__inverse__nonzero,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( invers3762989301784728874d_real @ A )
!= zero_z580800474297136991d_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_67_nonzero__imp__inverse__nonzero,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_68_nonzero__inverse__inverse__eq,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( invers3762989301784728874d_real @ ( invers3762989301784728874d_real @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_69_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_70_nonzero__inverse__eq__imp__eq,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( invers3762989301784728874d_real @ A )
= ( invers3762989301784728874d_real @ B ) )
=> ( ( A != zero_z580800474297136991d_real )
=> ( ( B != zero_z580800474297136991d_real )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_71_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_72_inverse__zero__imp__zero,axiom,
! [A: dedekind_real] :
( ( ( invers3762989301784728874d_real @ A )
= zero_z580800474297136991d_real )
=> ( A = zero_z580800474297136991d_real ) ) ).
% inverse_zero_imp_zero
thf(fact_73_inverse__zero__imp__zero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_74_field__class_Ofield__inverse__zero,axiom,
( ( invers3762989301784728874d_real @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% field_class.field_inverse_zero
thf(fact_75_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_76_mem__Collect__eq,axiom,
! [A: dedekind_real,P: dedekind_real > $o] :
( ( member_Dedekind_real @ A @ ( collec1873404778653292075d_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A2: set_Dedekind_real] :
( ( collec1873404778653292075d_real
@ ^ [X3: dedekind_real] : ( member_Dedekind_real @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_78_positive__imp__inverse__positive,axiom,
! [A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( invers3762989301784728874d_real @ A ) ) ) ).
% positive_imp_inverse_positive
thf(fact_79_positive__imp__inverse__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% positive_imp_inverse_positive
thf(fact_80_negative__imp__inverse__negative,axiom,
! [A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( invers3762989301784728874d_real @ A ) @ zero_z580800474297136991d_real ) ) ).
% negative_imp_inverse_negative
thf(fact_81_negative__imp__inverse__negative,axiom,
! [A: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% negative_imp_inverse_negative
thf(fact_82_inverse__positive__imp__positive,axiom,
! [A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( invers3762989301784728874d_real @ A ) )
=> ( ( A != zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A ) ) ) ).
% inverse_positive_imp_positive
thf(fact_83_inverse__positive__imp__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
=> ( ( A != zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% inverse_positive_imp_positive
thf(fact_84_order__refl,axiom,
! [X4: dedekind_real] : ( ord_le2716243287969276982d_real @ X4 @ X4 ) ).
% order_refl
thf(fact_85_order__refl,axiom,
! [X4: real] : ( ord_less_eq_real @ X4 @ X4 ) ).
% order_refl
thf(fact_86_order__refl,axiom,
! [X4: num] : ( ord_less_eq_num @ X4 @ X4 ) ).
% order_refl
thf(fact_87_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_88_order__refl,axiom,
! [X4: int] : ( ord_less_eq_int @ X4 @ X4 ) ).
% order_refl
thf(fact_89_dual__order_Orefl,axiom,
! [A: dedekind_real] : ( ord_le2716243287969276982d_real @ A @ A ) ).
% dual_order.refl
thf(fact_90_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_91_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_92_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_93_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_94_field__lbound__gt__zero,axiom,
! [D1: dedekind_real,D2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ D1 )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ D2 )
=> ? [E: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ E )
& ( ord_le2991122432403439658d_real @ E @ D1 )
& ( ord_le2991122432403439658d_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_95_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_96_less__numeral__extra_I3_J,axiom,
~ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ zero_z580800474297136991d_real ) ).
% less_numeral_extra(3)
thf(fact_97_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_98_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_99_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_100_order__le__imp__less__or__eq,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X4 @ Y4 )
=> ( ( ord_le5708704896291381698_preal @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_101_order__le__imp__less__or__eq,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X4 @ Y4 )
=> ( ( ord_le2991122432403439658d_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_102_order__le__imp__less__or__eq,axiom,
! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_103_order__le__imp__less__or__eq,axiom,
! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ( ord_less_num @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_104_order__le__imp__less__or__eq,axiom,
! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_105_order__le__imp__less__or__eq,axiom,
! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ( ord_less_int @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_106_linorder__le__less__linear,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X4 @ Y4 )
| ( ord_le5708704896291381698_preal @ Y4 @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_107_linorder__le__less__linear,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X4 @ Y4 )
| ( ord_le2991122432403439658d_real @ Y4 @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_108_linorder__le__less__linear,axiom,
! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
| ( ord_less_real @ Y4 @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_109_linorder__le__less__linear,axiom,
! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
| ( ord_less_num @ Y4 @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_110_linorder__le__less__linear,axiom,
! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
| ( ord_less_nat @ Y4 @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_111_linorder__le__less__linear,axiom,
! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
| ( ord_less_int @ Y4 @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_112_order__less__le__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_113_order__less__le__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_114_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_115_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > dedekind_preal,C: dedekind_preal] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_116_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > dedekind_preal,C: dedekind_preal] :
( ( ord_less_real @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_117_order__less__le__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_118_order__less__le__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_119_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_120_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > dedekind_real,C: dedekind_real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_121_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > dedekind_real,C: dedekind_real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_122_order__less__le__subst1,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_123_order__less__le__subst1,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_124_order__less__le__subst1,axiom,
! [A: real,F: dedekind_real > real,B: dedekind_real,C: dedekind_real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_125_order__less__le__subst1,axiom,
! [A: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_126_order__less__le__subst1,axiom,
! [A: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_127_order__less__le__subst1,axiom,
! [A: int,F: dedekind_real > int,B: dedekind_real,C: dedekind_real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_128_order__less__le__subst1,axiom,
! [A: dedekind_preal,F: real > dedekind_preal,B: real,C: real] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_129_order__less__le__subst1,axiom,
! [A: dedekind_real,F: real > dedekind_real,B: real,C: real] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_130_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_131_order__less__le__subst1,axiom,
! [A: num,F: real > num,B: real,C: real] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_132_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_133_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_134_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > real,C: real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_135_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_136_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_137_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_138_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_139_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > dedekind_real,C: dedekind_real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_140_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_141_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > num,C: num] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_142_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_143_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_144_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_145_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_146_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: real > dedekind_preal,B: real,C: real] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_147_order__le__less__subst1,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_148_order__le__less__subst1,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_149_order__le__less__subst1,axiom,
! [A: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_150_order__le__less__subst1,axiom,
! [A: dedekind_real,F: int > dedekind_real,B: int,C: int] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_151_order__le__less__subst1,axiom,
! [A: dedekind_real,F: real > dedekind_real,B: real,C: real] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_152_order__less__le__trans,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal,Z: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Y4 )
=> ( ( ord_le5604041210740703414_preal @ Y4 @ Z )
=> ( ord_le5708704896291381698_preal @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_153_order__less__le__trans,axiom,
! [X4: dedekind_real,Y4: dedekind_real,Z: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ( ( ord_le2716243287969276982d_real @ Y4 @ Z )
=> ( ord_le2991122432403439658d_real @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_154_order__less__le__trans,axiom,
! [X4: real,Y4: real,Z: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ Z )
=> ( ord_less_real @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_155_order__less__le__trans,axiom,
! [X4: num,Y4: num,Z: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ Z )
=> ( ord_less_num @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_156_order__less__le__trans,axiom,
! [X4: nat,Y4: nat,Z: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z )
=> ( ord_less_nat @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_157_order__less__le__trans,axiom,
! [X4: int,Y4: int,Z: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z )
=> ( ord_less_int @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_158_order__le__less__trans,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal,Z: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X4 @ Y4 )
=> ( ( ord_le5708704896291381698_preal @ Y4 @ Z )
=> ( ord_le5708704896291381698_preal @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_159_order__le__less__trans,axiom,
! [X4: dedekind_real,Y4: dedekind_real,Z: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X4 @ Y4 )
=> ( ( ord_le2991122432403439658d_real @ Y4 @ Z )
=> ( ord_le2991122432403439658d_real @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_160_order__le__less__trans,axiom,
! [X4: real,Y4: real,Z: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ( ord_less_real @ Y4 @ Z )
=> ( ord_less_real @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_161_order__le__less__trans,axiom,
! [X4: num,Y4: num,Z: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ( ord_less_num @ Y4 @ Z )
=> ( ord_less_num @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_162_order__le__less__trans,axiom,
! [X4: nat,Y4: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z )
=> ( ord_less_nat @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_163_order__le__less__trans,axiom,
! [X4: int,Y4: int,Z: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ( ord_less_int @ Y4 @ Z )
=> ( ord_less_int @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_164_order__antisym__conv,axiom,
! [Y4: dedekind_real,X4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Y4 @ X4 )
=> ( ( ord_le2716243287969276982d_real @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_165_order__antisym__conv,axiom,
! [Y4: real,X4: real] :
( ( ord_less_eq_real @ Y4 @ X4 )
=> ( ( ord_less_eq_real @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_166_order__antisym__conv,axiom,
! [Y4: num,X4: num] :
( ( ord_less_eq_num @ Y4 @ X4 )
=> ( ( ord_less_eq_num @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_167_order__antisym__conv,axiom,
! [Y4: nat,X4: nat] :
( ( ord_less_eq_nat @ Y4 @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_168_order__antisym__conv,axiom,
! [Y4: int,X4: int] :
( ( ord_less_eq_int @ Y4 @ X4 )
=> ( ( ord_less_eq_int @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_169_linorder__le__cases,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ~ ( ord_le2716243287969276982d_real @ X4 @ Y4 )
=> ( ord_le2716243287969276982d_real @ Y4 @ X4 ) ) ).
% linorder_le_cases
thf(fact_170_linorder__le__cases,axiom,
! [X4: real,Y4: real] :
( ~ ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ Y4 @ X4 ) ) ).
% linorder_le_cases
thf(fact_171_linorder__le__cases,axiom,
! [X4: num,Y4: num] :
( ~ ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ Y4 @ X4 ) ) ).
% linorder_le_cases
thf(fact_172_linorder__le__cases,axiom,
! [X4: nat,Y4: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ).
% linorder_le_cases
thf(fact_173_linorder__le__cases,axiom,
! [X4: int,Y4: int] :
( ~ ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X4 ) ) ).
% linorder_le_cases
thf(fact_174_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_175_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > real,C: real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_176_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_177_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_178_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_179_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > dedekind_real,C: dedekind_real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_180_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_181_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > num,C: num] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_182_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_183_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_184_ord__eq__le__subst,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_185_ord__eq__le__subst,axiom,
! [A: real,F: dedekind_real > real,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_186_ord__eq__le__subst,axiom,
! [A: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_187_ord__eq__le__subst,axiom,
! [A: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_188_ord__eq__le__subst,axiom,
! [A: int,F: dedekind_real > int,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_189_ord__eq__le__subst,axiom,
! [A: dedekind_real,F: real > dedekind_real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_190_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_191_ord__eq__le__subst,axiom,
! [A: num,F: real > num,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_192_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_193_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_194_linorder__linear,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X4 @ Y4 )
| ( ord_le2716243287969276982d_real @ Y4 @ X4 ) ) ).
% linorder_linear
thf(fact_195_linorder__linear,axiom,
! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
| ( ord_less_eq_real @ Y4 @ X4 ) ) ).
% linorder_linear
thf(fact_196_linorder__linear,axiom,
! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
| ( ord_less_eq_num @ Y4 @ X4 ) ) ).
% linorder_linear
thf(fact_197_linorder__linear,axiom,
! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X4 ) ) ).
% linorder_linear
thf(fact_198_linorder__linear,axiom,
! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
| ( ord_less_eq_int @ Y4 @ X4 ) ) ).
% linorder_linear
thf(fact_199_order__eq__refl,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( X4 = Y4 )
=> ( ord_le2716243287969276982d_real @ X4 @ Y4 ) ) ).
% order_eq_refl
thf(fact_200_order__eq__refl,axiom,
! [X4: real,Y4: real] :
( ( X4 = Y4 )
=> ( ord_less_eq_real @ X4 @ Y4 ) ) ).
% order_eq_refl
thf(fact_201_order__eq__refl,axiom,
! [X4: num,Y4: num] :
( ( X4 = Y4 )
=> ( ord_less_eq_num @ X4 @ Y4 ) ) ).
% order_eq_refl
thf(fact_202_order__eq__refl,axiom,
! [X4: nat,Y4: nat] :
( ( X4 = Y4 )
=> ( ord_less_eq_nat @ X4 @ Y4 ) ) ).
% order_eq_refl
thf(fact_203_order__eq__refl,axiom,
! [X4: int,Y4: int] :
( ( X4 = Y4 )
=> ( ord_less_eq_int @ X4 @ Y4 ) ) ).
% order_eq_refl
thf(fact_204_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_205_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > real,C: real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_206_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_207_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_208_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_209_order__subst2,axiom,
! [A: real,B: real,F: real > dedekind_real,C: dedekind_real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_210_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_211_order__subst2,axiom,
! [A: real,B: real,F: real > num,C: num] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_212_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_213_order__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_214_order__subst1,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_215_order__subst1,axiom,
! [A: dedekind_real,F: real > dedekind_real,B: real,C: real] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_216_order__subst1,axiom,
! [A: dedekind_real,F: num > dedekind_real,B: num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y3: num] :
( ( ord_less_eq_num @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_217_order__subst1,axiom,
! [A: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_218_order__subst1,axiom,
! [A: dedekind_real,F: int > dedekind_real,B: int,C: int] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_219_order__subst1,axiom,
! [A: real,F: dedekind_real > real,B: dedekind_real,C: dedekind_real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_220_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_221_order__subst1,axiom,
! [A: real,F: num > real,B: num,C: num] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y3: num] :
( ( ord_less_eq_num @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_222_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_223_order__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_224_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: dedekind_real,Z2: dedekind_real] : ( Y5 = Z2 ) )
= ( ^ [A3: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A3 @ B2 )
& ( ord_le2716243287969276982d_real @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_225_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_226_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z2: num] : ( Y5 = Z2 ) )
= ( ^ [A3: num,B2: num] :
( ( ord_less_eq_num @ A3 @ B2 )
& ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_227_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_228_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_229_antisym,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_230_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_231_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_232_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_233_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_234_dual__order_Otrans,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A )
=> ( ( ord_le2716243287969276982d_real @ C @ B )
=> ( ord_le2716243287969276982d_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_235_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_236_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_237_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_238_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_239_dual__order_Oantisym,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A )
=> ( ( ord_le2716243287969276982d_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_240_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_241_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_242_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_243_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_244_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: dedekind_real,Z2: dedekind_real] : ( Y5 = Z2 ) )
= ( ^ [A3: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B2 @ A3 )
& ( ord_le2716243287969276982d_real @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_245_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_246_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: num,Z2: num] : ( Y5 = Z2 ) )
= ( ^ [A3: num,B2: num] :
( ( ord_less_eq_num @ B2 @ A3 )
& ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_247_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_248_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_249_linorder__wlog,axiom,
! [P: dedekind_real > dedekind_real > $o,A: dedekind_real,B: dedekind_real] :
( ! [A4: dedekind_real,B3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: dedekind_real,B3: dedekind_real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_250_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_251_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: num,B3: num] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_252_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_253_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_254_order__trans,axiom,
! [X4: dedekind_real,Y4: dedekind_real,Z: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X4 @ Y4 )
=> ( ( ord_le2716243287969276982d_real @ Y4 @ Z )
=> ( ord_le2716243287969276982d_real @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_255_order__trans,axiom,
! [X4: real,Y4: real,Z: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ Z )
=> ( ord_less_eq_real @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_256_order__trans,axiom,
! [X4: num,Y4: num,Z: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ Z )
=> ( ord_less_eq_num @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_257_order__trans,axiom,
! [X4: nat,Y4: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z )
=> ( ord_less_eq_nat @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_258_order__trans,axiom,
! [X4: int,Y4: int,Z: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z )
=> ( ord_less_eq_int @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_259_order_Otrans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2716243287969276982d_real @ A @ C ) ) ) ).
% order.trans
thf(fact_260_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_261_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_262_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_263_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_264_order__antisym,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X4 @ Y4 )
=> ( ( ord_le2716243287969276982d_real @ Y4 @ X4 )
=> ( X4 = Y4 ) ) ) ).
% order_antisym
thf(fact_265_order__antisym,axiom,
! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ X4 )
=> ( X4 = Y4 ) ) ) ).
% order_antisym
thf(fact_266_order__antisym,axiom,
! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ X4 )
=> ( X4 = Y4 ) ) ) ).
% order_antisym
thf(fact_267_order__antisym,axiom,
! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X4 )
=> ( X4 = Y4 ) ) ) ).
% order_antisym
thf(fact_268_order__antisym,axiom,
! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ X4 )
=> ( X4 = Y4 ) ) ) ).
% order_antisym
thf(fact_269_ord__le__eq__trans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( B = C )
=> ( ord_le2716243287969276982d_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_270_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_271_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_272_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_273_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_274_ord__eq__le__trans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A = B )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2716243287969276982d_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_275_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_276_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_277_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_278_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_279_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: dedekind_real,Z2: dedekind_real] : ( Y5 = Z2 ) )
= ( ^ [X3: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y2 )
& ( ord_le2716243287969276982d_real @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_280_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
& ( ord_less_eq_real @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_281_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z2: num] : ( Y5 = Z2 ) )
= ( ^ [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
& ( ord_less_eq_num @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_282_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_283_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
& ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_284_le__cases3,axiom,
! [X4: dedekind_real,Y4: dedekind_real,Z: dedekind_real] :
( ( ( ord_le2716243287969276982d_real @ X4 @ Y4 )
=> ~ ( ord_le2716243287969276982d_real @ Y4 @ Z ) )
=> ( ( ( ord_le2716243287969276982d_real @ Y4 @ X4 )
=> ~ ( ord_le2716243287969276982d_real @ X4 @ Z ) )
=> ( ( ( ord_le2716243287969276982d_real @ X4 @ Z )
=> ~ ( ord_le2716243287969276982d_real @ Z @ Y4 ) )
=> ( ( ( ord_le2716243287969276982d_real @ Z @ Y4 )
=> ~ ( ord_le2716243287969276982d_real @ Y4 @ X4 ) )
=> ( ( ( ord_le2716243287969276982d_real @ Y4 @ Z )
=> ~ ( ord_le2716243287969276982d_real @ Z @ X4 ) )
=> ~ ( ( ord_le2716243287969276982d_real @ Z @ X4 )
=> ~ ( ord_le2716243287969276982d_real @ X4 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_285_le__cases3,axiom,
! [X4: real,Y4: real,Z: real] :
( ( ( ord_less_eq_real @ X4 @ Y4 )
=> ~ ( ord_less_eq_real @ Y4 @ Z ) )
=> ( ( ( ord_less_eq_real @ Y4 @ X4 )
=> ~ ( ord_less_eq_real @ X4 @ Z ) )
=> ( ( ( ord_less_eq_real @ X4 @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y4 ) )
=> ( ( ( ord_less_eq_real @ Z @ Y4 )
=> ~ ( ord_less_eq_real @ Y4 @ X4 ) )
=> ( ( ( ord_less_eq_real @ Y4 @ Z )
=> ~ ( ord_less_eq_real @ Z @ X4 ) )
=> ~ ( ( ord_less_eq_real @ Z @ X4 )
=> ~ ( ord_less_eq_real @ X4 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_286_le__cases3,axiom,
! [X4: num,Y4: num,Z: num] :
( ( ( ord_less_eq_num @ X4 @ Y4 )
=> ~ ( ord_less_eq_num @ Y4 @ Z ) )
=> ( ( ( ord_less_eq_num @ Y4 @ X4 )
=> ~ ( ord_less_eq_num @ X4 @ Z ) )
=> ( ( ( ord_less_eq_num @ X4 @ Z )
=> ~ ( ord_less_eq_num @ Z @ Y4 ) )
=> ( ( ( ord_less_eq_num @ Z @ Y4 )
=> ~ ( ord_less_eq_num @ Y4 @ X4 ) )
=> ( ( ( ord_less_eq_num @ Y4 @ Z )
=> ~ ( ord_less_eq_num @ Z @ X4 ) )
=> ~ ( ( ord_less_eq_num @ Z @ X4 )
=> ~ ( ord_less_eq_num @ X4 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_287_le__cases3,axiom,
! [X4: nat,Y4: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_288_le__cases3,axiom,
! [X4: int,Y4: int,Z: int] :
( ( ( ord_less_eq_int @ X4 @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ Z ) )
=> ( ( ( ord_less_eq_int @ Y4 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ Z ) )
=> ( ( ( ord_less_eq_int @ X4 @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y4 ) )
=> ( ( ( ord_less_eq_int @ Z @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ X4 ) )
=> ( ( ( ord_less_eq_int @ Y4 @ Z )
=> ~ ( ord_less_eq_int @ Z @ X4 ) )
=> ~ ( ( ord_less_eq_int @ Z @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_289_nle__le,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ~ ( ord_le2716243287969276982d_real @ A @ B ) )
= ( ( ord_le2716243287969276982d_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_290_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_291_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_292_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_293_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_294_lt__ex,axiom,
! [X4: dedekind_real] :
? [Y3: dedekind_real] : ( ord_le2991122432403439658d_real @ Y3 @ X4 ) ).
% lt_ex
thf(fact_295_lt__ex,axiom,
! [X4: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X4 ) ).
% lt_ex
thf(fact_296_lt__ex,axiom,
! [X4: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X4 ) ).
% lt_ex
thf(fact_297_gt__ex,axiom,
! [X4: dedekind_real] :
? [X_1: dedekind_real] : ( ord_le2991122432403439658d_real @ X4 @ X_1 ) ).
% gt_ex
thf(fact_298_gt__ex,axiom,
! [X4: nat] :
? [X_1: nat] : ( ord_less_nat @ X4 @ X_1 ) ).
% gt_ex
thf(fact_299_gt__ex,axiom,
! [X4: int] :
? [X_1: int] : ( ord_less_int @ X4 @ X_1 ) ).
% gt_ex
thf(fact_300_gt__ex,axiom,
! [X4: real] :
? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% gt_ex
thf(fact_301_dense,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ? [Z3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Z3 )
& ( ord_le2991122432403439658d_real @ Z3 @ Y4 ) ) ) ).
% dense
thf(fact_302_dense,axiom,
! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ? [Z3: real] :
( ( ord_less_real @ X4 @ Z3 )
& ( ord_less_real @ Z3 @ Y4 ) ) ) ).
% dense
thf(fact_303_less__imp__neq,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ( X4 != Y4 ) ) ).
% less_imp_neq
thf(fact_304_less__imp__neq,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Y4 )
=> ( X4 != Y4 ) ) ).
% less_imp_neq
thf(fact_305_less__imp__neq,axiom,
! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( X4 != Y4 ) ) ).
% less_imp_neq
thf(fact_306_less__imp__neq,axiom,
! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( X4 != Y4 ) ) ).
% less_imp_neq
thf(fact_307_less__imp__neq,axiom,
! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( X4 != Y4 ) ) ).
% less_imp_neq
thf(fact_308_order_Oasym,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ~ ( ord_le2991122432403439658d_real @ B @ A ) ) ).
% order.asym
thf(fact_309_order_Oasym,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ~ ( ord_le5708704896291381698_preal @ B @ A ) ) ).
% order.asym
thf(fact_310_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_311_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_312_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_313_ord__eq__less__trans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A = B )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_314_ord__eq__less__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A = B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_315_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_316_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_317_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_318_ord__less__eq__trans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( B = C )
=> ( ord_le2991122432403439658d_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_319_ord__less__eq__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( B = C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_320_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_321_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_322_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_323_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y: nat] :
( ( ord_less_nat @ Y @ X )
=> ( P @ Y ) )
=> ( P @ X ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_324_antisym__conv3,axiom,
! [Y4: dedekind_real,X4: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ Y4 @ X4 )
=> ( ( ~ ( ord_le2991122432403439658d_real @ X4 @ Y4 ) )
= ( X4 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_325_antisym__conv3,axiom,
! [Y4: dedekind_preal,X4: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ Y4 @ X4 )
=> ( ( ~ ( ord_le5708704896291381698_preal @ X4 @ Y4 ) )
= ( X4 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_326_antisym__conv3,axiom,
! [Y4: nat,X4: nat] :
( ~ ( ord_less_nat @ Y4 @ X4 )
=> ( ( ~ ( ord_less_nat @ X4 @ Y4 ) )
= ( X4 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_327_antisym__conv3,axiom,
! [Y4: int,X4: int] :
( ~ ( ord_less_int @ Y4 @ X4 )
=> ( ( ~ ( ord_less_int @ X4 @ Y4 ) )
= ( X4 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_328_antisym__conv3,axiom,
! [Y4: real,X4: real] :
( ~ ( ord_less_real @ Y4 @ X4 )
=> ( ( ~ ( ord_less_real @ X4 @ Y4 ) )
= ( X4 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_329_linorder__cases,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ( ( X4 != Y4 )
=> ( ord_le2991122432403439658d_real @ Y4 @ X4 ) ) ) ).
% linorder_cases
thf(fact_330_linorder__cases,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X4 @ Y4 )
=> ( ( X4 != Y4 )
=> ( ord_le5708704896291381698_preal @ Y4 @ X4 ) ) ) ).
% linorder_cases
thf(fact_331_linorder__cases,axiom,
! [X4: nat,Y4: nat] :
( ~ ( ord_less_nat @ X4 @ Y4 )
=> ( ( X4 != Y4 )
=> ( ord_less_nat @ Y4 @ X4 ) ) ) ).
% linorder_cases
thf(fact_332_linorder__cases,axiom,
! [X4: int,Y4: int] :
( ~ ( ord_less_int @ X4 @ Y4 )
=> ( ( X4 != Y4 )
=> ( ord_less_int @ Y4 @ X4 ) ) ) ).
% linorder_cases
thf(fact_333_linorder__cases,axiom,
! [X4: real,Y4: real] :
( ~ ( ord_less_real @ X4 @ Y4 )
=> ( ( X4 != Y4 )
=> ( ord_less_real @ Y4 @ X4 ) ) ) ).
% linorder_cases
thf(fact_334_dual__order_Oasym,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ~ ( ord_le2991122432403439658d_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_335_dual__order_Oasym,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ~ ( ord_le5708704896291381698_preal @ A @ B ) ) ).
% dual_order.asym
thf(fact_336_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_337_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_338_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_339_dual__order_Oirrefl,axiom,
! [A: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_340_dual__order_Oirrefl,axiom,
! [A: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ A @ A ) ).
% dual_order.irrefl
thf(fact_341_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_342_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_343_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_344_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_345_linorder__less__wlog,axiom,
! [P: dedekind_real > dedekind_real > $o,A: dedekind_real,B: dedekind_real] :
( ! [A4: dedekind_real,B3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: dedekind_real] : ( P @ A4 @ A4 )
=> ( ! [A4: dedekind_real,B3: dedekind_real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_346_linorder__less__wlog,axiom,
! [P: dedekind_preal > dedekind_preal > $o,A: dedekind_preal,B: dedekind_preal] :
( ! [A4: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: dedekind_preal] : ( P @ A4 @ A4 )
=> ( ! [A4: dedekind_preal,B3: dedekind_preal] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_347_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_348_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_349_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_350_order_Ostrict__trans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_351_order_Ostrict__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_352_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_353_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_354_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_355_not__less__iff__gr__or__eq,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ~ ( ord_le2991122432403439658d_real @ X4 @ Y4 ) )
= ( ( ord_le2991122432403439658d_real @ Y4 @ X4 )
| ( X4 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_356_not__less__iff__gr__or__eq,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ X4 @ Y4 ) )
= ( ( ord_le5708704896291381698_preal @ Y4 @ X4 )
| ( X4 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_357_not__less__iff__gr__or__eq,axiom,
! [X4: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y4 ) )
= ( ( ord_less_nat @ Y4 @ X4 )
| ( X4 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_358_not__less__iff__gr__or__eq,axiom,
! [X4: int,Y4: int] :
( ( ~ ( ord_less_int @ X4 @ Y4 ) )
= ( ( ord_less_int @ Y4 @ X4 )
| ( X4 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_359_not__less__iff__gr__or__eq,axiom,
! [X4: real,Y4: real] :
( ( ~ ( ord_less_real @ X4 @ Y4 ) )
= ( ( ord_less_real @ Y4 @ X4 )
| ( X4 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_360_dual__order_Ostrict__trans,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( ( ord_le2991122432403439658d_real @ C @ B )
=> ( ord_le2991122432403439658d_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_361_dual__order_Ostrict__trans,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( ( ord_le5708704896291381698_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_362_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_363_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_364_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_365_order_Ostrict__implies__not__eq,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_366_order_Ostrict__implies__not__eq,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_367_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_368_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_369_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_370_dual__order_Ostrict__implies__not__eq,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_371_dual__order_Ostrict__implies__not__eq,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_372_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_373_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_374_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_375_linorder__neqE,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( X4 != Y4 )
=> ( ~ ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ( ord_le2991122432403439658d_real @ Y4 @ X4 ) ) ) ).
% linorder_neqE
thf(fact_376_linorder__neqE,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( X4 != Y4 )
=> ( ~ ( ord_le5708704896291381698_preal @ X4 @ Y4 )
=> ( ord_le5708704896291381698_preal @ Y4 @ X4 ) ) ) ).
% linorder_neqE
thf(fact_377_linorder__neqE,axiom,
! [X4: nat,Y4: nat] :
( ( X4 != Y4 )
=> ( ~ ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ Y4 @ X4 ) ) ) ).
% linorder_neqE
thf(fact_378_linorder__neqE,axiom,
! [X4: int,Y4: int] :
( ( X4 != Y4 )
=> ( ~ ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ Y4 @ X4 ) ) ) ).
% linorder_neqE
thf(fact_379_linorder__neqE,axiom,
! [X4: real,Y4: real] :
( ( X4 != Y4 )
=> ( ~ ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ Y4 @ X4 ) ) ) ).
% linorder_neqE
thf(fact_380_order__less__asym,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ~ ( ord_le2991122432403439658d_real @ Y4 @ X4 ) ) ).
% order_less_asym
thf(fact_381_order__less__asym,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Y4 )
=> ~ ( ord_le5708704896291381698_preal @ Y4 @ X4 ) ) ).
% order_less_asym
thf(fact_382_order__less__asym,axiom,
! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X4 ) ) ).
% order_less_asym
thf(fact_383_order__less__asym,axiom,
! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X4 ) ) ).
% order_less_asym
thf(fact_384_order__less__asym,axiom,
! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ~ ( ord_less_real @ Y4 @ X4 ) ) ).
% order_less_asym
thf(fact_385_linorder__neq__iff,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( X4 != Y4 )
= ( ( ord_le2991122432403439658d_real @ X4 @ Y4 )
| ( ord_le2991122432403439658d_real @ Y4 @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_386_linorder__neq__iff,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( X4 != Y4 )
= ( ( ord_le5708704896291381698_preal @ X4 @ Y4 )
| ( ord_le5708704896291381698_preal @ Y4 @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_387_linorder__neq__iff,axiom,
! [X4: nat,Y4: nat] :
( ( X4 != Y4 )
= ( ( ord_less_nat @ X4 @ Y4 )
| ( ord_less_nat @ Y4 @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_388_linorder__neq__iff,axiom,
! [X4: int,Y4: int] :
( ( X4 != Y4 )
= ( ( ord_less_int @ X4 @ Y4 )
| ( ord_less_int @ Y4 @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_389_linorder__neq__iff,axiom,
! [X4: real,Y4: real] :
( ( X4 != Y4 )
= ( ( ord_less_real @ X4 @ Y4 )
| ( ord_less_real @ Y4 @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_390_order__less__asym_H,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ~ ( ord_le2991122432403439658d_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_391_order__less__asym_H,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ~ ( ord_le5708704896291381698_preal @ B @ A ) ) ).
% order_less_asym'
thf(fact_392_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_393_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_394_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_395_order__less__trans,axiom,
! [X4: dedekind_real,Y4: dedekind_real,Z: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ( ( ord_le2991122432403439658d_real @ Y4 @ Z )
=> ( ord_le2991122432403439658d_real @ X4 @ Z ) ) ) ).
% order_less_trans
thf(fact_396_order__less__trans,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal,Z: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Y4 )
=> ( ( ord_le5708704896291381698_preal @ Y4 @ Z )
=> ( ord_le5708704896291381698_preal @ X4 @ Z ) ) ) ).
% order_less_trans
thf(fact_397_order__less__trans,axiom,
! [X4: nat,Y4: nat,Z: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z )
=> ( ord_less_nat @ X4 @ Z ) ) ) ).
% order_less_trans
thf(fact_398_order__less__trans,axiom,
! [X4: int,Y4: int,Z: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ( ord_less_int @ Y4 @ Z )
=> ( ord_less_int @ X4 @ Z ) ) ) ).
% order_less_trans
thf(fact_399_order__less__trans,axiom,
! [X4: real,Y4: real,Z: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ( ord_less_real @ Y4 @ Z )
=> ( ord_less_real @ X4 @ Z ) ) ) ).
% order_less_trans
thf(fact_400_ord__eq__less__subst,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_401_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_402_ord__eq__less__subst,axiom,
! [A: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_403_ord__eq__less__subst,axiom,
! [A: int,F: dedekind_real > int,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_404_ord__eq__less__subst,axiom,
! [A: real,F: dedekind_real > real,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_405_ord__eq__less__subst,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_406_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_407_ord__eq__less__subst,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_408_ord__eq__less__subst,axiom,
! [A: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_409_ord__eq__less__subst,axiom,
! [A: real,F: dedekind_preal > real,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_410_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_411_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_412_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_413_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_414_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > real,C: real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_415_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_416_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_417_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_418_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_419_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > real,C: real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_420_order__less__irrefl,axiom,
! [X4: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_421_order__less__irrefl,axiom,
! [X4: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_422_order__less__irrefl,axiom,
! [X4: nat] :
~ ( ord_less_nat @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_423_order__less__irrefl,axiom,
! [X4: int] :
~ ( ord_less_int @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_424_order__less__irrefl,axiom,
! [X4: real] :
~ ( ord_less_real @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_425_order__less__subst1,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_426_order__less__subst1,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_427_order__less__subst1,axiom,
! [A: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_428_order__less__subst1,axiom,
! [A: dedekind_real,F: int > dedekind_real,B: int,C: int] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_429_order__less__subst1,axiom,
! [A: dedekind_real,F: real > dedekind_real,B: real,C: real] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_430_order__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_431_order__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_432_order__less__subst1,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_433_order__less__subst1,axiom,
! [A: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_434_order__less__subst1,axiom,
! [A: dedekind_preal,F: real > dedekind_preal,B: real,C: real] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_435_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_436_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_437_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_438_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_439_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > real,C: real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_440_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_441_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_442_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_443_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_444_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > real,C: real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_445_order__less__not__sym,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ~ ( ord_le2991122432403439658d_real @ Y4 @ X4 ) ) ).
% order_less_not_sym
thf(fact_446_order__less__not__sym,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Y4 )
=> ~ ( ord_le5708704896291381698_preal @ Y4 @ X4 ) ) ).
% order_less_not_sym
thf(fact_447_order__less__not__sym,axiom,
! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X4 ) ) ).
% order_less_not_sym
thf(fact_448_order__less__not__sym,axiom,
! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X4 ) ) ).
% order_less_not_sym
thf(fact_449_order__less__not__sym,axiom,
! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ~ ( ord_less_real @ Y4 @ X4 ) ) ).
% order_less_not_sym
thf(fact_450_order__less__imp__triv,axiom,
! [X4: dedekind_real,Y4: dedekind_real,P: $o] :
( ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ( ( ord_le2991122432403439658d_real @ Y4 @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_451_order__less__imp__triv,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal,P: $o] :
( ( ord_le5708704896291381698_preal @ X4 @ Y4 )
=> ( ( ord_le5708704896291381698_preal @ Y4 @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_452_order__less__imp__triv,axiom,
! [X4: nat,Y4: nat,P: $o] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_453_order__less__imp__triv,axiom,
! [X4: int,Y4: int,P: $o] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ( ord_less_int @ Y4 @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_454_order__less__imp__triv,axiom,
! [X4: real,Y4: real,P: $o] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ( ord_less_real @ Y4 @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_455_linorder__less__linear,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Y4 )
| ( X4 = Y4 )
| ( ord_le2991122432403439658d_real @ Y4 @ X4 ) ) ).
% linorder_less_linear
thf(fact_456_linorder__less__linear,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Y4 )
| ( X4 = Y4 )
| ( ord_le5708704896291381698_preal @ Y4 @ X4 ) ) ).
% linorder_less_linear
thf(fact_457_linorder__less__linear,axiom,
! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 )
| ( ord_less_nat @ Y4 @ X4 ) ) ).
% linorder_less_linear
thf(fact_458_linorder__less__linear,axiom,
! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
| ( X4 = Y4 )
| ( ord_less_int @ Y4 @ X4 ) ) ).
% linorder_less_linear
thf(fact_459_linorder__less__linear,axiom,
! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 )
| ( ord_less_real @ Y4 @ X4 ) ) ).
% linorder_less_linear
thf(fact_460_order__less__imp__not__eq,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ( X4 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_461_order__less__imp__not__eq,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Y4 )
=> ( X4 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_462_order__less__imp__not__eq,axiom,
! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( X4 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_463_order__less__imp__not__eq,axiom,
! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( X4 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_464_order__less__imp__not__eq,axiom,
! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( X4 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_465_order__less__imp__not__eq2,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ( Y4 != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_466_order__less__imp__not__eq2,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Y4 )
=> ( Y4 != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_467_order__less__imp__not__eq2,axiom,
! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( Y4 != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_468_order__less__imp__not__eq2,axiom,
! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( Y4 != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_469_order__less__imp__not__eq2,axiom,
! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( Y4 != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_470_order__less__imp__not__less,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ~ ( ord_le2991122432403439658d_real @ Y4 @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_471_order__less__imp__not__less,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Y4 )
=> ~ ( ord_le5708704896291381698_preal @ Y4 @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_472_order__less__imp__not__less,axiom,
! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_473_order__less__imp__not__less,axiom,
! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_474_order__less__imp__not__less,axiom,
! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ~ ( ord_less_real @ Y4 @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_475_le__numeral__extra_I3_J,axiom,
ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ zero_z580800474297136991d_real ).
% le_numeral_extra(3)
thf(fact_476_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_477_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_478_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_479_leD,axiom,
! [Y4: dedekind_preal,X4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ Y4 @ X4 )
=> ~ ( ord_le5708704896291381698_preal @ X4 @ Y4 ) ) ).
% leD
thf(fact_480_leD,axiom,
! [Y4: dedekind_real,X4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Y4 @ X4 )
=> ~ ( ord_le2991122432403439658d_real @ X4 @ Y4 ) ) ).
% leD
thf(fact_481_leD,axiom,
! [Y4: real,X4: real] :
( ( ord_less_eq_real @ Y4 @ X4 )
=> ~ ( ord_less_real @ X4 @ Y4 ) ) ).
% leD
thf(fact_482_leD,axiom,
! [Y4: num,X4: num] :
( ( ord_less_eq_num @ Y4 @ X4 )
=> ~ ( ord_less_num @ X4 @ Y4 ) ) ).
% leD
thf(fact_483_leD,axiom,
! [Y4: nat,X4: nat] :
( ( ord_less_eq_nat @ Y4 @ X4 )
=> ~ ( ord_less_nat @ X4 @ Y4 ) ) ).
% leD
thf(fact_484_leD,axiom,
! [Y4: int,X4: int] :
( ( ord_less_eq_int @ Y4 @ X4 )
=> ~ ( ord_less_int @ X4 @ Y4 ) ) ).
% leD
thf(fact_485_leI,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X4 @ Y4 )
=> ( ord_le5604041210740703414_preal @ Y4 @ X4 ) ) ).
% leI
thf(fact_486_leI,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ( ord_le2716243287969276982d_real @ Y4 @ X4 ) ) ).
% leI
thf(fact_487_leI,axiom,
! [X4: real,Y4: real] :
( ~ ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ Y4 @ X4 ) ) ).
% leI
thf(fact_488_leI,axiom,
! [X4: num,Y4: num] :
( ~ ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ Y4 @ X4 ) ) ).
% leI
thf(fact_489_leI,axiom,
! [X4: nat,Y4: nat] :
( ~ ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ).
% leI
thf(fact_490_leI,axiom,
! [X4: int,Y4: int] :
( ~ ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X4 ) ) ).
% leI
thf(fact_491_nless__le,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ A @ B ) )
= ( ~ ( ord_le5604041210740703414_preal @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_492_nless__le,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ~ ( ord_le2991122432403439658d_real @ A @ B ) )
= ( ~ ( ord_le2716243287969276982d_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_493_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_494_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_495_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_496_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_497_antisym__conv1,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X4 @ Y4 )
=> ( ( ord_le5604041210740703414_preal @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_498_antisym__conv1,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ( ( ord_le2716243287969276982d_real @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_499_antisym__conv1,axiom,
! [X4: real,Y4: real] :
( ~ ( ord_less_real @ X4 @ Y4 )
=> ( ( ord_less_eq_real @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_500_antisym__conv1,axiom,
! [X4: num,Y4: num] :
( ~ ( ord_less_num @ X4 @ Y4 )
=> ( ( ord_less_eq_num @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_501_antisym__conv1,axiom,
! [X4: nat,Y4: nat] :
( ~ ( ord_less_nat @ X4 @ Y4 )
=> ( ( ord_less_eq_nat @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_502_antisym__conv1,axiom,
! [X4: int,Y4: int] :
( ~ ( ord_less_int @ X4 @ Y4 )
=> ( ( ord_less_eq_int @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_503_antisym__conv2,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X4 @ Y4 )
=> ( ( ~ ( ord_le5708704896291381698_preal @ X4 @ Y4 ) )
= ( X4 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_504_antisym__conv2,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X4 @ Y4 )
=> ( ( ~ ( ord_le2991122432403439658d_real @ X4 @ Y4 ) )
= ( X4 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_505_antisym__conv2,axiom,
! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ( ~ ( ord_less_real @ X4 @ Y4 ) )
= ( X4 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_506_antisym__conv2,axiom,
! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ( ~ ( ord_less_num @ X4 @ Y4 ) )
= ( X4 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_507_antisym__conv2,axiom,
! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ( ~ ( ord_less_nat @ X4 @ Y4 ) )
= ( X4 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_508_antisym__conv2,axiom,
! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ( ~ ( ord_less_int @ X4 @ Y4 ) )
= ( X4 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_509_dense__ge,axiom,
! [Z: dedekind_real,Y4: dedekind_real] :
( ! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X )
=> ( ord_le2716243287969276982d_real @ Y4 @ X ) )
=> ( ord_le2716243287969276982d_real @ Y4 @ Z ) ) ).
% dense_ge
thf(fact_510_dense__ge,axiom,
! [Z: real,Y4: real] :
( ! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( ord_less_eq_real @ Y4 @ X ) )
=> ( ord_less_eq_real @ Y4 @ Z ) ) ).
% dense_ge
thf(fact_511_dense__le,axiom,
! [Y4: dedekind_real,Z: dedekind_real] :
( ! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y4 )
=> ( ord_le2716243287969276982d_real @ X @ Z ) )
=> ( ord_le2716243287969276982d_real @ Y4 @ Z ) ) ).
% dense_le
thf(fact_512_dense__le,axiom,
! [Y4: real,Z: real] :
( ! [X: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_eq_real @ X @ Z ) )
=> ( ord_less_eq_real @ Y4 @ Z ) ) ).
% dense_le
thf(fact_513_less__le__not__le,axiom,
( ord_le5708704896291381698_preal
= ( ^ [X3: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y2 )
& ~ ( ord_le5604041210740703414_preal @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_514_less__le__not__le,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X3: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y2 )
& ~ ( ord_le2716243287969276982d_real @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_515_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
& ~ ( ord_less_eq_real @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_516_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
& ~ ( ord_less_eq_num @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_517_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_518_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
& ~ ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_519_not__le__imp__less,axiom,
! [Y4: dedekind_preal,X4: dedekind_preal] :
( ~ ( ord_le5604041210740703414_preal @ Y4 @ X4 )
=> ( ord_le5708704896291381698_preal @ X4 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_520_not__le__imp__less,axiom,
! [Y4: dedekind_real,X4: dedekind_real] :
( ~ ( ord_le2716243287969276982d_real @ Y4 @ X4 )
=> ( ord_le2991122432403439658d_real @ X4 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_521_not__le__imp__less,axiom,
! [Y4: real,X4: real] :
( ~ ( ord_less_eq_real @ Y4 @ X4 )
=> ( ord_less_real @ X4 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_522_not__le__imp__less,axiom,
! [Y4: num,X4: num] :
( ~ ( ord_less_eq_num @ Y4 @ X4 )
=> ( ord_less_num @ X4 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_523_not__le__imp__less,axiom,
! [Y4: nat,X4: nat] :
( ~ ( ord_less_eq_nat @ Y4 @ X4 )
=> ( ord_less_nat @ X4 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_524_not__le__imp__less,axiom,
! [Y4: int,X4: int] :
( ~ ( ord_less_eq_int @ Y4 @ X4 )
=> ( ord_less_int @ X4 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_525_order_Oorder__iff__strict,axiom,
( ord_le5604041210740703414_preal
= ( ^ [A3: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_526_order_Oorder__iff__strict,axiom,
( ord_le2716243287969276982d_real
= ( ^ [A3: dedekind_real,B2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_527_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_528_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A3: num,B2: num] :
( ( ord_less_num @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_529_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_530_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_531_order_Ostrict__iff__order,axiom,
( ord_le5708704896291381698_preal
= ( ^ [A3: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_532_order_Ostrict__iff__order,axiom,
( ord_le2991122432403439658d_real
= ( ^ [A3: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_533_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_534_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A3: num,B2: num] :
( ( ord_less_eq_num @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_535_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_536_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_537_order_Ostrict__trans1,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_538_order_Ostrict__trans1,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_539_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_540_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_541_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_542_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_543_order_Ostrict__trans2,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_544_order_Ostrict__trans2,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_545_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_546_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_547_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_548_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_549_order_Ostrict__iff__not,axiom,
( ord_le5708704896291381698_preal
= ( ^ [A3: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A3 @ B2 )
& ~ ( ord_le5604041210740703414_preal @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_550_order_Ostrict__iff__not,axiom,
( ord_le2991122432403439658d_real
= ( ^ [A3: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A3 @ B2 )
& ~ ( ord_le2716243287969276982d_real @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_551_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_552_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A3: num,B2: num] :
( ( ord_less_eq_num @ A3 @ B2 )
& ~ ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_553_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_554_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_555_dense__ge__bounded,axiom,
! [Z: dedekind_real,X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X4 )
=> ( ! [W2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ W2 )
=> ( ( ord_le2991122432403439658d_real @ W2 @ X4 )
=> ( ord_le2716243287969276982d_real @ Y4 @ W2 ) ) )
=> ( ord_le2716243287969276982d_real @ Y4 @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_556_dense__ge__bounded,axiom,
! [Z: real,X4: real,Y4: real] :
( ( ord_less_real @ Z @ X4 )
=> ( ! [W2: real] :
( ( ord_less_real @ Z @ W2 )
=> ( ( ord_less_real @ W2 @ X4 )
=> ( ord_less_eq_real @ Y4 @ W2 ) ) )
=> ( ord_less_eq_real @ Y4 @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_557_dense__le__bounded,axiom,
! [X4: dedekind_real,Y4: dedekind_real,Z: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ( ! [W2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ W2 )
=> ( ( ord_le2991122432403439658d_real @ W2 @ Y4 )
=> ( ord_le2716243287969276982d_real @ W2 @ Z ) ) )
=> ( ord_le2716243287969276982d_real @ Y4 @ Z ) ) ) ).
% dense_le_bounded
thf(fact_558_dense__le__bounded,axiom,
! [X4: real,Y4: real,Z: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ! [W2: real] :
( ( ord_less_real @ X4 @ W2 )
=> ( ( ord_less_real @ W2 @ Y4 )
=> ( ord_less_eq_real @ W2 @ Z ) ) )
=> ( ord_less_eq_real @ Y4 @ Z ) ) ) ).
% dense_le_bounded
thf(fact_559_dual__order_Oorder__iff__strict,axiom,
( ord_le5604041210740703414_preal
= ( ^ [B2: dedekind_preal,A3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_560_dual__order_Oorder__iff__strict,axiom,
( ord_le2716243287969276982d_real
= ( ^ [B2: dedekind_real,A3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_561_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_real @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_562_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B2: num,A3: num] :
( ( ord_less_num @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_563_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_564_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_int @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_565_dual__order_Ostrict__iff__order,axiom,
( ord_le5708704896291381698_preal
= ( ^ [B2: dedekind_preal,A3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_566_dual__order_Ostrict__iff__order,axiom,
( ord_le2991122432403439658d_real
= ( ^ [B2: dedekind_real,A3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_567_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_568_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B2: num,A3: num] :
( ( ord_less_eq_num @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_569_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_570_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_571_dual__order_Ostrict__trans1,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A )
=> ( ( ord_le5708704896291381698_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_572_dual__order_Ostrict__trans1,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A )
=> ( ( ord_le2991122432403439658d_real @ C @ B )
=> ( ord_le2991122432403439658d_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_573_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_574_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_575_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_576_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_577_dual__order_Ostrict__trans2,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( ( ord_le5604041210740703414_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_578_dual__order_Ostrict__trans2,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( ( ord_le2716243287969276982d_real @ C @ B )
=> ( ord_le2991122432403439658d_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_579_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_580_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_581_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_582_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_583_dual__order_Ostrict__iff__not,axiom,
( ord_le5708704896291381698_preal
= ( ^ [B2: dedekind_preal,A3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B2 @ A3 )
& ~ ( ord_le5604041210740703414_preal @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_584_dual__order_Ostrict__iff__not,axiom,
( ord_le2991122432403439658d_real
= ( ^ [B2: dedekind_real,A3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B2 @ A3 )
& ~ ( ord_le2716243287969276982d_real @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_585_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_586_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B2: num,A3: num] :
( ( ord_less_eq_num @ B2 @ A3 )
& ~ ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_587_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_588_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_589_order_Ostrict__implies__order,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ord_le5604041210740703414_preal @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_590_order_Ostrict__implies__order,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ord_le2716243287969276982d_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_591_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_592_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_593_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_594_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_595_dual__order_Ostrict__implies__order,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( ord_le5604041210740703414_preal @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_596_dual__order_Ostrict__implies__order,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( ord_le2716243287969276982d_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_597_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_598_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_599_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_600_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_601_order__le__less,axiom,
( ord_le5604041210740703414_preal
= ( ^ [X3: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_602_order__le__less,axiom,
( ord_le2716243287969276982d_real
= ( ^ [X3: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_603_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_604_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_605_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_606_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_607_order__less__le,axiom,
( ord_le5708704896291381698_preal
= ( ^ [X3: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_608_order__less__le,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X3: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_609_order__less__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_610_order__less__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_611_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_612_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_613_linorder__not__le,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ X4 @ Y4 ) )
= ( ord_le5708704896291381698_preal @ Y4 @ X4 ) ) ).
% linorder_not_le
thf(fact_614_linorder__not__le,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ~ ( ord_le2716243287969276982d_real @ X4 @ Y4 ) )
= ( ord_le2991122432403439658d_real @ Y4 @ X4 ) ) ).
% linorder_not_le
thf(fact_615_linorder__not__le,axiom,
! [X4: real,Y4: real] :
( ( ~ ( ord_less_eq_real @ X4 @ Y4 ) )
= ( ord_less_real @ Y4 @ X4 ) ) ).
% linorder_not_le
thf(fact_616_linorder__not__le,axiom,
! [X4: num,Y4: num] :
( ( ~ ( ord_less_eq_num @ X4 @ Y4 ) )
= ( ord_less_num @ Y4 @ X4 ) ) ).
% linorder_not_le
thf(fact_617_linorder__not__le,axiom,
! [X4: nat,Y4: nat] :
( ( ~ ( ord_less_eq_nat @ X4 @ Y4 ) )
= ( ord_less_nat @ Y4 @ X4 ) ) ).
% linorder_not_le
thf(fact_618_linorder__not__le,axiom,
! [X4: int,Y4: int] :
( ( ~ ( ord_less_eq_int @ X4 @ Y4 ) )
= ( ord_less_int @ Y4 @ X4 ) ) ).
% linorder_not_le
thf(fact_619_linorder__not__less,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ X4 @ Y4 ) )
= ( ord_le5604041210740703414_preal @ Y4 @ X4 ) ) ).
% linorder_not_less
thf(fact_620_linorder__not__less,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ~ ( ord_le2991122432403439658d_real @ X4 @ Y4 ) )
= ( ord_le2716243287969276982d_real @ Y4 @ X4 ) ) ).
% linorder_not_less
thf(fact_621_linorder__not__less,axiom,
! [X4: real,Y4: real] :
( ( ~ ( ord_less_real @ X4 @ Y4 ) )
= ( ord_less_eq_real @ Y4 @ X4 ) ) ).
% linorder_not_less
thf(fact_622_linorder__not__less,axiom,
! [X4: num,Y4: num] :
( ( ~ ( ord_less_num @ X4 @ Y4 ) )
= ( ord_less_eq_num @ Y4 @ X4 ) ) ).
% linorder_not_less
thf(fact_623_linorder__not__less,axiom,
! [X4: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y4 ) )
= ( ord_less_eq_nat @ Y4 @ X4 ) ) ).
% linorder_not_less
thf(fact_624_linorder__not__less,axiom,
! [X4: int,Y4: int] :
( ( ~ ( ord_less_int @ X4 @ Y4 ) )
= ( ord_less_eq_int @ Y4 @ X4 ) ) ).
% linorder_not_less
thf(fact_625_order__less__imp__le,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Y4 )
=> ( ord_le5604041210740703414_preal @ X4 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_626_order__less__imp__le,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ( ord_le2716243287969276982d_real @ X4 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_627_order__less__imp__le,axiom,
! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ X4 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_628_order__less__imp__le,axiom,
! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ X4 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_629_order__less__imp__le,axiom,
! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ X4 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_630_order__less__imp__le,axiom,
! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ X4 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_631_order__le__neq__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( A != B )
=> ( ord_le5708704896291381698_preal @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_632_order__le__neq__trans,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( A != B )
=> ( ord_le2991122432403439658d_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_633_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_634_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_635_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_636_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_637_order__neq__le__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( A != B )
=> ( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ord_le5708704896291381698_preal @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_638_order__neq__le__trans,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != B )
=> ( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ord_le2991122432403439658d_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_639_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_640_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_641_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_642_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_643_minf_I8_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z3 )
=> ~ ( ord_le5604041210740703414_preal @ T @ X2 ) ) ).
% minf(8)
thf(fact_644_minf_I8_J,axiom,
! [T: dedekind_real] :
? [Z3: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z3 )
=> ~ ( ord_le2716243287969276982d_real @ T @ X2 ) ) ).
% minf(8)
thf(fact_645_minf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z3 )
=> ~ ( ord_less_eq_real @ T @ X2 ) ) ).
% minf(8)
thf(fact_646_minf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z3 )
=> ~ ( ord_less_eq_num @ T @ X2 ) ) ).
% minf(8)
thf(fact_647_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X2 ) ) ).
% minf(8)
thf(fact_648_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% minf(8)
thf(fact_649_minf_I6_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z3 )
=> ( ord_le5604041210740703414_preal @ X2 @ T ) ) ).
% minf(6)
thf(fact_650_minf_I6_J,axiom,
! [T: dedekind_real] :
? [Z3: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z3 )
=> ( ord_le2716243287969276982d_real @ X2 @ T ) ) ).
% minf(6)
thf(fact_651_minf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z3 )
=> ( ord_less_eq_real @ X2 @ T ) ) ).
% minf(6)
thf(fact_652_minf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z3 )
=> ( ord_less_eq_num @ X2 @ T ) ) ).
% minf(6)
thf(fact_653_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ T ) ) ).
% minf(6)
thf(fact_654_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( ord_less_eq_int @ X2 @ T ) ) ).
% minf(6)
thf(fact_655_pinf_I8_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X2 )
=> ( ord_le5604041210740703414_preal @ T @ X2 ) ) ).
% pinf(8)
thf(fact_656_pinf_I8_J,axiom,
! [T: dedekind_real] :
? [Z3: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ X2 )
=> ( ord_le2716243287969276982d_real @ T @ X2 ) ) ).
% pinf(8)
thf(fact_657_pinf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( ord_less_eq_real @ T @ X2 ) ) ).
% pinf(8)
thf(fact_658_pinf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X2: num] :
( ( ord_less_num @ Z3 @ X2 )
=> ( ord_less_eq_num @ T @ X2 ) ) ).
% pinf(8)
thf(fact_659_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ T @ X2 ) ) ).
% pinf(8)
thf(fact_660_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ( ord_less_eq_int @ T @ X2 ) ) ).
% pinf(8)
thf(fact_661_pinf_I6_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X2 )
=> ~ ( ord_le5604041210740703414_preal @ X2 @ T ) ) ).
% pinf(6)
thf(fact_662_pinf_I6_J,axiom,
! [T: dedekind_real] :
? [Z3: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ X2 )
=> ~ ( ord_le2716243287969276982d_real @ X2 @ T ) ) ).
% pinf(6)
thf(fact_663_pinf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ T ) ) ).
% pinf(6)
thf(fact_664_pinf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X2: num] :
( ( ord_less_num @ Z3 @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ T ) ) ).
% pinf(6)
thf(fact_665_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ T ) ) ).
% pinf(6)
thf(fact_666_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% pinf(6)
thf(fact_667_verit__comp__simplify1_I3_J,axiom,
! [B4: dedekind_preal,A5: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ B4 @ A5 ) )
= ( ord_le5708704896291381698_preal @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_668_verit__comp__simplify1_I3_J,axiom,
! [B4: dedekind_real,A5: dedekind_real] :
( ( ~ ( ord_le2716243287969276982d_real @ B4 @ A5 ) )
= ( ord_le2991122432403439658d_real @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_669_verit__comp__simplify1_I3_J,axiom,
! [B4: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B4 @ A5 ) )
= ( ord_less_real @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_670_verit__comp__simplify1_I3_J,axiom,
! [B4: num,A5: num] :
( ( ~ ( ord_less_eq_num @ B4 @ A5 ) )
= ( ord_less_num @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_671_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_672_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
= ( ord_less_int @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_673_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B )
& ! [X2: real] :
( ( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_real @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D: real] :
( ! [X: real] :
( ( ( ord_less_eq_real @ A @ X )
& ( ord_less_real @ X @ D ) )
=> ( P @ X ) )
=> ( ord_less_eq_real @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_674_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A @ X )
& ( ord_less_nat @ X @ D ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_675_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X2: int] :
( ( ( ord_less_eq_int @ A @ X2 )
& ( ord_less_int @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D: int] :
( ! [X: int] :
( ( ( ord_less_eq_int @ A @ X )
& ( ord_less_int @ X @ D ) )
=> ( P @ X ) )
=> ( ord_less_eq_int @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_676_one__le__inverse,axiom,
! [A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ A @ one_on6069100329679821595d_real )
=> ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ ( invers3762989301784728874d_real @ A ) ) ) ) ).
% one_le_inverse
thf(fact_677_one__le__inverse,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% one_le_inverse
thf(fact_678_inverse__less__1__iff,axiom,
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( invers3762989301784728874d_real @ X4 ) @ one_on6069100329679821595d_real )
= ( ( ord_le2716243287969276982d_real @ X4 @ zero_z580800474297136991d_real )
| ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ X4 ) ) ) ).
% inverse_less_1_iff
thf(fact_679_inverse__less__1__iff,axiom,
! [X4: real] :
( ( ord_less_real @ ( inverse_inverse_real @ X4 ) @ one_one_real )
= ( ( ord_less_eq_real @ X4 @ zero_zero_real )
| ( ord_less_real @ one_one_real @ X4 ) ) ) ).
% inverse_less_1_iff
thf(fact_680_one__le__inverse__iff,axiom,
! [X4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ ( invers3762989301784728874d_real @ X4 ) )
= ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ X4 )
& ( ord_le2716243287969276982d_real @ X4 @ one_on6069100329679821595d_real ) ) ) ).
% one_le_inverse_iff
thf(fact_681_one__le__inverse__iff,axiom,
! [X4: real] :
( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X4 ) )
= ( ( ord_less_real @ zero_zero_real @ X4 )
& ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ).
% one_le_inverse_iff
thf(fact_682_inverse__le__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( invers3762989301784728874d_real @ A ) @ ( invers3762989301784728874d_real @ B ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) )
=> ( ord_le2716243287969276982d_real @ B @ A ) )
& ( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ B ) @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ A @ B ) ) ) ) ).
% inverse_le_iff
thf(fact_683_inverse__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ B @ A ) )
& ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
=> ( ord_less_eq_real @ A @ B ) ) ) ) ).
% inverse_le_iff
thf(fact_684_mult__1,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ A )
= A ) ).
% mult_1
thf(fact_685_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_686_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_687_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_688_mult_Oright__neutral,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ one_on6069100329679821595d_real )
= A ) ).
% mult.right_neutral
thf(fact_689_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_690_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_691_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_692_inverse__mult__distrib,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( invers3762989301784728874d_real @ ( times_2157731159493324635d_real @ A @ B ) )
= ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ A ) @ ( invers3762989301784728874d_real @ B ) ) ) ).
% inverse_mult_distrib
thf(fact_693_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_694_inverse__1,axiom,
( ( invers3762989301784728874d_real @ one_on6069100329679821595d_real )
= one_on6069100329679821595d_real ) ).
% inverse_1
thf(fact_695_inverse__1,axiom,
( ( inverse_inverse_real @ one_one_real )
= one_one_real ) ).
% inverse_1
thf(fact_696_inverse__eq__1__iff,axiom,
! [X4: dedekind_real] :
( ( ( invers3762989301784728874d_real @ X4 )
= one_on6069100329679821595d_real )
= ( X4 = one_on6069100329679821595d_real ) ) ).
% inverse_eq_1_iff
thf(fact_697_inverse__eq__1__iff,axiom,
! [X4: real] :
( ( ( inverse_inverse_real @ X4 )
= one_one_real )
= ( X4 = one_one_real ) ) ).
% inverse_eq_1_iff
thf(fact_698_right__inverse,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ A @ ( invers3762989301784728874d_real @ A ) )
= one_on6069100329679821595d_real ) ) ).
% right_inverse
thf(fact_699_right__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
= one_one_real ) ) ).
% right_inverse
thf(fact_700_left__inverse,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ A ) @ A )
= one_on6069100329679821595d_real ) ) ).
% left_inverse
thf(fact_701_left__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
= one_one_real ) ) ).
% left_inverse
thf(fact_702_one__reorient,axiom,
! [X4: dedekind_real] :
( ( one_on6069100329679821595d_real = X4 )
= ( X4 = one_on6069100329679821595d_real ) ) ).
% one_reorient
thf(fact_703_one__reorient,axiom,
! [X4: nat] :
( ( one_one_nat = X4 )
= ( X4 = one_one_nat ) ) ).
% one_reorient
thf(fact_704_one__reorient,axiom,
! [X4: int] :
( ( one_one_int = X4 )
= ( X4 = one_one_int ) ) ).
% one_reorient
thf(fact_705_one__reorient,axiom,
! [X4: real] :
( ( one_one_real = X4 )
= ( X4 = one_one_real ) ) ).
% one_reorient
thf(fact_706_mult_Oleft__commute,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ B @ ( times_2157731159493324635d_real @ A @ C ) )
= ( times_2157731159493324635d_real @ A @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_707_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_708_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_709_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_710_mult_Ocomm__neutral,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ one_on6069100329679821595d_real )
= A ) ).
% mult.comm_neutral
thf(fact_711_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_712_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_713_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_714_mult_Ocommute,axiom,
( times_2157731159493324635d_real
= ( ^ [A3: dedekind_real,B2: dedekind_real] : ( times_2157731159493324635d_real @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_715_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_716_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_717_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_718_mult_Oassoc,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ A @ B ) @ C )
= ( times_2157731159493324635d_real @ A @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_719_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_720_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_721_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_722_comm__monoid__mult__class_Omult__1,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_723_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_724_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_725_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_726_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ A @ B ) @ C )
= ( times_2157731159493324635d_real @ A @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_727_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_728_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_729_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_730_inverse__unique,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ B )
= one_on6069100329679821595d_real )
=> ( ( invers3762989301784728874d_real @ A )
= B ) ) ).
% inverse_unique
thf(fact_731_inverse__unique,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= one_one_real )
=> ( ( inverse_inverse_real @ A )
= B ) ) ).
% inverse_unique
thf(fact_732_real__mult__inverse__left__ex,axiom,
! [X4: dedekind_real] :
( ( X4 != zero_z580800474297136991d_real )
=> ~ ! [Y3: dedekind_real] :
( ( times_2157731159493324635d_real @ Y3 @ X4 )
!= one_on6069100329679821595d_real ) ) ).
% real_mult_inverse_left_ex
thf(fact_733_field__class_Ofield__inverse,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ A ) @ A )
= one_on6069100329679821595d_real ) ) ).
% field_class.field_inverse
thf(fact_734_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_735_le__numeral__extra_I4_J,axiom,
ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ one_on6069100329679821595d_real ).
% le_numeral_extra(4)
thf(fact_736_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_737_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_738_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_739_less__numeral__extra_I4_J,axiom,
~ ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ one_on6069100329679821595d_real ) ).
% less_numeral_extra(4)
thf(fact_740_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_741_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_742_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_743_mult__commute__imp__mult__inverse__commute,axiom,
! [Y4: dedekind_real,X4: dedekind_real] :
( ( ( times_2157731159493324635d_real @ Y4 @ X4 )
= ( times_2157731159493324635d_real @ X4 @ Y4 ) )
=> ( ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ Y4 ) @ X4 )
= ( times_2157731159493324635d_real @ X4 @ ( invers3762989301784728874d_real @ Y4 ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_744_mult__commute__imp__mult__inverse__commute,axiom,
! [Y4: real,X4: real] :
( ( ( times_times_real @ Y4 @ X4 )
= ( times_times_real @ X4 @ Y4 ) )
=> ( ( times_times_real @ ( inverse_inverse_real @ Y4 ) @ X4 )
= ( times_times_real @ X4 @ ( inverse_inverse_real @ Y4 ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_745_field__le__mult__one__interval,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ! [Z3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Z3 )
=> ( ( ord_le2991122432403439658d_real @ Z3 @ one_on6069100329679821595d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ Z3 @ X4 ) @ Y4 ) ) )
=> ( ord_le2716243287969276982d_real @ X4 @ Y4 ) ) ).
% field_le_mult_one_interval
thf(fact_746_field__le__mult__one__interval,axiom,
! [X4: real,Y4: real] :
( ! [Z3: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_real @ Z3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z3 @ X4 ) @ Y4 ) ) )
=> ( ord_less_eq_real @ X4 @ Y4 ) ) ).
% field_le_mult_one_interval
thf(fact_747_real__zero__not__eq__one,axiom,
zero_z580800474297136991d_real != one_on6069100329679821595d_real ).
% real_zero_not_eq_one
thf(fact_748_real__mult__inverse__left,axiom,
! [X4: dedekind_real] :
( ( X4 != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ X4 ) @ X4 )
= one_on6069100329679821595d_real ) ) ).
% real_mult_inverse_left
thf(fact_749_nonzero__inverse__mult__distrib,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( B != zero_z580800474297136991d_real )
=> ( ( invers3762989301784728874d_real @ ( times_2157731159493324635d_real @ A @ B ) )
= ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ B ) @ ( invers3762989301784728874d_real @ A ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_750_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_751_less__numeral__extra_I1_J,axiom,
ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ one_on6069100329679821595d_real ).
% less_numeral_extra(1)
thf(fact_752_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_753_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_754_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_755_real__mult__less__mono2,axiom,
! [Z: dedekind_real,X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Z )
=> ( ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ Z @ X4 ) @ ( times_2157731159493324635d_real @ Z @ Y4 ) ) ) ) ).
% real_mult_less_mono2
thf(fact_756_real__mult__order,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ X4 )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Y4 )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ X4 @ Y4 ) ) ) ) ).
% real_mult_order
thf(fact_757_verit__comp__simplify1_I2_J,axiom,
! [A: dedekind_real] : ( ord_le2716243287969276982d_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_758_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_759_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_760_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_761_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_762_verit__la__disequality,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A = B )
| ~ ( ord_le2716243287969276982d_real @ A @ B )
| ~ ( ord_le2716243287969276982d_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_763_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_764_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_765_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_766_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_767_ex__gt__or__lt,axiom,
! [A: real] :
? [B3: real] :
( ( ord_less_real @ A @ B3 )
| ( ord_less_real @ B3 @ A ) ) ).
% ex_gt_or_lt
thf(fact_768_verit__comp__simplify1_I1_J,axiom,
! [A: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_769_verit__comp__simplify1_I1_J,axiom,
! [A: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_770_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_771_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_772_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_773_pinf_I1_J,axiom,
! [P: dedekind_real > $o,P4: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z4: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_774_pinf_I1_J,axiom,
! [P: dedekind_preal > $o,P4: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z4: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_775_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z3 @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_776_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_777_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X: real] :
( ( ord_less_real @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: real] :
! [X: real] :
( ( ord_less_real @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_778_pinf_I2_J,axiom,
! [P: dedekind_real > $o,P4: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z4: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_779_pinf_I2_J,axiom,
! [P: dedekind_preal > $o,P4: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z4: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_780_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z3 @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_781_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_782_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X: real] :
( ( ord_less_real @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: real] :
! [X: real] :
( ( ord_less_real @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_783_pinf_I3_J,axiom,
! [T: dedekind_real] :
? [Z3: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_784_pinf_I3_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_785_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z3 @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_786_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_787_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_788_pinf_I4_J,axiom,
! [T: dedekind_real] :
? [Z3: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_789_pinf_I4_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_790_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z3 @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_791_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_792_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_793_pinf_I5_J,axiom,
! [T: dedekind_real] :
? [Z3: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ X2 )
=> ~ ( ord_le2991122432403439658d_real @ X2 @ T ) ) ).
% pinf(5)
thf(fact_794_pinf_I5_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X2 )
=> ~ ( ord_le5708704896291381698_preal @ X2 @ T ) ) ).
% pinf(5)
thf(fact_795_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z3 @ X2 )
=> ~ ( ord_less_nat @ X2 @ T ) ) ).
% pinf(5)
thf(fact_796_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ~ ( ord_less_int @ X2 @ T ) ) ).
% pinf(5)
thf(fact_797_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ~ ( ord_less_real @ X2 @ T ) ) ).
% pinf(5)
thf(fact_798_pinf_I7_J,axiom,
! [T: dedekind_real] :
? [Z3: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ X2 )
=> ( ord_le2991122432403439658d_real @ T @ X2 ) ) ).
% pinf(7)
thf(fact_799_pinf_I7_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X2 )
=> ( ord_le5708704896291381698_preal @ T @ X2 ) ) ).
% pinf(7)
thf(fact_800_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z3 @ X2 )
=> ( ord_less_nat @ T @ X2 ) ) ).
% pinf(7)
thf(fact_801_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ( ord_less_int @ T @ X2 ) ) ).
% pinf(7)
thf(fact_802_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( ord_less_real @ T @ X2 ) ) ).
% pinf(7)
thf(fact_803_minf_I1_J,axiom,
! [P: dedekind_real > $o,P4: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z4: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z3 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_804_minf_I1_J,axiom,
! [P: dedekind_preal > $o,P4: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z4: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z3 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_805_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z3 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_806_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_807_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X: real] :
( ( ord_less_real @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: real] :
! [X: real] :
( ( ord_less_real @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z3 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_808_minf_I2_J,axiom,
! [P: dedekind_real > $o,P4: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z4: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z3 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_809_minf_I2_J,axiom,
! [P: dedekind_preal > $o,P4: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z4: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z3 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_810_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z3 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_811_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_812_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X: real] :
( ( ord_less_real @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: real] :
! [X: real] :
( ( ord_less_real @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z3 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_813_minf_I3_J,axiom,
! [T: dedekind_real] :
? [Z3: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z3 )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_814_minf_I3_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z3 )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_815_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z3 )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_816_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_817_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z3 )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_818_minf_I4_J,axiom,
! [T: dedekind_real] :
? [Z3: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z3 )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_819_minf_I4_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z3 )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_820_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z3 )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_821_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_822_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z3 )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_823_minf_I5_J,axiom,
! [T: dedekind_real] :
? [Z3: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z3 )
=> ( ord_le2991122432403439658d_real @ X2 @ T ) ) ).
% minf(5)
thf(fact_824_minf_I5_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z3 )
=> ( ord_le5708704896291381698_preal @ X2 @ T ) ) ).
% minf(5)
thf(fact_825_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z3 )
=> ( ord_less_nat @ X2 @ T ) ) ).
% minf(5)
thf(fact_826_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( ord_less_int @ X2 @ T ) ) ).
% minf(5)
thf(fact_827_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z3 )
=> ( ord_less_real @ X2 @ T ) ) ).
% minf(5)
thf(fact_828_minf_I7_J,axiom,
! [T: dedekind_real] :
? [Z3: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z3 )
=> ~ ( ord_le2991122432403439658d_real @ T @ X2 ) ) ).
% minf(7)
thf(fact_829_minf_I7_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z3 )
=> ~ ( ord_le5708704896291381698_preal @ T @ X2 ) ) ).
% minf(7)
thf(fact_830_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z3 )
=> ~ ( ord_less_nat @ T @ X2 ) ) ).
% minf(7)
thf(fact_831_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ~ ( ord_less_int @ T @ X2 ) ) ).
% minf(7)
thf(fact_832_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z3 )
=> ~ ( ord_less_real @ T @ X2 ) ) ).
% minf(7)
thf(fact_833_inverse__le__1__iff,axiom,
! [X4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( invers3762989301784728874d_real @ X4 ) @ one_on6069100329679821595d_real )
= ( ( ord_le2716243287969276982d_real @ X4 @ zero_z580800474297136991d_real )
| ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ X4 ) ) ) ).
% inverse_le_1_iff
thf(fact_834_inverse__le__1__iff,axiom,
! [X4: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ X4 ) @ one_one_real )
= ( ( ord_less_eq_real @ X4 @ zero_zero_real )
| ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).
% inverse_le_1_iff
thf(fact_835_one__less__inverse__iff,axiom,
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ ( invers3762989301784728874d_real @ X4 ) )
= ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ X4 )
& ( ord_le2991122432403439658d_real @ X4 @ one_on6069100329679821595d_real ) ) ) ).
% one_less_inverse_iff
thf(fact_836_one__less__inverse__iff,axiom,
! [X4: real] :
( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X4 ) )
= ( ( ord_less_real @ zero_zero_real @ X4 )
& ( ord_less_real @ X4 @ one_one_real ) ) ) ).
% one_less_inverse_iff
thf(fact_837_one__less__inverse,axiom,
! [A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ A @ one_on6069100329679821595d_real )
=> ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ ( invers3762989301784728874d_real @ A ) ) ) ) ).
% one_less_inverse
thf(fact_838_one__less__inverse,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% one_less_inverse
thf(fact_839_inverse__less__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( invers3762989301784728874d_real @ A ) @ ( invers3762989301784728874d_real @ B ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) )
=> ( ord_le2991122432403439658d_real @ B @ A ) )
& ( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ B ) @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ A @ B ) ) ) ) ).
% inverse_less_iff
thf(fact_840_inverse__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ B @ A ) )
& ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
=> ( ord_less_real @ A @ B ) ) ) ) ).
% inverse_less_iff
thf(fact_841_mult__cancel__right2,axiom,
! [A: dedekind_real,C: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ C )
= C )
= ( ( C = zero_z580800474297136991d_real )
| ( A = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_right2
thf(fact_842_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_843_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_844_mult__cancel__right1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( C
= ( times_2157731159493324635d_real @ B @ C ) )
= ( ( C = zero_z580800474297136991d_real )
| ( B = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_right1
thf(fact_845_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_846_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_847_mult__cancel__left2,axiom,
! [C: dedekind_real,A: dedekind_real] :
( ( ( times_2157731159493324635d_real @ C @ A )
= C )
= ( ( C = zero_z580800474297136991d_real )
| ( A = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_left2
thf(fact_848_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_849_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_850_mult__cancel__left1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( C
= ( times_2157731159493324635d_real @ C @ B ) )
= ( ( C = zero_z580800474297136991d_real )
| ( B = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_left1
thf(fact_851_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_852_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_853_mult__less__cancel__right2,axiom,
! [A: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ C ) @ C )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ A @ one_on6069100329679821595d_real ) )
& ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_854_mult__less__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_855_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_856_mult__less__cancel__right1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ C @ ( times_2157731159493324635d_real @ B @ C ) )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ B ) )
& ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ B @ one_on6069100329679821595d_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_857_mult__less__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_858_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_859_mult__less__cancel__left2,axiom,
! [C: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ C )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ A @ one_on6069100329679821595d_real ) )
& ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_860_mult__less__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_861_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_862_mult__less__cancel__left1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ C @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ B ) )
& ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ B @ one_on6069100329679821595d_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_863_mult__less__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_864_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_865_mult__zero__left,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ zero_z580800474297136991d_real @ A )
= zero_z580800474297136991d_real ) ).
% mult_zero_left
thf(fact_866_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_867_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_868_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_869_mult__zero__right,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% mult_zero_right
thf(fact_870_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_871_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_872_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_873_mult__eq__0__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ B )
= zero_z580800474297136991d_real )
= ( ( A = zero_z580800474297136991d_real )
| ( B = zero_z580800474297136991d_real ) ) ) ).
% mult_eq_0_iff
thf(fact_874_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_875_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_876_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_877_mult__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ C @ A )
= ( times_2157731159493324635d_real @ C @ B ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_878_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_879_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_880_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_881_mult__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ C )
= ( times_2157731159493324635d_real @ B @ C ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_882_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_883_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_884_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_885_real__mult__1,axiom,
! [Z: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ Z )
= Z ) ).
% real_mult_1
thf(fact_886_real__mult__assoc,axiom,
! [Z1: dedekind_real,Z22: dedekind_real,Z32: dedekind_real] :
( ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ Z1 @ Z22 ) @ Z32 )
= ( times_2157731159493324635d_real @ Z1 @ ( times_2157731159493324635d_real @ Z22 @ Z32 ) ) ) ).
% real_mult_assoc
thf(fact_887_real__mult__commute,axiom,
( times_2157731159493324635d_real
= ( ^ [Z5: dedekind_real,W3: dedekind_real] : ( times_2157731159493324635d_real @ W3 @ Z5 ) ) ) ).
% real_mult_commute
thf(fact_888_linorder__neqE__linordered__idom,axiom,
! [X4: dedekind_real,Y4: dedekind_real] :
( ( X4 != Y4 )
=> ( ~ ( ord_le2991122432403439658d_real @ X4 @ Y4 )
=> ( ord_le2991122432403439658d_real @ Y4 @ X4 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_889_linorder__neqE__linordered__idom,axiom,
! [X4: int,Y4: int] :
( ( X4 != Y4 )
=> ( ~ ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ Y4 @ X4 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_890_linorder__neqE__linordered__idom,axiom,
! [X4: real,Y4: real] :
( ( X4 != Y4 )
=> ( ~ ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ Y4 @ X4 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_891_mult__not__zero,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ B )
!= zero_z580800474297136991d_real )
=> ( ( A != zero_z580800474297136991d_real )
& ( B != zero_z580800474297136991d_real ) ) ) ).
% mult_not_zero
thf(fact_892_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_893_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_894_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_895_divisors__zero,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ B )
= zero_z580800474297136991d_real )
=> ( ( A = zero_z580800474297136991d_real )
| ( B = zero_z580800474297136991d_real ) ) ) ).
% divisors_zero
thf(fact_896_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_897_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_898_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_899_no__zero__divisors,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( B != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ A @ B )
!= zero_z580800474297136991d_real ) ) ) ).
% no_zero_divisors
thf(fact_900_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_901_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_902_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_903_mult__left__cancel,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( ( times_2157731159493324635d_real @ C @ A )
= ( times_2157731159493324635d_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_904_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_905_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_906_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_907_mult__right__cancel,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( ( times_2157731159493324635d_real @ A @ C )
= ( times_2157731159493324635d_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_908_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_909_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_910_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_911_zero__neq__one,axiom,
zero_z580800474297136991d_real != one_on6069100329679821595d_real ).
% zero_neq_one
thf(fact_912_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_913_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_914_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_915_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_916_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_917_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_918_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_919_zero__le__mult__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
& ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) )
| ( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
& ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_920_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_921_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_922_mult__nonneg__nonpos2,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ B @ A ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_923_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_924_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_925_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_926_mult__nonpos__nonneg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_927_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_928_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_929_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_930_mult__nonneg__nonpos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_931_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_932_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_933_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_934_mult__nonneg__nonneg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_935_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_936_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_937_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_938_split__mult__neg__le,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
& ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real ) )
| ( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
& ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) ) )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ).
% split_mult_neg_le
thf(fact_939_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_940_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_941_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_942_mult__le__0__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ B ) @ zero_z580800474297136991d_real )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
& ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real ) )
| ( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
& ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_943_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_944_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_945_mult__right__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_946_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_947_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_948_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_949_mult__right__mono__neg,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A )
=> ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_950_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_951_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_952_mult__left__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_953_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_954_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_955_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_956_mult__nonpos__nonpos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_957_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_958_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_959_mult__left__mono__neg,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A )
=> ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_960_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_961_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_962_split__mult__pos__le,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
& ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) )
| ( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
& ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real ) ) )
=> ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_963_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_964_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_965_zero__le__square,axiom,
! [A: dedekind_real] : ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ A ) ) ).
% zero_le_square
thf(fact_966_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_967_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_968_mult__mono_H,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ C @ D3 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_969_mult__mono_H,axiom,
! [A: real,B: real,C: real,D3: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D3 )
=> ( ( 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 @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_970_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ( 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 @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_971_mult__mono_H,axiom,
! [A: int,B: int,C: int,D3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D3 )
=> ( ( 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 @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_972_mult__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ C @ D3 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_973_mult__mono,axiom,
! [A: real,B: real,C: real,D3: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D3 )
=> ( ( 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 @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_974_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ( 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 @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_975_mult__mono,axiom,
! [A: int,B: int,C: int,D3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D3 )
=> ( ( 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 @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_976_not__one__le__zero,axiom,
~ ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ zero_z580800474297136991d_real ) ).
% not_one_le_zero
thf(fact_977_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_978_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_979_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_980_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ one_on6069100329679821595d_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_981_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_982_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_983_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_984_zero__less__one__class_Ozero__le__one,axiom,
ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ one_on6069100329679821595d_real ).
% zero_less_one_class.zero_le_one
thf(fact_985_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_986_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_987_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_988_mult__neg__neg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_989_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_990_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_991_not__square__less__zero,axiom,
! [A: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ A ) @ zero_z580800474297136991d_real ) ).
% not_square_less_zero
thf(fact_992_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_993_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_994_mult__less__0__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ B ) @ zero_z580800474297136991d_real )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
& ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real ) )
| ( ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real )
& ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_995_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_996_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_997_mult__neg__pos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_neg_pos
thf(fact_998_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_999_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_1000_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_1001_mult__pos__neg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_pos_neg
thf(fact_1002_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_1003_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_1004_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_1005_mult__pos__pos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1006_mult__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 @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1007_mult__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 @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1008_mult__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 @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1009_mult__pos__neg2,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ B @ A ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_pos_neg2
thf(fact_1010_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_1011_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_1012_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_1013_zero__less__mult__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
& ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) )
| ( ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real )
& ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1014_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1015_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1016_zero__less__mult__pos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1017_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1018_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1019_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1020_zero__less__mult__pos2,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ B @ A ) )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1021_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1022_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1023_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1024_mult__less__cancel__left__neg,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ord_le2991122432403439658d_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1025_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1026_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1027_mult__less__cancel__left__pos,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ord_le2991122432403439658d_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1028_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1029_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1030_mult__strict__left__mono__neg,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1031_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1032_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1033_mult__strict__left__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1034_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1035_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1036_mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1037_mult__less__cancel__left__disj,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
& ( ord_le2991122432403439658d_real @ A @ B ) )
| ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
& ( ord_le2991122432403439658d_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1038_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1039_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1040_mult__strict__right__mono__neg,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1041_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1042_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1043_mult__strict__right__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1044_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1045_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1046_mult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1047_mult__less__cancel__right__disj,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
& ( ord_le2991122432403439658d_real @ A @ B ) )
| ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
& ( ord_le2991122432403439658d_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1048_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1049_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1050_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1051_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1052_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1053_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1054_zero__less__one,axiom,
ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ one_on6069100329679821595d_real ).
% zero_less_one
thf(fact_1055_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1056_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_1057_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_1058_not__one__less__zero,axiom,
~ ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ zero_z580800474297136991d_real ) ).
% not_one_less_zero
thf(fact_1059_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1060_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_1061_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_1062_less__1__mult,axiom,
! [M: dedekind_real,N: dedekind_real] :
( ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ M )
=> ( ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ N )
=> ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ ( times_2157731159493324635d_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1063_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1064_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1065_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1066_mult__le__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ A @ B ) )
& ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1067_mult__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1068_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1069_mult__le__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ A @ B ) )
& ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1070_mult__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1071_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1072_mult__left__less__imp__less,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1073_mult__left__less__imp__less,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1074_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1075_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1076_mult__strict__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ C @ D3 )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ D3 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1077_mult__strict__mono,axiom,
! [A: real,B: real,C: real,D3: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D3 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D3 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1078_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D3 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1079_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D3: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D3 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D3 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1080_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1081_real__add__mult__distrib,axiom,
! [Z1: dedekind_real,Z22: dedekind_real,W: dedekind_real] :
( ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ Z1 @ Z22 ) @ W )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ Z1 @ W ) @ ( times_2157731159493324635d_real @ Z22 @ W ) ) ) ).
% real_add_mult_distrib
thf(fact_1082_real__add__left__mono,axiom,
! [X4: dedekind_real,Y4: dedekind_real,Z: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X4 @ Y4 )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ Z @ X4 ) @ ( plus_p4060926892116697567d_real @ Z @ Y4 ) ) ) ).
% real_add_left_mono
thf(fact_1083_real__divide__def,axiom,
( divide9119111104558704680d_real
= ( ^ [R: dedekind_real,S2: dedekind_real] : ( times_2157731159493324635d_real @ R @ ( invers3762989301784728874d_real @ S2 ) ) ) ) ).
% real_divide_def
thf(fact_1084_preal__add__less__cancel__left,axiom,
! [T: dedekind_preal,R2: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S3 ) )
= ( ord_le5708704896291381698_preal @ R2 @ S3 ) ) ).
% preal_add_less_cancel_left
thf(fact_1085_preal__add__less__cancel__right,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S3 @ T ) )
= ( ord_le5708704896291381698_preal @ R2 @ S3 ) ) ).
% preal_add_less_cancel_right
thf(fact_1086_preal__add__le__cancel__right,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S3 @ T ) )
= ( ord_le5604041210740703414_preal @ R2 @ S3 ) ) ).
% preal_add_le_cancel_right
thf(fact_1087_preal__add__le__cancel__left,axiom,
! [T: dedekind_preal,R2: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S3 ) )
= ( ord_le5604041210740703414_preal @ R2 @ S3 ) ) ).
% preal_add_le_cancel_left
thf(fact_1088_preal__add__mult__distrib,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal,W: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ X4 @ Y4 ) @ W )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X4 @ W ) @ ( times_3000655703912201937_preal @ Y4 @ W ) ) ) ).
% preal_add_mult_distrib
thf(fact_1089_preal__add__mult__distrib2,axiom,
! [W: dedekind_preal,X4: dedekind_preal,Y4: dedekind_preal] :
( ( times_3000655703912201937_preal @ W @ ( plus_p3173629198307831117_preal @ X4 @ Y4 ) )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ W @ X4 ) @ ( times_3000655703912201937_preal @ W @ Y4 ) ) ) ).
% preal_add_mult_distrib2
thf(fact_1090_real__mult__congruent2__lemma,axiom,
! [X1: dedekind_preal,Y22: dedekind_preal,X22: dedekind_preal,Y1: dedekind_preal,X4: dedekind_preal,Y4: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ X1 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y1 ) )
=> ( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X4 @ X1 ) @ ( times_3000655703912201937_preal @ Y4 @ Y1 ) ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X4 @ Y22 ) @ ( times_3000655703912201937_preal @ Y4 @ X22 ) ) )
= ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X4 @ X22 ) @ ( times_3000655703912201937_preal @ Y4 @ Y22 ) ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X4 @ Y1 ) @ ( times_3000655703912201937_preal @ Y4 @ X1 ) ) ) ) ) ).
% real_mult_congruent2_lemma
thf(fact_1091_preal__add__less2__mono1,axiom,
! [R2: dedekind_preal,S3: dedekind_preal,T: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R2 @ S3 )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S3 @ T ) ) ) ).
% preal_add_less2_mono1
thf(fact_1092_preal__add__less2__mono2,axiom,
! [R2: dedekind_preal,S3: dedekind_preal,T: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R2 @ S3 )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S3 ) ) ) ).
% preal_add_less2_mono2
thf(fact_1093_preal__self__less__add__left,axiom,
! [R2: dedekind_preal,S3: dedekind_preal] : ( ord_le5708704896291381698_preal @ R2 @ ( plus_p3173629198307831117_preal @ R2 @ S3 ) ) ).
% preal_self_less_add_left
thf(fact_1094_preal__add__left__less__cancel,axiom,
! [T: dedekind_preal,R2: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S3 ) )
=> ( ord_le5708704896291381698_preal @ R2 @ S3 ) ) ).
% preal_add_left_less_cancel
thf(fact_1095_preal__add__right__less__cancel,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S3 @ T ) )
=> ( ord_le5708704896291381698_preal @ R2 @ S3 ) ) ).
% preal_add_right_less_cancel
thf(fact_1096_preal__add__assoc,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal,Z: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ X4 @ Y4 ) @ Z )
= ( plus_p3173629198307831117_preal @ X4 @ ( plus_p3173629198307831117_preal @ Y4 @ Z ) ) ) ).
% preal_add_assoc
thf(fact_1097_preal__add__commute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [X3: dedekind_preal,Y2: dedekind_preal] : ( plus_p3173629198307831117_preal @ Y2 @ X3 ) ) ) ).
% preal_add_commute
thf(fact_1098_preal__trans__lemma,axiom,
! [X4: dedekind_preal,Y1: dedekind_preal,X1: dedekind_preal,Y4: dedekind_preal,Y22: dedekind_preal,X22: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ X4 @ Y1 )
= ( plus_p3173629198307831117_preal @ X1 @ Y4 ) )
=> ( ( ( plus_p3173629198307831117_preal @ X4 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y4 ) )
=> ( ( plus_p3173629198307831117_preal @ X1 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y1 ) ) ) ) ).
% preal_trans_lemma
thf(fact_1099_preal__add__left__cancel,axiom,
! [C: dedekind_preal,A: dedekind_preal,B: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ C @ A )
= ( plus_p3173629198307831117_preal @ C @ B ) )
=> ( A = B ) ) ).
% preal_add_left_cancel
thf(fact_1100_preal__add__right__cancel,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S3: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ R2 @ T )
= ( plus_p3173629198307831117_preal @ S3 @ T ) )
=> ( R2 = S3 ) ) ).
% preal_add_right_cancel
thf(fact_1101_preal__eq__le__imp__le,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal,D3: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ A @ B )
= ( plus_p3173629198307831117_preal @ C @ D3 ) )
=> ( ( ord_le5604041210740703414_preal @ C @ A )
=> ( ord_le5604041210740703414_preal @ B @ D3 ) ) ) ).
% preal_eq_le_imp_le
thf(fact_1102_real__le__lemma,axiom,
! [U1: dedekind_preal,V2: dedekind_preal,U2: dedekind_preal,V1: dedekind_preal,X1: dedekind_preal,Y1: dedekind_preal,X22: dedekind_preal,Y22: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U1 @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ V1 ) )
=> ( ( ( plus_p3173629198307831117_preal @ X1 @ V1 )
= ( plus_p3173629198307831117_preal @ U1 @ Y1 ) )
=> ( ( ( plus_p3173629198307831117_preal @ X22 @ V2 )
= ( plus_p3173629198307831117_preal @ U2 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X1 @ Y22 ) @ ( plus_p3173629198307831117_preal @ X22 @ Y1 ) ) ) ) ) ).
% real_le_lemma
thf(fact_1103_divide__preal__def,axiom,
( divide4190755330972744004_preal
= ( ^ [R: dedekind_preal,S2: dedekind_preal] : ( times_3000655703912201937_preal @ R @ ( invers3090987106763476162_preal @ S2 ) ) ) ) ).
% divide_preal_def
thf(fact_1104_real__of__preal__le__iff,axiom,
! [M1: dedekind_preal,M22: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ ( dedeki829075424511776921_preal @ M1 ) @ ( dedeki829075424511776921_preal @ M22 ) )
= ( ord_le5604041210740703414_preal @ M1 @ M22 ) ) ).
% real_of_preal_le_iff
thf(fact_1105_real__of__preal__less__iff,axiom,
! [M1: dedekind_preal,M22: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ ( dedeki829075424511776921_preal @ M1 ) @ ( dedeki829075424511776921_preal @ M22 ) )
= ( ord_le5708704896291381698_preal @ M1 @ M22 ) ) ).
% real_of_preal_less_iff
thf(fact_1106_real__of__preal__mult,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( dedeki829075424511776921_preal @ ( times_3000655703912201937_preal @ X4 @ Y4 ) )
= ( times_2157731159493324635d_real @ ( dedeki829075424511776921_preal @ X4 ) @ ( dedeki829075424511776921_preal @ Y4 ) ) ) ).
% real_of_preal_mult
thf(fact_1107_real__of__preal__zero__less,axiom,
! [M: dedekind_preal] : ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( dedeki829075424511776921_preal @ M ) ) ).
% real_of_preal_zero_less
thf(fact_1108_real__gt__zero__preal__Ex,axiom,
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ X4 )
= ( ? [Y2: dedekind_preal] :
( X4
= ( dedeki829075424511776921_preal @ Y2 ) ) ) ) ).
% real_gt_zero_preal_Ex
thf(fact_1109_real__of__preal__add,axiom,
! [X4: dedekind_preal,Y4: dedekind_preal] :
( ( dedeki829075424511776921_preal @ ( plus_p3173629198307831117_preal @ X4 @ Y4 ) )
= ( plus_p4060926892116697567d_real @ ( dedeki829075424511776921_preal @ X4 ) @ ( dedeki829075424511776921_preal @ Y4 ) ) ) ).
% real_of_preal_add
thf(fact_1110_real__of__preal__trichotomy,axiom,
! [X4: dedekind_real] :
? [M3: dedekind_preal] :
( ( X4
= ( dedeki829075424511776921_preal @ M3 ) )
| ( X4 = zero_z580800474297136991d_real )
| ( X4
= ( uminus7714077491378687647d_real @ ( dedeki829075424511776921_preal @ M3 ) ) ) ) ).
% real_of_preal_trichotomy
thf(fact_1111_real__less__sum__gt__zero,axiom,
! [W: dedekind_real,S3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ W @ S3 )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ S3 @ ( uminus7714077491378687647d_real @ W ) ) ) ) ).
% real_less_sum_gt_zero
thf(fact_1112_real__sum__gt__zero__less,axiom,
! [S3: dedekind_real,W: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ S3 @ ( uminus7714077491378687647d_real @ W ) ) )
=> ( ord_le2991122432403439658d_real @ W @ S3 ) ) ).
% real_sum_gt_zero_less
thf(fact_1113_add__inc,axiom,
! [X4: num,Y4: num] :
( ( plus_plus_num @ X4 @ ( inc @ Y4 ) )
= ( inc @ ( plus_plus_num @ X4 @ Y4 ) ) ) ).
% add_inc
thf(fact_1114_mult__inc,axiom,
! [X4: num,Y4: num] :
( ( times_times_num @ X4 @ ( inc @ Y4 ) )
= ( plus_plus_num @ ( times_times_num @ X4 @ Y4 ) @ X4 ) ) ).
% mult_inc
thf(fact_1115_num__induct,axiom,
! [P: num > $o,X4: num] :
( ( P @ one )
=> ( ! [X: num] :
( ( P @ X )
=> ( P @ ( inc @ X ) ) )
=> ( P @ X4 ) ) ) ).
% num_induct
thf(fact_1116_le__num__One__iff,axiom,
! [X4: num] :
( ( ord_less_eq_num @ X4 @ one )
= ( X4 = one ) ) ).
% le_num_One_iff
thf(fact_1117_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1118_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1119_add__One,axiom,
! [X4: num] :
( ( plus_plus_num @ X4 @ one )
= ( inc @ X4 ) ) ).
% add_One
thf(fact_1120_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1121_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1122_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1123_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1124_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1125_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1126_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1127_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1128_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I2: nat,J2: nat] :
( ( ( ord_less_nat @ J2 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_div
thf(fact_1129_div__less__iff__less__mult,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1130_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1131_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1132_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1133_div__mult2__eq,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q3 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q3 ) ) ).
% div_mult2_eq
thf(fact_1134_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1135_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1136_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1137_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1138_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1139_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1140_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1141_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1142_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1143_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1144_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1145_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1146_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1147_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1148_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1149_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1150_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_1151_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_1152_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_1153_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1154_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1155_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1156_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1157_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1158_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1159_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1160_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1161_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1162_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1163_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1164_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1165_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_1166_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1167_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1168_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_1169_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_1170_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: 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 @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1171_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1172_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
= ( P @ B3 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1173_zdiv__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_1174_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_1175_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_1176_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_1177_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_1178_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_1179_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1180_add__self__div__2,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M ) ).
% add_self_div_2
thf(fact_1181_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_1182_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_1183_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K3: int] :
( ( P @ K3 )
=> ( ( K3 != zero_zero_int )
=> ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K3: int] :
( ( P @ K3 )
=> ( ( K3
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_1184_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_1185_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_1186_plus__inverse__ge__2,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_1187_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_1188_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
thf(fact_1189_ln__less__cancel__iff,axiom,
! [X4: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y4 ) )
= ( ord_less_real @ X4 @ Y4 ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_1190_ln__inj__iff,axiom,
! [X4: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ( ln_ln_real @ X4 )
= ( ln_ln_real @ Y4 ) )
= ( X4 = Y4 ) ) ) ) ).
% ln_inj_iff
thf(fact_1191_ln__less__zero__iff,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ ( ln_ln_real @ X4 ) @ zero_zero_real )
= ( ord_less_real @ X4 @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_1192_ln__gt__zero__iff,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
= ( ord_less_real @ one_one_real @ X4 ) ) ) ).
% ln_gt_zero_iff
thf(fact_1193_ln__eq__zero__iff,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ( ln_ln_real @ X4 )
= zero_zero_real )
= ( X4 = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_1194_real__add__minus__iff,axiom,
! [X4: real,A: real] :
( ( ( plus_plus_real @ X4 @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X4 = A ) ) ).
% real_add_minus_iff
thf(fact_1195_ln__ge__zero__iff,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
= ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).
% ln_ge_zero_iff
thf(fact_1196_ln__le__zero__iff,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ zero_zero_real )
= ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_1197_ln__le__cancel__iff,axiom,
! [X4: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y4 ) )
= ( ord_less_eq_real @ X4 @ Y4 ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_1198_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1199_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1200_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1201_ln__add__one__self__le__self2,axiom,
! [X4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) ).
% ln_add_one_self_le_self2
thf(fact_1202_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( divide_divide_int @ K @ L )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1203_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1204_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1205_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1206_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1207_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_1208_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1209_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1210_ln__ge__zero__imp__ge__one,axiom,
! [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
=> ( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_1211_ln__add__one__self__le__self,axiom,
! [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) ).
% ln_add_one_self_le_self
thf(fact_1212_ln__ge__zero,axiom,
! [X4: real] :
( ( ord_less_eq_real @ one_one_real @ X4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) ) ) ).
% ln_ge_zero
thf(fact_1213_ln__mult,axiom,
! [X4: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ln_ln_real @ ( times_times_real @ X4 @ Y4 ) )
= ( plus_plus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y4 ) ) ) ) ) ).
% ln_mult
thf(fact_1214_ln__bound,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ).
% ln_bound
thf(fact_1215_incr__mult__lemma,axiom,
! [D3: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ! [X: int] :
( ( P @ X )
=> ( P @ ( plus_plus_int @ X @ D3 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ ( times_times_int @ K @ D3 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1216_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1217_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1218_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1219_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1220_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1221_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1222_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1223_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1224_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1225_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1226_zdiv__mono2__neg,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1227_zdiv__mono1__neg,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1228_int__div__pos__eq,axiom,
! [A: int,B: int,Q3: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( divide_divide_int @ A @ B )
= Q3 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1229_int__div__neg__eq,axiom,
! [A: int,B: int,Q3: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( divide_divide_int @ A @ B )
= Q3 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1230_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1231_zdiv__mono2,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1232_zdiv__mono1,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1233_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I2: int,J2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
& ( ord_less_int @ J2 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I2: int,J2: int] :
( ( ( ord_less_int @ K @ J2 )
& ( ord_less_eq_int @ J2 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_zdiv
thf(fact_1234_divide__real__def,axiom,
( divide_divide_real
= ( ^ [X3: real,Y2: real] : ( times_times_real @ X3 @ ( inverse_inverse_real @ Y2 ) ) ) ) ).
% divide_real_def
thf(fact_1235_verit__less__mono__div__int2,axiom,
! [A2: int,B5: int,N: int] :
( ( ord_less_eq_int @ A2 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1236_real__add__le__0__iff,axiom,
! [X4: real,Y4: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X4 @ Y4 ) @ zero_zero_real )
= ( ord_less_eq_real @ Y4 @ ( uminus_uminus_real @ X4 ) ) ) ).
% real_add_le_0_iff
thf(fact_1237_real__0__le__add__iff,axiom,
! [X4: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X4 @ Y4 ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X4 ) @ Y4 ) ) ).
% real_0_le_add_iff
thf(fact_1238_real__0__less__add__iff,axiom,
! [X4: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X4 @ Y4 ) )
= ( ord_less_real @ ( uminus_uminus_real @ X4 ) @ Y4 ) ) ).
% real_0_less_add_iff
thf(fact_1239_real__add__less__0__iff,axiom,
! [X4: real,Y4: real] :
( ( ord_less_real @ ( plus_plus_real @ X4 @ Y4 ) @ zero_zero_real )
= ( ord_less_real @ Y4 @ ( uminus_uminus_real @ X4 ) ) ) ).
% real_add_less_0_iff
thf(fact_1240_real__minus__mult__self__le,axiom,
! [U: real,X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X4 @ X4 ) ) ).
% real_minus_mult_self_le
thf(fact_1241_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1242_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1243_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1244_ln__inverse,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ln_ln_real @ ( inverse_inverse_real @ X4 ) )
= ( uminus_uminus_real @ ( ln_ln_real @ X4 ) ) ) ) ).
% ln_inverse
thf(fact_1245_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1246_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1247_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1248_ln__less__self,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ord_less_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ).
% ln_less_self
thf(fact_1249_ln__gt__zero__imp__gt__one,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
=> ( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ord_less_real @ one_one_real @ X4 ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_1250_ln__less__zero,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ X4 @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X4 ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_1251_ln__gt__zero,axiom,
! [X4: real] :
( ( ord_less_real @ one_one_real @ X4 )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) ) ) ).
% ln_gt_zero
thf(fact_1252_int__div__less__self,axiom,
! [X4: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X4 )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X4 @ K ) @ X4 ) ) ) ).
% int_div_less_self
thf(fact_1253_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_1254_arsinh__minus__real,axiom,
! [X4: real] :
( ( arsinh_real @ ( uminus_uminus_real @ X4 ) )
= ( uminus_uminus_real @ ( arsinh_real @ X4 ) ) ) ).
% arsinh_minus_real
thf(fact_1255_real__divide__square__eq,axiom,
! [R2: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
= ( divide_divide_real @ A @ R2 ) ) ).
% real_divide_square_eq
thf(fact_1256_cosh__ln__real,axiom,
! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ( ( cosh_real @ ( ln_ln_real @ X4 ) )
= ( divide_divide_real @ ( plus_plus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% cosh_ln_real
thf(fact_1257_floor__minus__one__divide__eq__div__numeral,axiom,
! [B: num] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
= ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% floor_minus_one_divide_eq_div_numeral
thf(fact_1258_floor__divide__eq__div__numeral,axiom,
! [A: num,B: num] :
( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).
% floor_divide_eq_div_numeral
thf(fact_1259_floor__one__divide__eq__div__numeral,axiom,
! [B: num] :
( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
= ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).
% floor_one_divide_eq_div_numeral
thf(fact_1260_floor__minus__divide__eq__div__numeral,axiom,
! [A: num,B: num] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
= ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% floor_minus_divide_eq_div_numeral
thf(fact_1261_arcosh__cosh__real,axiom,
! [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( ( arcosh_real @ ( cosh_real @ X4 ) )
= X4 ) ) ).
% arcosh_cosh_real
thf(fact_1262_cosh__real__nonneg,axiom,
! [X4: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X4 ) ) ).
% cosh_real_nonneg
thf(fact_1263_cosh__real__nonneg__le__iff,axiom,
! [X4: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_eq_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y4 ) )
= ( ord_less_eq_real @ X4 @ Y4 ) ) ) ) ).
% cosh_real_nonneg_le_iff
thf(fact_1264_cosh__real__nonpos__le__iff,axiom,
! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y4 ) )
= ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% cosh_real_nonpos_le_iff
thf(fact_1265_cosh__real__nonzero,axiom,
! [X4: real] :
( ( cosh_real @ X4 )
!= zero_zero_real ) ).
% cosh_real_nonzero
thf(fact_1266_cosh__real__pos,axiom,
! [X4: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X4 ) ) ).
% cosh_real_pos
thf(fact_1267_cosh__real__strict__mono,axiom,
! [X4: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y4 ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_1268_cosh__real__nonneg__less__iff,axiom,
! [X4: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y4 ) )
= ( ord_less_real @ X4 @ Y4 ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_1269_cosh__real__nonpos__less__iff,axiom,
! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
=> ( ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y4 ) )
= ( ord_less_real @ Y4 @ X4 ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_1270_cosh__real__ge__1,axiom,
! [X4: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X4 ) ) ).
% cosh_real_ge_1
% Conjectures (1)
thf(conj_0,conjecture,
ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( invers3762989301784728874d_real @ x ) ).
%------------------------------------------------------------------------------