TPTP Problem File: SLH0662^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Dedekind_Real/0000_Dedekind_Real/prob_01370_044704__5756468_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1345 ( 455 unt; 74 typ; 0 def)
% Number of atoms : 3759 (1087 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10419 ( 367 ~; 107 |; 169 &;8041 @)
% ( 0 <=>;1735 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 343 ( 343 >; 0 *; 0 +; 0 <<)
% Number of symbols : 67 ( 66 usr; 14 con; 0-2 aty)
% Number of variables : 3686 ( 180 ^;3351 !; 155 ?;3686 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:30:25.511
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_n_t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
set_Dedekind_preal: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_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 (66)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(sy_c_Dedekind__Real_Opsup,type,
dedekind_psup: set_Dedekind_preal > dedekind_preal ).
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__Opreal,type,
one_on9143529541772854033_preal: dedekind_preal ).
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_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
bot_bo4848840305443107682_preal: set_Dedekind_preal ).
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_Orderings_Oorder__class_OGreatest_001t__Dedekind____Real__Oreal,type,
order_1105151008113787197d_real: ( dedekind_real > $o ) > dedekind_real ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Int__Oint,type,
order_Greatest_int: ( int > $o ) > int ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Num__Onum,type,
order_Greatest_num: ( num > $o ) > num ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal,type,
order_Greatest_real: ( real > $o ) > real ).
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_Set_OCollect_001t__Dedekind____Real__Opreal,type,
collec1132657498972982273_preal: ( dedekind_preal > $o ) > set_Dedekind_preal ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_member_001t__Dedekind____Real__Opreal,type,
member6871284927547481791_preal: dedekind_preal > set_Dedekind_preal > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_m1,type,
m1: dedekind_preal ).
thf(sy_v_m2,type,
m2: dedekind_preal ).
% Relevant facts (1270)
thf(fact_0_minf_I7_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z )
=> ~ ( ord_le2991122432403439658d_real @ T @ X ) ) ).
% minf(7)
thf(fact_1_minf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ~ ( ord_less_real @ T @ X ) ) ).
% minf(7)
thf(fact_2_minf_I7_J,axiom,
! [T: num] :
? [Z: num] :
! [X: num] :
( ( ord_less_num @ X @ Z )
=> ~ ( ord_less_num @ T @ X ) ) ).
% minf(7)
thf(fact_3_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ~ ( ord_less_nat @ T @ X ) ) ).
% minf(7)
thf(fact_4_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ~ ( ord_less_int @ T @ X ) ) ).
% minf(7)
thf(fact_5_minf_I5_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z )
=> ( ord_le2991122432403439658d_real @ X @ T ) ) ).
% minf(5)
thf(fact_6_minf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( ord_less_real @ X @ T ) ) ).
% minf(5)
thf(fact_7_minf_I5_J,axiom,
! [T: num] :
? [Z: num] :
! [X: num] :
( ( ord_less_num @ X @ Z )
=> ( ord_less_num @ X @ T ) ) ).
% minf(5)
thf(fact_8_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ord_less_nat @ X @ T ) ) ).
% minf(5)
thf(fact_9_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( ord_less_int @ X @ T ) ) ).
% minf(5)
thf(fact_10_minf_I4_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z )
=> ( X != T ) ) ).
% minf(4)
thf(fact_11_minf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( X != T ) ) ).
% minf(4)
thf(fact_12_minf_I4_J,axiom,
! [T: num] :
? [Z: num] :
! [X: num] :
( ( ord_less_num @ X @ Z )
=> ( X != T ) ) ).
% minf(4)
thf(fact_13_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( X != T ) ) ).
% minf(4)
thf(fact_14_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( X != T ) ) ).
% minf(4)
thf(fact_15_minf_I3_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z )
=> ( X != T ) ) ).
% minf(3)
thf(fact_16_minf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( X != T ) ) ).
% minf(3)
thf(fact_17_minf_I3_J,axiom,
! [T: num] :
? [Z: num] :
! [X: num] :
( ( ord_less_num @ X @ Z )
=> ( X != T ) ) ).
% minf(3)
thf(fact_18_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( X != T ) ) ).
% minf(3)
thf(fact_19_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( X != T ) ) ).
% minf(3)
thf(fact_20_minf_I2_J,axiom,
! [P: dedekind_real > $o,P2: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_21_minf_I2_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z2: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_22_minf_I2_J,axiom,
! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: num] :
! [X: num] :
( ( ord_less_num @ X @ Z )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_23_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_24_minf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_25_minf_I1_J,axiom,
! [P: dedekind_real > $o,P2: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_26_minf_I1_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z2: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_27_minf_I1_J,axiom,
! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: num] :
! [X: num] :
( ( ord_less_num @ X @ Z )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_28_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_29_minf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_30_pinf_I7_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X )
=> ( ord_le2991122432403439658d_real @ T @ X ) ) ).
% pinf(7)
thf(fact_31_pinf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( ord_less_real @ T @ X ) ) ).
% pinf(7)
thf(fact_32_pinf_I7_J,axiom,
! [T: num] :
? [Z: num] :
! [X: num] :
( ( ord_less_num @ Z @ X )
=> ( ord_less_num @ T @ X ) ) ).
% pinf(7)
thf(fact_33_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( ord_less_nat @ T @ X ) ) ).
% pinf(7)
thf(fact_34_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( ord_less_int @ T @ X ) ) ).
% pinf(7)
thf(fact_35_pinf_I5_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X )
=> ~ ( ord_le2991122432403439658d_real @ X @ T ) ) ).
% pinf(5)
thf(fact_36_pinf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ~ ( ord_less_real @ X @ T ) ) ).
% pinf(5)
thf(fact_37_pinf_I5_J,axiom,
! [T: num] :
? [Z: num] :
! [X: num] :
( ( ord_less_num @ Z @ X )
=> ~ ( ord_less_num @ X @ T ) ) ).
% pinf(5)
thf(fact_38_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ~ ( ord_less_nat @ X @ T ) ) ).
% pinf(5)
thf(fact_39_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ~ ( ord_less_int @ X @ T ) ) ).
% pinf(5)
thf(fact_40_pinf_I4_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_41_pinf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_42_pinf_I4_J,axiom,
! [T: num] :
? [Z: num] :
! [X: num] :
( ( ord_less_num @ Z @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_43_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_44_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_45_pinf_I3_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_46_pinf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_47_pinf_I3_J,axiom,
! [T: num] :
? [Z: num] :
! [X: num] :
( ( ord_less_num @ Z @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_48_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_49_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_50_pinf_I1_J,axiom,
! [P: dedekind_real > $o,P2: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_51_pinf_I1_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z2: real] :
! [X2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: real] :
! [X2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_52_pinf_I1_J,axiom,
! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: num] :
! [X: num] :
( ( ord_less_num @ Z @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_53_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_54_pinf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_55_pinf_I2_J,axiom,
! [P: dedekind_real > $o,P2: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_56_pinf_I2_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z2: real] :
! [X2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: real] :
! [X2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_57_pinf_I2_J,axiom,
! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: num] :
! [X: num] :
( ( ord_less_num @ Z @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_58_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_59_pinf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_60_order__less__imp__not__less,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ~ ( ord_le2991122432403439658d_real @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_61_order__less__imp__not__less,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ~ ( ord_less_real @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_62_order__less__imp__not__less,axiom,
! [X3: num,Y: num] :
( ( ord_less_num @ X3 @ Y )
=> ~ ( ord_less_num @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_63_order__less__imp__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_64_order__less__imp__not__less,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_65_order__less__imp__not__eq2,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_66_order__less__imp__not__eq2,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_67_order__less__imp__not__eq2,axiom,
! [X3: num,Y: num] :
( ( ord_less_num @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_68_order__less__imp__not__eq2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_69_order__less__imp__not__eq2,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_70_order__less__imp__not__eq,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_71_order__less__imp__not__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_72_order__less__imp__not__eq,axiom,
! [X3: num,Y: num] :
( ( ord_less_num @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_73_order__less__imp__not__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_74_order__less__imp__not__eq,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_75_linorder__less__linear,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y )
| ( X3 = Y )
| ( ord_le2991122432403439658d_real @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_76_linorder__less__linear,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_real @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_77_linorder__less__linear,axiom,
! [X3: num,Y: num] :
( ( ord_less_num @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_num @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_78_linorder__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_79_linorder__less__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_int @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_80_order__less__imp__triv,axiom,
! [X3: dedekind_real,Y: dedekind_real,P: $o] :
( ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ( ( ord_le2991122432403439658d_real @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_81_order__less__imp__triv,axiom,
! [X3: real,Y: real,P: $o] :
( ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_real @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_82_order__less__imp__triv,axiom,
! [X3: num,Y: num,P: $o] :
( ( ord_less_num @ X3 @ Y )
=> ( ( ord_less_num @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_83_order__less__imp__triv,axiom,
! [X3: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_84_order__less__imp__triv,axiom,
! [X3: int,Y: int,P: $o] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_85_order__less__not__sym,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ~ ( ord_le2991122432403439658d_real @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_86_order__less__not__sym,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ~ ( ord_less_real @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_87_order__less__not__sym,axiom,
! [X3: num,Y: num] :
( ( ord_less_num @ X3 @ Y )
=> ~ ( ord_less_num @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_88_order__less__not__sym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_89_order__less__not__sym,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_90_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_91_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_92_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_93_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_94_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_95_order__less__subst2,axiom,
! [A: real,B: real,F: real > dedekind_real,C: dedekind_real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_96_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_97_order__less__subst2,axiom,
! [A: real,B: real,F: real > num,C: num] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_98_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_99_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_100_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_101_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_102_order__less__subst1,axiom,
! [A: dedekind_real,F: num > dedekind_real,B: num,C: num] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_103_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_104_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 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_105_order__less__subst1,axiom,
! [A: real,F: dedekind_real > real,B: dedekind_real,C: dedekind_real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_106_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_107_order__less__subst1,axiom,
! [A: real,F: num > real,B: num,C: num] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_108_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_109_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_110_order__less__irrefl,axiom,
! [X3: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_111_order__less__irrefl,axiom,
! [X3: real] :
~ ( ord_less_real @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_112_order__less__irrefl,axiom,
! [X3: num] :
~ ( ord_less_num @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_113_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_114_order__less__irrefl,axiom,
! [X3: int] :
~ ( ord_less_int @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_115_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_116_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_117_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_118_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_119_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_120_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > dedekind_real,C: dedekind_real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_121_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_122_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > num,C: num] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_123_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_124_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_125_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_126_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_127_ord__eq__less__subst,axiom,
! [A: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_128_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_129_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_130_ord__eq__less__subst,axiom,
! [A: dedekind_real,F: real > dedekind_real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_131_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_132_ord__eq__less__subst,axiom,
! [A: num,F: real > num,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_133_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_134_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_135_order__less__trans,axiom,
! [X3: dedekind_real,Y: dedekind_real,Z3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ( ( ord_le2991122432403439658d_real @ Y @ Z3 )
=> ( ord_le2991122432403439658d_real @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_136_order__less__trans,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_137_order__less__trans,axiom,
! [X3: num,Y: num,Z3: num] :
( ( ord_less_num @ X3 @ Y )
=> ( ( ord_less_num @ Y @ Z3 )
=> ( ord_less_num @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_138_order__less__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_139_order__less__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_140_lt__ex,axiom,
! [X3: dedekind_real] :
? [Y2: dedekind_real] : ( ord_le2991122432403439658d_real @ Y2 @ X3 ) ).
% lt_ex
thf(fact_141_lt__ex,axiom,
! [X3: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X3 ) ).
% lt_ex
thf(fact_142_lt__ex,axiom,
! [X3: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X3 ) ).
% lt_ex
thf(fact_143_gt__ex,axiom,
! [X3: dedekind_real] :
? [X_1: dedekind_real] : ( ord_le2991122432403439658d_real @ X3 @ X_1 ) ).
% gt_ex
thf(fact_144_gt__ex,axiom,
! [X3: real] :
? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).
% gt_ex
thf(fact_145_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_146_gt__ex,axiom,
! [X3: int] :
? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% gt_ex
thf(fact_147_dense,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ? [Z: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Z )
& ( ord_le2991122432403439658d_real @ Z @ Y ) ) ) ).
% dense
thf(fact_148_dense,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ? [Z: real] :
( ( ord_less_real @ X3 @ Z )
& ( ord_less_real @ Z @ Y ) ) ) ).
% dense
thf(fact_149_less__imp__neq,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_150_less__imp__neq,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_151_less__imp__neq,axiom,
! [X3: num,Y: num] :
( ( ord_less_num @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_152_less__imp__neq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_153_less__imp__neq,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_154_order_Oasym,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ~ ( ord_le2991122432403439658d_real @ B @ A ) ) ).
% order.asym
thf(fact_155_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_156_order_Oasym,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order.asym
thf(fact_157_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_158_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_159_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_160_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_161_ord__eq__less__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_162_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_163_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_164_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_165_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_166_ord__less__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_167_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_168_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_169_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X2 )
=> ( P @ Y3 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_170_antisym__conv3,axiom,
! [Y: dedekind_real,X3: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ Y @ X3 )
=> ( ( ~ ( ord_le2991122432403439658d_real @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_171_antisym__conv3,axiom,
! [Y: real,X3: real] :
( ~ ( ord_less_real @ Y @ X3 )
=> ( ( ~ ( ord_less_real @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_172_antisym__conv3,axiom,
! [Y: num,X3: num] :
( ~ ( ord_less_num @ Y @ X3 )
=> ( ( ~ ( ord_less_num @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_173_antisym__conv3,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_nat @ Y @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_174_antisym__conv3,axiom,
! [Y: int,X3: int] :
( ~ ( ord_less_int @ Y @ X3 )
=> ( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_175_linorder__cases,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_le2991122432403439658d_real @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_176_linorder__cases,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_real @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_real @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_177_linorder__cases,axiom,
! [X3: num,Y: num] :
( ~ ( ord_less_num @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_num @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_178_linorder__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_179_linorder__cases,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_180_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_181_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_182_dual__order_Oasym,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ A @ B ) ) ).
% dual_order.asym
thf(fact_183_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_184_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_185_dual__order_Oirrefl,axiom,
! [A: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_186_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_187_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_188_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_189_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_190_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X4: nat] : ( P3 @ X4 ) )
= ( ^ [P4: nat > $o] :
? [N: nat] :
( ( P4 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P4 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_191_linorder__less__wlog,axiom,
! [P: dedekind_real > dedekind_real > $o,A: dedekind_real,B: dedekind_real] :
( ! [A2: dedekind_real,B2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: dedekind_real] : ( P @ A2 @ A2 )
=> ( ! [A2: dedekind_real,B2: dedekind_real] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_192_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: real] : ( P @ A2 @ A2 )
=> ( ! [A2: real,B2: real] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_193_linorder__less__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: num] : ( P @ A2 @ A2 )
=> ( ! [A2: num,B2: num] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_194_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ A2 )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_195_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: int] : ( P @ A2 @ A2 )
=> ( ! [A2: int,B2: int] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_196_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_197_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_198_order_Ostrict__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_199_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_200_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_201_not__less__iff__gr__or__eq,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ~ ( ord_le2991122432403439658d_real @ X3 @ Y ) )
= ( ( ord_le2991122432403439658d_real @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_202_not__less__iff__gr__or__eq,axiom,
! [X3: real,Y: real] :
( ( ~ ( ord_less_real @ X3 @ Y ) )
= ( ( ord_less_real @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_203_not__less__iff__gr__or__eq,axiom,
! [X3: num,Y: num] :
( ( ~ ( ord_less_num @ X3 @ Y ) )
= ( ( ord_less_num @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_204_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ( ord_less_nat @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_205_not__less__iff__gr__or__eq,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( ( ord_less_int @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_206_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_207_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_208_dual__order_Ostrict__trans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_209_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_210_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_211_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_212_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_213_order_Ostrict__implies__not__eq,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_214_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_215_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_216_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_217_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_218_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_219_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_220_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_221_mem__Collect__eq,axiom,
! [A: dedekind_preal,P: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ A @ ( collec1132657498972982273_preal @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_222_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_223_Collect__mem__eq,axiom,
! [A3: set_Dedekind_preal] :
( ( collec1132657498972982273_preal
@ ^ [X5: dedekind_preal] : ( member6871284927547481791_preal @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_224_Collect__mem__eq,axiom,
! [A3: set_real] :
( ( collect_real
@ ^ [X5: real] : ( member_real @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_225_linorder__neqE,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( X3 != Y )
=> ( ~ ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ( ord_le2991122432403439658d_real @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_226_linorder__neqE,axiom,
! [X3: real,Y: real] :
( ( X3 != Y )
=> ( ~ ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_227_linorder__neqE,axiom,
! [X3: num,Y: num] :
( ( X3 != Y )
=> ( ~ ( ord_less_num @ X3 @ Y )
=> ( ord_less_num @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_228_linorder__neqE,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_229_linorder__neqE,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
=> ( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_230_order__less__asym,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ~ ( ord_le2991122432403439658d_real @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_231_order__less__asym,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ~ ( ord_less_real @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_232_order__less__asym,axiom,
! [X3: num,Y: num] :
( ( ord_less_num @ X3 @ Y )
=> ~ ( ord_less_num @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_233_order__less__asym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_234_order__less__asym,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_235_linorder__neq__iff,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( X3 != Y )
= ( ( ord_le2991122432403439658d_real @ X3 @ Y )
| ( ord_le2991122432403439658d_real @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_236_linorder__neq__iff,axiom,
! [X3: real,Y: real] :
( ( X3 != Y )
= ( ( ord_less_real @ X3 @ Y )
| ( ord_less_real @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_237_linorder__neq__iff,axiom,
! [X3: num,Y: num] :
( ( X3 != Y )
= ( ( ord_less_num @ X3 @ Y )
| ( ord_less_num @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_238_linorder__neq__iff,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
= ( ( ord_less_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_239_linorder__neq__iff,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
= ( ( ord_less_int @ X3 @ Y )
| ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_240_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_241_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_242_order__less__asym_H,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order_less_asym'
thf(fact_243_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_244_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_245_verit__comp__simplify1_I1_J,axiom,
! [A: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_246_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_247_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_248_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_249_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_250_linordered__field__no__lb,axiom,
! [X: dedekind_real] :
? [Y2: dedekind_real] : ( ord_le2991122432403439658d_real @ Y2 @ X ) ).
% linordered_field_no_lb
thf(fact_251_linordered__field__no__lb,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% linordered_field_no_lb
thf(fact_252_linordered__field__no__ub,axiom,
! [X: dedekind_real] :
? [X_1: dedekind_real] : ( ord_le2991122432403439658d_real @ X @ X_1 ) ).
% linordered_field_no_ub
thf(fact_253_linordered__field__no__ub,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% linordered_field_no_ub
thf(fact_254_linorder__neqE__linordered__idom,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( X3 != Y )
=> ( ~ ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ( ord_le2991122432403439658d_real @ Y @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_255_linorder__neqE__linordered__idom,axiom,
! [X3: real,Y: real] :
( ( X3 != Y )
=> ( ~ ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ Y @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_256_linorder__neqE__linordered__idom,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
=> ( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_257_ex__gt__or__lt,axiom,
! [A: real] :
? [B2: real] :
( ( ord_less_real @ A @ B2 )
| ( ord_less_real @ B2 @ A ) ) ).
% ex_gt_or_lt
thf(fact_258_preal__add__less__cancel__left,axiom,
! [T: dedekind_preal,R: dedekind_preal,S: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R ) @ ( plus_p3173629198307831117_preal @ T @ S ) )
= ( ord_le5708704896291381698_preal @ R @ S ) ) ).
% preal_add_less_cancel_left
thf(fact_259_preal__add__less__cancel__right,axiom,
! [R: dedekind_preal,T: dedekind_preal,S: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R @ T ) @ ( plus_p3173629198307831117_preal @ S @ T ) )
= ( ord_le5708704896291381698_preal @ R @ S ) ) ).
% preal_add_less_cancel_right
thf(fact_260_real__less__def,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
& ( X5 != Y4 ) ) ) ) ).
% real_less_def
thf(fact_261_dual__order_Orefl,axiom,
! [A: dedekind_real] : ( ord_le2716243287969276982d_real @ A @ A ) ).
% dual_order.refl
thf(fact_262_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_263_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_264_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_265_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_266_order__refl,axiom,
! [X3: dedekind_real] : ( ord_le2716243287969276982d_real @ X3 @ X3 ) ).
% order_refl
thf(fact_267_order__refl,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ X3 ) ).
% order_refl
thf(fact_268_order__refl,axiom,
! [X3: num] : ( ord_less_eq_num @ X3 @ X3 ) ).
% order_refl
thf(fact_269_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_270_order__refl,axiom,
! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% order_refl
thf(fact_271_order__antisym__conv,axiom,
! [Y: dedekind_real,X3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Y @ X3 )
=> ( ( ord_le2716243287969276982d_real @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_272_order__antisym__conv,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ Y @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_273_order__antisym__conv,axiom,
! [Y: num,X3: num] :
( ( ord_less_eq_num @ Y @ X3 )
=> ( ( ord_less_eq_num @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_274_order__antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_275_order__antisym__conv,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ( ( ord_less_eq_int @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_276_linorder__le__cases,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ~ ( ord_le2716243287969276982d_real @ X3 @ Y )
=> ( ord_le2716243287969276982d_real @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_277_linorder__le__cases,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_278_linorder__le__cases,axiom,
! [X3: num,Y: num] :
( ~ ( ord_less_eq_num @ X3 @ Y )
=> ( ord_less_eq_num @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_279_linorder__le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_280_linorder__le__cases,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_281_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_282_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_283_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_284_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_285_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_286_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_287_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_288_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > num,C: num] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_289_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_290_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_291_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_292_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_293_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_294_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_295_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_296_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_297_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_298_ord__eq__le__subst,axiom,
! [A: num,F: real > num,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_299_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_300_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_301_linorder__linear,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y )
| ( ord_le2716243287969276982d_real @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_302_linorder__linear,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
| ( ord_less_eq_real @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_303_linorder__linear,axiom,
! [X3: num,Y: num] :
( ( ord_less_eq_num @ X3 @ Y )
| ( ord_less_eq_num @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_304_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_305_linorder__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
| ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_306_order__eq__refl,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( X3 = Y )
=> ( ord_le2716243287969276982d_real @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_307_order__eq__refl,axiom,
! [X3: real,Y: real] :
( ( X3 = Y )
=> ( ord_less_eq_real @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_308_order__eq__refl,axiom,
! [X3: num,Y: num] :
( ( X3 = Y )
=> ( ord_less_eq_num @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_309_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_310_order__eq__refl,axiom,
! [X3: int,Y: int] :
( ( X3 = Y )
=> ( ord_less_eq_int @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_311_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_312_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_313_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_314_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_315_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_316_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_317_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_318_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_319_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_320_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_321_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_322_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_323_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 )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_324_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_325_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 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_326_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_327_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_328_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 )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_329_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_330_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 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_331_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: dedekind_real,Z4: dedekind_real] : ( Y5 = Z4 ) )
= ( ^ [A4: dedekind_real,B3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A4 @ B3 )
& ( ord_le2716243287969276982d_real @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_332_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_333_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_334_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_335_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_336_antisym,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_337_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_338_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_339_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_340_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_341_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_342_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_343_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_344_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_345_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_346_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_347_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_348_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_349_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_350_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_351_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: dedekind_real,Z4: dedekind_real] : ( Y5 = Z4 ) )
= ( ^ [A4: dedekind_real,B3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B3 @ A4 )
& ( ord_le2716243287969276982d_real @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_352_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_353_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_354_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_355_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_356_linorder__wlog,axiom,
! [P: dedekind_real > dedekind_real > $o,A: dedekind_real,B: dedekind_real] :
( ! [A2: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: dedekind_real,B2: dedekind_real] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_357_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: real,B2: real] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_358_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: num,B2: num] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_359_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_360_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: int,B2: int] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_361_order__trans,axiom,
! [X3: dedekind_real,Y: dedekind_real,Z3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y )
=> ( ( ord_le2716243287969276982d_real @ Y @ Z3 )
=> ( ord_le2716243287969276982d_real @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_362_order__trans,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_eq_real @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_363_order__trans,axiom,
! [X3: num,Y: num,Z3: num] :
( ( ord_less_eq_num @ X3 @ Y )
=> ( ( ord_less_eq_num @ Y @ Z3 )
=> ( ord_less_eq_num @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_364_order__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_365_order__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_366_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_367_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_368_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_369_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_370_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_371_order__antisym,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y )
=> ( ( ord_le2716243287969276982d_real @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_372_order__antisym,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_373_order__antisym,axiom,
! [X3: num,Y: num] :
( ( ord_less_eq_num @ X3 @ Y )
=> ( ( ord_less_eq_num @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_374_order__antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_375_order__antisym,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_376_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_377_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_378_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_379_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_380_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_381_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_382_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_383_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_384_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_385_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_386_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: dedekind_real,Z4: dedekind_real] : ( Y5 = Z4 ) )
= ( ^ [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
& ( ord_le2716243287969276982d_real @ Y4 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_387_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
= ( ^ [X5: real,Y4: real] :
( ( ord_less_eq_real @ X5 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_388_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
= ( ^ [X5: num,Y4: num] :
( ( ord_less_eq_num @ X5 @ Y4 )
& ( ord_less_eq_num @ Y4 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_389_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_390_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_391_le__cases3,axiom,
! [X3: dedekind_real,Y: dedekind_real,Z3: dedekind_real] :
( ( ( ord_le2716243287969276982d_real @ X3 @ Y )
=> ~ ( ord_le2716243287969276982d_real @ Y @ Z3 ) )
=> ( ( ( ord_le2716243287969276982d_real @ Y @ X3 )
=> ~ ( ord_le2716243287969276982d_real @ X3 @ Z3 ) )
=> ( ( ( ord_le2716243287969276982d_real @ X3 @ Z3 )
=> ~ ( ord_le2716243287969276982d_real @ Z3 @ Y ) )
=> ( ( ( ord_le2716243287969276982d_real @ Z3 @ Y )
=> ~ ( ord_le2716243287969276982d_real @ Y @ X3 ) )
=> ( ( ( ord_le2716243287969276982d_real @ Y @ Z3 )
=> ~ ( ord_le2716243287969276982d_real @ Z3 @ X3 ) )
=> ~ ( ( ord_le2716243287969276982d_real @ Z3 @ X3 )
=> ~ ( ord_le2716243287969276982d_real @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_392_le__cases3,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ( ord_less_eq_real @ X3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X3 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X3 ) )
=> ( ( ( ord_less_eq_real @ Y @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_393_le__cases3,axiom,
! [X3: num,Y: num,Z3: num] :
( ( ( ord_less_eq_num @ X3 @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_num @ Y @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_num @ X3 @ Z3 )
=> ~ ( ord_less_eq_num @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_num @ Z3 @ Y )
=> ~ ( ord_less_eq_num @ Y @ X3 ) )
=> ( ( ( ord_less_eq_num @ Y @ Z3 )
=> ~ ( ord_less_eq_num @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_num @ Z3 @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_394_le__cases3,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_395_le__cases3,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X3 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X3 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_396_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_397_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_398_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_399_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_400_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_401_verit__comp__simplify1_I2_J,axiom,
! [A: dedekind_real] : ( ord_le2716243287969276982d_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_402_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_403_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_404_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_405_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_406_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_407_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_408_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_409_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_410_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_411_real__le__refl,axiom,
! [W: dedekind_real] : ( ord_le2716243287969276982d_real @ W @ W ) ).
% real_le_refl
thf(fact_412_add__eq__exists,axiom,
! [A: dedekind_real,B: dedekind_real] :
? [X2: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ X2 )
= B ) ).
% add_eq_exists
thf(fact_413_add__eq__exists,axiom,
! [A: real,B: real] :
? [X2: real] :
( ( plus_plus_real @ A @ X2 )
= B ) ).
% add_eq_exists
thf(fact_414_add__eq__exists,axiom,
! [A: int,B: int] :
? [X2: int] :
( ( plus_plus_int @ A @ X2 )
= B ) ).
% add_eq_exists
thf(fact_415_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_416_preal__add__assoc,axiom,
! [X3: dedekind_preal,Y: dedekind_preal,Z3: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ X3 @ Y ) @ Z3 )
= ( plus_p3173629198307831117_preal @ X3 @ ( plus_p3173629198307831117_preal @ Y @ Z3 ) ) ) ).
% preal_add_assoc
thf(fact_417_real__le__antisym,axiom,
! [Z3: dedekind_real,W: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Z3 @ W )
=> ( ( ord_le2716243287969276982d_real @ W @ Z3 )
=> ( Z3 = W ) ) ) ).
% real_le_antisym
thf(fact_418_preal__add__commute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [X5: dedekind_preal,Y4: dedekind_preal] : ( plus_p3173629198307831117_preal @ Y4 @ X5 ) ) ) ).
% preal_add_commute
thf(fact_419_preal__trans__lemma,axiom,
! [X3: dedekind_preal,Y1: dedekind_preal,X1: dedekind_preal,Y: dedekind_preal,Y22: dedekind_preal,X22: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ X3 @ Y1 )
= ( plus_p3173629198307831117_preal @ X1 @ Y ) )
=> ( ( ( plus_p3173629198307831117_preal @ X3 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y ) )
=> ( ( plus_p3173629198307831117_preal @ X1 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y1 ) ) ) ) ).
% preal_trans_lemma
thf(fact_420_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_421_preal__add__right__cancel,axiom,
! [R: dedekind_preal,T: dedekind_preal,S: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ R @ T )
= ( plus_p3173629198307831117_preal @ S @ T ) )
=> ( R = S ) ) ).
% preal_add_right_cancel
thf(fact_422_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 )
& ! [X: real] :
( ( ( ord_less_eq_real @ A @ X )
& ( ord_less_real @ X @ C2 ) )
=> ( P @ X ) )
& ! [D: real] :
( ! [X2: real] :
( ( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_real @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_real @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_423_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 )
& ! [X: nat] :
( ( ( ord_less_eq_nat @ A @ X )
& ( ord_less_nat @ X @ C2 ) )
=> ( P @ X ) )
& ! [D: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_424_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 )
& ! [X: int] :
( ( ( ord_less_eq_int @ A @ X )
& ( ord_less_int @ X @ C2 ) )
=> ( P @ X ) )
& ! [D: int] :
( ! [X2: int] :
( ( ( ord_less_eq_int @ A @ X2 )
& ( ord_less_int @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_int @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_425_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_426_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_427_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_428_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_429_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_430_pinf_I6_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X )
=> ~ ( ord_le2716243287969276982d_real @ X @ T ) ) ).
% pinf(6)
thf(fact_431_pinf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ T ) ) ).
% pinf(6)
thf(fact_432_pinf_I6_J,axiom,
! [T: num] :
? [Z: num] :
! [X: num] :
( ( ord_less_num @ Z @ X )
=> ~ ( ord_less_eq_num @ X @ T ) ) ).
% pinf(6)
thf(fact_433_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ T ) ) ).
% pinf(6)
thf(fact_434_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ T ) ) ).
% pinf(6)
thf(fact_435_pinf_I8_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X )
=> ( ord_le2716243287969276982d_real @ T @ X ) ) ).
% pinf(8)
thf(fact_436_pinf_I8_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( ord_less_eq_real @ T @ X ) ) ).
% pinf(8)
thf(fact_437_pinf_I8_J,axiom,
! [T: num] :
? [Z: num] :
! [X: num] :
( ( ord_less_num @ Z @ X )
=> ( ord_less_eq_num @ T @ X ) ) ).
% pinf(8)
thf(fact_438_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( ord_less_eq_nat @ T @ X ) ) ).
% pinf(8)
thf(fact_439_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( ord_less_eq_int @ T @ X ) ) ).
% pinf(8)
thf(fact_440_minf_I6_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z )
=> ( ord_le2716243287969276982d_real @ X @ T ) ) ).
% minf(6)
thf(fact_441_minf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ( ord_less_eq_real @ X @ T ) ) ).
% minf(6)
thf(fact_442_minf_I6_J,axiom,
! [T: num] :
? [Z: num] :
! [X: num] :
( ( ord_less_num @ X @ Z )
=> ( ord_less_eq_num @ X @ T ) ) ).
% minf(6)
thf(fact_443_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ T ) ) ).
% minf(6)
thf(fact_444_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( ord_less_eq_int @ X @ T ) ) ).
% minf(6)
thf(fact_445_minf_I8_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z )
=> ~ ( ord_le2716243287969276982d_real @ T @ X ) ) ).
% minf(8)
thf(fact_446_minf_I8_J,axiom,
! [T: real] :
? [Z: real] :
! [X: real] :
( ( ord_less_real @ X @ Z )
=> ~ ( ord_less_eq_real @ T @ X ) ) ).
% minf(8)
thf(fact_447_minf_I8_J,axiom,
! [T: num] :
? [Z: num] :
! [X: num] :
( ( ord_less_num @ X @ Z )
=> ~ ( ord_less_eq_num @ T @ X ) ) ).
% minf(8)
thf(fact_448_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ T @ X ) ) ).
% minf(8)
thf(fact_449_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ~ ( ord_less_eq_int @ T @ X ) ) ).
% minf(8)
thf(fact_450_order__le__imp__less__or__eq,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y )
=> ( ( ord_le2991122432403439658d_real @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_451_order__le__imp__less__or__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_real @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_452_order__le__imp__less__or__eq,axiom,
! [X3: num,Y: num] :
( ( ord_less_eq_num @ X3 @ Y )
=> ( ( ord_less_num @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_453_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_454_order__le__imp__less__or__eq,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_int @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_455_linorder__le__less__linear,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y )
| ( ord_le2991122432403439658d_real @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_456_linorder__le__less__linear,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
| ( ord_less_real @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_457_linorder__le__less__linear,axiom,
! [X3: num,Y: num] :
( ( ord_less_eq_num @ X3 @ Y )
| ( ord_less_num @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_458_linorder__le__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_459_linorder__le__less__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
| ( ord_less_int @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_460_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_461_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_462_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > dedekind_real,C: dedekind_real] :
( ( ord_less_num @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_463_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_464_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 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_465_order__less__le__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > real,C: real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_466_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_467_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > real,C: real] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_468_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_469_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_470_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_471_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_472_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_473_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_474_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_475_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_476_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_477_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_478_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_479_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_480_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_481_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_482_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_483_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_484_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_485_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_486_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_487_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_488_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_489_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_490_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 )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_491_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 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_492_order__le__less__subst1,axiom,
! [A: dedekind_real,F: num > dedekind_real,B: num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_493_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_494_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 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_495_order__le__less__subst1,axiom,
! [A: real,F: dedekind_real > real,B: dedekind_real,C: dedekind_real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_496_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_497_order__le__less__subst1,axiom,
! [A: real,F: num > real,B: num,C: num] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_498_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_499_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_500_order__less__le__trans,axiom,
! [X3: dedekind_real,Y: dedekind_real,Z3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ( ( ord_le2716243287969276982d_real @ Y @ Z3 )
=> ( ord_le2991122432403439658d_real @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_501_order__less__le__trans,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_real @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_502_order__less__le__trans,axiom,
! [X3: num,Y: num,Z3: num] :
( ( ord_less_num @ X3 @ Y )
=> ( ( ord_less_eq_num @ Y @ Z3 )
=> ( ord_less_num @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_503_order__less__le__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_504_order__less__le__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_505_order__le__less__trans,axiom,
! [X3: dedekind_real,Y: dedekind_real,Z3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y )
=> ( ( ord_le2991122432403439658d_real @ Y @ Z3 )
=> ( ord_le2991122432403439658d_real @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_506_order__le__less__trans,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_507_order__le__less__trans,axiom,
! [X3: num,Y: num,Z3: num] :
( ( ord_less_eq_num @ X3 @ Y )
=> ( ( ord_less_num @ Y @ Z3 )
=> ( ord_less_num @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_508_order__le__less__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_509_order__le__less__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_510_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_511_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_512_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_513_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_514_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_515_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_516_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_517_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_518_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_519_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_520_order__less__imp__le,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ( ord_le2716243287969276982d_real @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_521_order__less__imp__le,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_522_order__less__imp__le,axiom,
! [X3: num,Y: num] :
( ( ord_less_num @ X3 @ Y )
=> ( ord_less_eq_num @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_523_order__less__imp__le,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_524_order__less__imp__le,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_eq_int @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_525_linorder__not__less,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ~ ( ord_le2991122432403439658d_real @ X3 @ Y ) )
= ( ord_le2716243287969276982d_real @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_526_linorder__not__less,axiom,
! [X3: real,Y: real] :
( ( ~ ( ord_less_real @ X3 @ Y ) )
= ( ord_less_eq_real @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_527_linorder__not__less,axiom,
! [X3: num,Y: num] :
( ( ~ ( ord_less_num @ X3 @ Y ) )
= ( ord_less_eq_num @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_528_linorder__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_529_linorder__not__less,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_530_linorder__not__le,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ~ ( ord_le2716243287969276982d_real @ X3 @ Y ) )
= ( ord_le2991122432403439658d_real @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_531_linorder__not__le,axiom,
! [X3: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X3 @ Y ) )
= ( ord_less_real @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_532_linorder__not__le,axiom,
! [X3: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X3 @ Y ) )
= ( ord_less_num @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_533_linorder__not__le,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y ) )
= ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_534_linorder__not__le,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X3 @ Y ) )
= ( ord_less_int @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_535_order__less__le,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
& ( X5 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_536_order__less__le,axiom,
( ord_less_real
= ( ^ [X5: real,Y4: real] :
( ( ord_less_eq_real @ X5 @ Y4 )
& ( X5 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_537_order__less__le,axiom,
( ord_less_num
= ( ^ [X5: num,Y4: num] :
( ( ord_less_eq_num @ X5 @ Y4 )
& ( X5 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_538_order__less__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
& ( X5 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_539_order__less__le,axiom,
( ord_less_int
= ( ^ [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
& ( X5 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_540_order__le__less,axiom,
( ord_le2716243287969276982d_real
= ( ^ [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
| ( X5 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_541_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X5: real,Y4: real] :
( ( ord_less_real @ X5 @ Y4 )
| ( X5 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_542_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X5: num,Y4: num] :
( ( ord_less_num @ X5 @ Y4 )
| ( X5 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_543_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
| ( X5 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_544_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
| ( X5 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_545_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_546_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_547_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_548_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_549_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_550_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_551_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_552_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_553_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_554_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_555_dual__order_Ostrict__iff__not,axiom,
( ord_le2991122432403439658d_real
= ( ^ [B3: dedekind_real,A4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B3 @ A4 )
& ~ ( ord_le2716243287969276982d_real @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_556_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_557_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ~ ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_558_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_559_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_560_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_561_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_562_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_563_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_564_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_565_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_566_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_567_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_568_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_569_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_570_dual__order_Ostrict__iff__order,axiom,
( ord_le2991122432403439658d_real
= ( ^ [B3: dedekind_real,A4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_571_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_572_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_573_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_574_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_575_dual__order_Oorder__iff__strict,axiom,
( ord_le2716243287969276982d_real
= ( ^ [B3: dedekind_real,A4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_576_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_real @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_577_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_num @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_578_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_579_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_int @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_580_dense__le__bounded,axiom,
! [X3: dedekind_real,Y: dedekind_real,Z3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ( ! [W2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ W2 )
=> ( ( ord_le2991122432403439658d_real @ W2 @ Y )
=> ( ord_le2716243287969276982d_real @ W2 @ Z3 ) ) )
=> ( ord_le2716243287969276982d_real @ Y @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_581_dense__le__bounded,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ! [W2: real] :
( ( ord_less_real @ X3 @ W2 )
=> ( ( ord_less_real @ W2 @ Y )
=> ( ord_less_eq_real @ W2 @ Z3 ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_582_dense__ge__bounded,axiom,
! [Z3: dedekind_real,X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ X3 )
=> ( ! [W2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ W2 )
=> ( ( ord_le2991122432403439658d_real @ W2 @ X3 )
=> ( ord_le2716243287969276982d_real @ Y @ W2 ) ) )
=> ( ord_le2716243287969276982d_real @ Y @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_583_dense__ge__bounded,axiom,
! [Z3: real,X3: real,Y: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ! [W2: real] :
( ( ord_less_real @ Z3 @ W2 )
=> ( ( ord_less_real @ W2 @ X3 )
=> ( ord_less_eq_real @ Y @ W2 ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_584_order_Ostrict__iff__not,axiom,
( ord_le2991122432403439658d_real
= ( ^ [A4: dedekind_real,B3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A4 @ B3 )
& ~ ( ord_le2716243287969276982d_real @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_585_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_586_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ~ ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_587_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_588_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_589_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_590_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_591_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_592_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_593_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_594_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_595_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_596_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_597_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_598_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_599_order_Ostrict__iff__order,axiom,
( ord_le2991122432403439658d_real
= ( ^ [A4: dedekind_real,B3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_600_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_601_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_602_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_603_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_604_order_Oorder__iff__strict,axiom,
( ord_le2716243287969276982d_real
= ( ^ [A4: dedekind_real,B3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_605_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_606_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_num @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_607_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_608_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_609_not__le__imp__less,axiom,
! [Y: dedekind_real,X3: dedekind_real] :
( ~ ( ord_le2716243287969276982d_real @ Y @ X3 )
=> ( ord_le2991122432403439658d_real @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_610_not__le__imp__less,axiom,
! [Y: real,X3: real] :
( ~ ( ord_less_eq_real @ Y @ X3 )
=> ( ord_less_real @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_611_not__le__imp__less,axiom,
! [Y: num,X3: num] :
( ~ ( ord_less_eq_num @ Y @ X3 )
=> ( ord_less_num @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_612_not__le__imp__less,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y @ X3 )
=> ( ord_less_nat @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_613_not__le__imp__less,axiom,
! [Y: int,X3: int] :
( ~ ( ord_less_eq_int @ Y @ X3 )
=> ( ord_less_int @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_614_less__le__not__le,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
& ~ ( ord_le2716243287969276982d_real @ Y4 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_615_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X5: real,Y4: real] :
( ( ord_less_eq_real @ X5 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_616_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X5: num,Y4: num] :
( ( ord_less_eq_num @ X5 @ Y4 )
& ~ ( ord_less_eq_num @ Y4 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_617_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_618_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_619_dense__le,axiom,
! [Y: dedekind_real,Z3: dedekind_real] :
( ! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y )
=> ( ord_le2716243287969276982d_real @ X2 @ Z3 ) )
=> ( ord_le2716243287969276982d_real @ Y @ Z3 ) ) ).
% dense_le
thf(fact_620_dense__le,axiom,
! [Y: real,Z3: real] :
( ! [X2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Z3 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_le
thf(fact_621_dense__ge,axiom,
! [Z3: dedekind_real,Y: dedekind_real] :
( ! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ X2 )
=> ( ord_le2716243287969276982d_real @ Y @ X2 ) )
=> ( ord_le2716243287969276982d_real @ Y @ Z3 ) ) ).
% dense_ge
thf(fact_622_dense__ge,axiom,
! [Z3: real,Y: real] :
( ! [X2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( ord_less_eq_real @ Y @ X2 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_ge
thf(fact_623_antisym__conv2,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y )
=> ( ( ~ ( ord_le2991122432403439658d_real @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_624_antisym__conv2,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ~ ( ord_less_real @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_625_antisym__conv2,axiom,
! [X3: num,Y: num] :
( ( ord_less_eq_num @ X3 @ Y )
=> ( ( ~ ( ord_less_num @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_626_antisym__conv2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_627_antisym__conv2,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_628_antisym__conv1,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ( ( ord_le2716243287969276982d_real @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_629_antisym__conv1,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_630_antisym__conv1,axiom,
! [X3: num,Y: num] :
( ~ ( ord_less_num @ X3 @ Y )
=> ( ( ord_less_eq_num @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_631_antisym__conv1,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_632_antisym__conv1,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_633_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_634_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_635_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_636_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_637_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_638_leI,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ( ord_le2716243287969276982d_real @ Y @ X3 ) ) ).
% leI
thf(fact_639_leI,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ Y @ X3 ) ) ).
% leI
thf(fact_640_leI,axiom,
! [X3: num,Y: num] :
( ~ ( ord_less_num @ X3 @ Y )
=> ( ord_less_eq_num @ Y @ X3 ) ) ).
% leI
thf(fact_641_leI,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% leI
thf(fact_642_leI,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_eq_int @ Y @ X3 ) ) ).
% leI
thf(fact_643_leD,axiom,
! [Y: dedekind_real,X3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Y @ X3 )
=> ~ ( ord_le2991122432403439658d_real @ X3 @ Y ) ) ).
% leD
thf(fact_644_leD,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ Y @ X3 )
=> ~ ( ord_less_real @ X3 @ Y ) ) ).
% leD
thf(fact_645_leD,axiom,
! [Y: num,X3: num] :
( ( ord_less_eq_num @ Y @ X3 )
=> ~ ( ord_less_num @ X3 @ Y ) ) ).
% leD
thf(fact_646_leD,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y ) ) ).
% leD
thf(fact_647_leD,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ~ ( ord_less_int @ X3 @ Y ) ) ).
% leD
thf(fact_648_preal__add__right__less__cancel,axiom,
! [R: dedekind_preal,T: dedekind_preal,S: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R @ T ) @ ( plus_p3173629198307831117_preal @ S @ T ) )
=> ( ord_le5708704896291381698_preal @ R @ S ) ) ).
% preal_add_right_less_cancel
thf(fact_649_preal__add__left__less__cancel,axiom,
! [T: dedekind_preal,R: dedekind_preal,S: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R ) @ ( plus_p3173629198307831117_preal @ T @ S ) )
=> ( ord_le5708704896291381698_preal @ R @ S ) ) ).
% preal_add_left_less_cancel
thf(fact_650_preal__self__less__add__left,axiom,
! [R: dedekind_preal,S: dedekind_preal] : ( ord_le5708704896291381698_preal @ R @ ( plus_p3173629198307831117_preal @ R @ S ) ) ).
% preal_self_less_add_left
thf(fact_651_preal__add__less2__mono2,axiom,
! [R: dedekind_preal,S: dedekind_preal,T: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R @ S )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R ) @ ( plus_p3173629198307831117_preal @ T @ S ) ) ) ).
% preal_add_less2_mono2
thf(fact_652_preal__add__less2__mono1,axiom,
! [R: dedekind_preal,S: dedekind_preal,T: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R @ S )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R @ T ) @ ( plus_p3173629198307831117_preal @ S @ T ) ) ) ).
% preal_add_less2_mono1
thf(fact_653_add__less__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( ord_le2991122432403439658d_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_654_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_655_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_656_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_657_add__less__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ C @ A ) @ ( plus_p4060926892116697567d_real @ C @ B ) )
= ( ord_le2991122432403439658d_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_658_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_659_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_660_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_661_add__le__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ C @ A ) @ ( plus_p4060926892116697567d_real @ C @ B ) )
= ( ord_le2716243287969276982d_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_662_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_663_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_664_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_665_add__le__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( ord_le2716243287969276982d_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_666_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_667_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_668_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_669_add__less__le__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ C @ D2 )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_670_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_671_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_672_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_673_add__le__less__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ C @ D2 )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_674_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D2 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_675_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_676_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_677_add__mono__thms__linordered__field_I3_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( ord_le2991122432403439658d_real @ I @ J )
& ( ord_le2716243287969276982d_real @ K @ L ) )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ I @ K ) @ ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_678_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_679_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_680_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_681_add__mono__thms__linordered__field_I4_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( ord_le2716243287969276982d_real @ I @ J )
& ( ord_le2991122432403439658d_real @ K @ L ) )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ I @ K ) @ ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_682_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_683_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_684_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_685_add__left__cancel,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ B )
= ( plus_p4060926892116697567d_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_686_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_687_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_688_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_689_add__right__cancel,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ B @ A )
= ( plus_p4060926892116697567d_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_690_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_691_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_692_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_693_preal__add__le__cancel__right,axiom,
! [R: dedekind_preal,T: dedekind_preal,S: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ R @ T ) @ ( plus_p3173629198307831117_preal @ S @ T ) )
= ( ord_le5604041210740703414_preal @ R @ S ) ) ).
% preal_add_le_cancel_right
thf(fact_694_preal__add__le__cancel__left,axiom,
! [T: dedekind_preal,R: dedekind_preal,S: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ T @ R ) @ ( plus_p3173629198307831117_preal @ T @ S ) )
= ( ord_le5604041210740703414_preal @ R @ S ) ) ).
% preal_add_le_cancel_left
thf(fact_695_preal__eq__le__imp__le,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal,D2: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ A @ B )
= ( plus_p3173629198307831117_preal @ C @ D2 ) )
=> ( ( ord_le5604041210740703414_preal @ C @ A )
=> ( ord_le5604041210740703414_preal @ B @ D2 ) ) ) ).
% preal_eq_le_imp_le
thf(fact_696_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_697_real__add__left__mono,axiom,
! [X3: dedekind_real,Y: dedekind_real,Z3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ Z3 @ X3 ) @ ( plus_p4060926892116697567d_real @ Z3 @ Y ) ) ) ).
% real_add_left_mono
thf(fact_698_real__trans__lemma,axiom,
! [X3: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V3: dedekind_preal,U3: dedekind_preal,X22: dedekind_real,V2: dedekind_real,U2: dedekind_real,Y22: dedekind_real] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X3 @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ V ) )
=> ( ( ( plus_p4060926892116697567d_real @ X22 @ V2 )
= ( plus_p4060926892116697567d_real @ U2 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X3 @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_699_real__trans__lemma,axiom,
! [X3: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V3: dedekind_preal,U3: dedekind_preal,X22: real,V2: real,U2: real,Y22: real] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X3 @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ V ) )
=> ( ( ( plus_plus_real @ X22 @ V2 )
= ( plus_plus_real @ U2 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X3 @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_700_real__trans__lemma,axiom,
! [X3: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V3: dedekind_preal,U3: dedekind_preal,X22: num,V2: num,U2: num,Y22: num] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X3 @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ V ) )
=> ( ( ( plus_plus_num @ X22 @ V2 )
= ( plus_plus_num @ U2 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X3 @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_701_real__trans__lemma,axiom,
! [X3: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V3: dedekind_preal,U3: dedekind_preal,X22: nat,V2: nat,U2: nat,Y22: nat] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X3 @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ V ) )
=> ( ( ( plus_plus_nat @ X22 @ V2 )
= ( plus_plus_nat @ U2 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X3 @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_702_real__trans__lemma,axiom,
! [X3: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V3: dedekind_preal,U3: dedekind_preal,X22: int,V2: int,U2: int,Y22: int] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X3 @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ V ) )
=> ( ( ( plus_plus_int @ X22 @ V2 )
= ( plus_plus_int @ U2 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X3 @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_703_real__of__preal__add,axiom,
! [X3: dedekind_preal,Y: dedekind_preal] :
( ( dedeki829075424511776921_preal @ ( plus_p3173629198307831117_preal @ X3 @ Y ) )
= ( plus_p4060926892116697567d_real @ ( dedeki829075424511776921_preal @ X3 ) @ ( dedeki829075424511776921_preal @ Y ) ) ) ).
% real_of_preal_add
thf(fact_704_add__right__imp__eq,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ B @ A )
= ( plus_p4060926892116697567d_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_705_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_706_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_707_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_708_add__left__imp__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ B )
= ( plus_p4060926892116697567d_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_709_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_710_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_711_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_712_add_Oleft__commute,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ B @ ( plus_p4060926892116697567d_real @ A @ C ) )
= ( plus_p4060926892116697567d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_713_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_714_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_715_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_716_add_Ocommute,axiom,
( plus_p4060926892116697567d_real
= ( ^ [A4: dedekind_real,B3: dedekind_real] : ( plus_p4060926892116697567d_real @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_717_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B3: real] : ( plus_plus_real @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_718_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_719_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_720_add_Oright__cancel,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ B @ A )
= ( plus_p4060926892116697567d_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_721_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_722_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_723_add_Oleft__cancel,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ B )
= ( plus_p4060926892116697567d_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_724_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_725_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_726_add_Oassoc,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_727_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_728_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_729_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_730_group__cancel_Oadd2,axiom,
! [B5: dedekind_real,K: dedekind_real,B: dedekind_real,A: dedekind_real] :
( ( B5
= ( plus_p4060926892116697567d_real @ K @ B ) )
=> ( ( plus_p4060926892116697567d_real @ A @ B5 )
= ( plus_p4060926892116697567d_real @ K @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_731_group__cancel_Oadd2,axiom,
! [B5: real,K: real,B: real,A: real] :
( ( B5
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B5 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_732_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_733_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_734_group__cancel_Oadd1,axiom,
! [A3: dedekind_real,K: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( A3
= ( plus_p4060926892116697567d_real @ K @ A ) )
=> ( ( plus_p4060926892116697567d_real @ A3 @ B )
= ( plus_p4060926892116697567d_real @ K @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_735_group__cancel_Oadd1,axiom,
! [A3: real,K: real,A: real,B: real] :
( ( A3
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A3 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_736_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_737_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_738_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_p4060926892116697567d_real @ I @ K )
= ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_739_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_740_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_741_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_742_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_743_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_744_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_745_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_746_add__le__imp__le__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ C ) )
=> ( ord_le2716243287969276982d_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_747_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_748_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_749_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_750_add__le__imp__le__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ C @ A ) @ ( plus_p4060926892116697567d_real @ C @ B ) )
=> ( ord_le2716243287969276982d_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_751_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_752_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_753_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_754_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_755_add__right__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_756_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_757_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_758_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_759_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_760_add__left__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ C @ A ) @ ( plus_p4060926892116697567d_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_761_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_762_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_763_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_764_add__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ C @ D2 )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_765_add__mono,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_766_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_767_add__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_768_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( ord_le2716243287969276982d_real @ I @ J )
& ( ord_le2716243287969276982d_real @ K @ L ) )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ I @ K ) @ ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_769_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_770_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_771_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_772_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( I = J )
& ( ord_le2716243287969276982d_real @ K @ L ) )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ I @ K ) @ ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_773_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_774_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_775_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_776_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( ord_le2716243287969276982d_real @ I @ J )
& ( K = L ) )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ I @ K ) @ ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_777_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_778_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_779_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_780_add__mono__thms__linordered__field_I5_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( ord_le2991122432403439658d_real @ I @ J )
& ( ord_le2991122432403439658d_real @ K @ L ) )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ I @ K ) @ ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_781_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_782_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_783_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_784_add__mono__thms__linordered__field_I2_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( I = J )
& ( ord_le2991122432403439658d_real @ K @ L ) )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ I @ K ) @ ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_785_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_786_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_787_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_788_add__mono__thms__linordered__field_I1_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( ord_le2991122432403439658d_real @ I @ J )
& ( K = L ) )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ I @ K ) @ ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_789_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_790_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_791_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_792_add__strict__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ C @ D2 )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_793_add__strict__mono,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D2 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_794_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_795_add__strict__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_796_add__strict__left__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ C @ A ) @ ( plus_p4060926892116697567d_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_797_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_798_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_799_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_800_add__strict__right__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_801_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_802_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_803_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_804_add__less__imp__less__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ C @ A ) @ ( plus_p4060926892116697567d_real @ C @ B ) )
=> ( ord_le2991122432403439658d_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_805_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_806_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_807_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_808_add__less__imp__less__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ C ) )
=> ( ord_le2991122432403439658d_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_809_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_810_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_811_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_812_GreatestI2__order,axiom,
! [P: dedekind_real > $o,X3: dedekind_real,Q: dedekind_real > $o] :
( ( P @ X3 )
=> ( ! [Y2: dedekind_real] :
( ( P @ Y2 )
=> ( ord_le2716243287969276982d_real @ Y2 @ X3 ) )
=> ( ! [X2: dedekind_real] :
( ( P @ X2 )
=> ( ! [Y3: dedekind_real] :
( ( P @ Y3 )
=> ( ord_le2716243287969276982d_real @ Y3 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_1105151008113787197d_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_813_GreatestI2__order,axiom,
! [P: real > $o,X3: real,Q: real > $o] :
( ( P @ X3 )
=> ( ! [Y2: real] :
( ( P @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X3 ) )
=> ( ! [X2: real] :
( ( P @ X2 )
=> ( ! [Y3: real] :
( ( P @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_814_GreatestI2__order,axiom,
! [P: num > $o,X3: num,Q: num > $o] :
( ( P @ X3 )
=> ( ! [Y2: num] :
( ( P @ Y2 )
=> ( ord_less_eq_num @ Y2 @ X3 ) )
=> ( ! [X2: num] :
( ( P @ X2 )
=> ( ! [Y3: num] :
( ( P @ Y3 )
=> ( ord_less_eq_num @ Y3 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_num @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_815_GreatestI2__order,axiom,
! [P: nat > $o,X3: nat,Q: nat > $o] :
( ( P @ X3 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_816_GreatestI2__order,axiom,
! [P: int > $o,X3: int,Q: int > $o] :
( ( P @ X3 )
=> ( ! [Y2: int] :
( ( P @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X3 ) )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( ! [Y3: int] :
( ( P @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_817_Greatest__equality,axiom,
! [P: dedekind_real > $o,X3: dedekind_real] :
( ( P @ X3 )
=> ( ! [Y2: dedekind_real] :
( ( P @ Y2 )
=> ( ord_le2716243287969276982d_real @ Y2 @ X3 ) )
=> ( ( order_1105151008113787197d_real @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_818_Greatest__equality,axiom,
! [P: real > $o,X3: real] :
( ( P @ X3 )
=> ( ! [Y2: real] :
( ( P @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X3 ) )
=> ( ( order_Greatest_real @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_819_Greatest__equality,axiom,
! [P: num > $o,X3: num] :
( ( P @ X3 )
=> ( ! [Y2: num] :
( ( P @ Y2 )
=> ( ord_less_eq_num @ Y2 @ X3 ) )
=> ( ( order_Greatest_num @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_820_Greatest__equality,axiom,
! [P: nat > $o,X3: nat] :
( ( P @ X3 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) )
=> ( ( order_Greatest_nat @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_821_Greatest__equality,axiom,
! [P: int > $o,X3: int] :
( ( P @ X3 )
=> ( ! [Y2: int] :
( ( P @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X3 ) )
=> ( ( order_Greatest_int @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_822_add__neg__nonpos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% add_neg_nonpos
thf(fact_823_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_824_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_825_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_826_add__nonneg__pos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_827_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_828_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_829_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_830_add__nonpos__neg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% add_nonpos_neg
thf(fact_831_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_832_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_833_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_834_add__pos__nonneg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_835_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_836_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_837_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_838_add__strict__increasing,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ B @ ( plus_p4060926892116697567d_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_839_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_840_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_841_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_842_add__strict__increasing2,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ B @ ( plus_p4060926892116697567d_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_843_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_844_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_845_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_846_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_847_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_848_add_Oright__neutral,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% add.right_neutral
thf(fact_849_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_850_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_851_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_852_double__zero__sym,axiom,
! [A: dedekind_real] :
( ( zero_z580800474297136991d_real
= ( plus_p4060926892116697567d_real @ A @ A ) )
= ( A = zero_z580800474297136991d_real ) ) ).
% double_zero_sym
thf(fact_853_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_854_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_855_add__cancel__left__left,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ B @ A )
= A )
= ( B = zero_z580800474297136991d_real ) ) ).
% add_cancel_left_left
thf(fact_856_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_857_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_858_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_859_add__cancel__left__right,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ B )
= A )
= ( B = zero_z580800474297136991d_real ) ) ).
% add_cancel_left_right
thf(fact_860_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_861_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_862_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_863_add__cancel__right__left,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A
= ( plus_p4060926892116697567d_real @ B @ A ) )
= ( B = zero_z580800474297136991d_real ) ) ).
% add_cancel_right_left
thf(fact_864_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_865_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_866_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_867_add__cancel__right__right,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A
= ( plus_p4060926892116697567d_real @ A @ B ) )
= ( B = zero_z580800474297136991d_real ) ) ).
% add_cancel_right_right
thf(fact_868_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_869_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_870_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_871_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_872_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y ) )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_873_add__0,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ zero_z580800474297136991d_real @ A )
= A ) ).
% add_0
thf(fact_874_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_875_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_876_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_877_add__le__same__cancel1,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ B @ A ) @ B )
= ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real ) ) ).
% add_le_same_cancel1
thf(fact_878_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_879_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_880_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_881_add__le__same__cancel2,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ B )
= ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real ) ) ).
% add_le_same_cancel2
thf(fact_882_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_883_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_884_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_885_le__add__same__cancel1,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ ( plus_p4060926892116697567d_real @ A @ B ) )
= ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_886_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_887_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_888_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_889_le__add__same__cancel2,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ ( plus_p4060926892116697567d_real @ B @ A ) )
= ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_890_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_891_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_892_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_893_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ A ) @ zero_z580800474297136991d_real )
= ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_894_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_895_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_896_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ A @ A ) )
= ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_897_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_898_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_899_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ A @ A ) )
= ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_900_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_901_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_902_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ A ) @ zero_z580800474297136991d_real )
= ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_903_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_904_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_905_less__add__same__cancel2,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ ( plus_p4060926892116697567d_real @ B @ A ) )
= ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_906_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_907_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_908_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_909_less__add__same__cancel1,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ ( plus_p4060926892116697567d_real @ A @ B ) )
= ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_910_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_911_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_912_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_913_add__less__same__cancel2,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ B )
= ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real ) ) ).
% add_less_same_cancel2
thf(fact_914_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_915_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_916_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_917_add__less__same__cancel1,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ B @ A ) @ B )
= ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real ) ) ).
% add_less_same_cancel1
thf(fact_918_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_919_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_920_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_921_zero__reorient,axiom,
! [X3: dedekind_real] :
( ( zero_z580800474297136991d_real = X3 )
= ( X3 = zero_z580800474297136991d_real ) ) ).
% zero_reorient
thf(fact_922_zero__reorient,axiom,
! [X3: int] :
( ( zero_zero_int = X3 )
= ( X3 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_923_zero__reorient,axiom,
! [X3: real] :
( ( zero_zero_real = X3 )
= ( X3 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_924_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_925_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_926_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_927_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_928_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_929_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_930_verit__sum__simplify,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% verit_sum_simplify
thf(fact_931_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_932_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_933_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_934_comm__monoid__add__class_Oadd__0,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ zero_z580800474297136991d_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_935_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_936_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_937_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_938_add_Ocomm__neutral,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% add.comm_neutral
thf(fact_939_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_940_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_941_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_942_add_Ogroup__left__neutral,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ zero_z580800474297136991d_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_943_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_944_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_945_add__decreasing,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ C @ B )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_946_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_947_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_948_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_949_add__increasing,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2716243287969276982d_real @ B @ ( plus_p4060926892116697567d_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_950_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_951_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_952_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_953_add__decreasing2,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_954_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_955_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_956_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_957_add__increasing2,axiom,
! [C: dedekind_real,B: dedekind_real,A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ( ord_le2716243287969276982d_real @ B @ A )
=> ( ord_le2716243287969276982d_real @ B @ ( plus_p4060926892116697567d_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_958_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_959_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_960_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_961_add__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 @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_962_add__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 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_963_add__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 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_964_add__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 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_965_add__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 @ ( plus_p4060926892116697567d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_966_add__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 @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_967_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_968_add__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 @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_969_add__nonneg__eq__0__iff,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ X3 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ Y )
=> ( ( ( plus_p4060926892116697567d_real @ X3 @ Y )
= zero_z580800474297136991d_real )
= ( ( X3 = zero_z580800474297136991d_real )
& ( Y = zero_z580800474297136991d_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_970_add__nonneg__eq__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X3 @ Y )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_971_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_972_add__nonneg__eq__0__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_973_add__nonpos__eq__0__iff,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ Y @ zero_z580800474297136991d_real )
=> ( ( ( plus_p4060926892116697567d_real @ X3 @ Y )
= zero_z580800474297136991d_real )
= ( ( X3 = zero_z580800474297136991d_real )
& ( Y = zero_z580800474297136991d_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_974_add__nonpos__eq__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X3 @ Y )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_975_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_976_add__nonpos__eq__0__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_977_add__less__zeroD,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ X3 @ Y ) @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ X3 @ zero_z580800474297136991d_real )
| ( ord_le2991122432403439658d_real @ Y @ zero_z580800474297136991d_real ) ) ) ).
% add_less_zeroD
thf(fact_978_add__less__zeroD,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X3 @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X3 @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_979_add__less__zeroD,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X3 @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X3 @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_980_pos__add__strict,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ B @ ( plus_p4060926892116697567d_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_981_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_982_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_983_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_984_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_985_add__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 @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_986_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_987_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_988_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_989_add__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 @ ( plus_p4060926892116697567d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% add_neg_neg
thf(fact_990_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_991_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_992_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_993_field__le__epsilon,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ! [E: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ E )
=> ( ord_le2716243287969276982d_real @ X3 @ ( plus_p4060926892116697567d_real @ Y @ E ) ) )
=> ( ord_le2716243287969276982d_real @ X3 @ Y ) ) ).
% field_le_epsilon
thf(fact_994_field__le__epsilon,axiom,
! [X3: real,Y: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ X3 @ ( plus_plus_real @ Y @ E ) ) )
=> ( ord_less_eq_real @ X3 @ Y ) ) ).
% field_le_epsilon
thf(fact_995_double__eq__0__iff,axiom,
! [A: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ A )
= zero_z580800474297136991d_real )
= ( A = zero_z580800474297136991d_real ) ) ).
% double_eq_0_iff
thf(fact_996_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_997_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_998_add__0__iff,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( B
= ( plus_p4060926892116697567d_real @ B @ A ) )
= ( A = zero_z580800474297136991d_real ) ) ).
% add_0_iff
thf(fact_999_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_1000_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_1001_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1002_less__numeral__extra_I3_J,axiom,
~ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ zero_z580800474297136991d_real ) ).
% less_numeral_extra(3)
thf(fact_1003_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_1004_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1005_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_1006_field__lbound__gt__zero,axiom,
! [D1: dedekind_real,D22: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ D1 )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ D22 )
=> ? [E: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ E )
& ( ord_le2991122432403439658d_real @ E @ D1 )
& ( ord_le2991122432403439658d_real @ E @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1007_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1008_le__numeral__extra_I3_J,axiom,
ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ zero_z580800474297136991d_real ).
% le_numeral_extra(3)
thf(fact_1009_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_1010_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1011_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_1012_preal__psup__le,axiom,
! [P: set_Dedekind_preal,Y6: dedekind_preal,X3: dedekind_preal] :
( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ P )
=> ( ord_le5604041210740703414_preal @ X6 @ Y6 ) )
=> ( ( member6871284927547481791_preal @ X3 @ P )
=> ( ord_le5604041210740703414_preal @ X3 @ ( dedekind_psup @ P ) ) ) ) ).
% preal_psup_le
thf(fact_1013_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_1014_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_1015_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_1016_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_1017_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_1018_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_1019_inverse__inverse__eq,axiom,
! [A: dedekind_real] :
( ( invers3762989301784728874d_real @ ( invers3762989301784728874d_real @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_1020_inverse__inverse__eq,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_1021_inverse__zero,axiom,
( ( invers3762989301784728874d_real @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% inverse_zero
thf(fact_1022_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_1023_inverse__nonzero__iff__nonzero,axiom,
! [A: dedekind_real] :
( ( ( invers3762989301784728874d_real @ A )
= zero_z580800474297136991d_real )
= ( A = zero_z580800474297136991d_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_1024_inverse__nonzero__iff__nonzero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_1025_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_1026_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_1027_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_1028_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_1029_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_1030_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_1031_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_1032_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_1033_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_1034_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_1035_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_1036_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_1037_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_1038_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_1039_field__class_Ofield__inverse__zero,axiom,
( ( invers3762989301784728874d_real @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% field_class.field_inverse_zero
thf(fact_1040_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_1041_inverse__zero__imp__zero,axiom,
! [A: dedekind_real] :
( ( ( invers3762989301784728874d_real @ A )
= zero_z580800474297136991d_real )
=> ( A = zero_z580800474297136991d_real ) ) ).
% inverse_zero_imp_zero
thf(fact_1042_inverse__zero__imp__zero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_1043_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_1044_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_1045_nonzero__inverse__inverse__eq,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( invers3762989301784728874d_real @ ( invers3762989301784728874d_real @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_1046_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_1047_nonzero__imp__inverse__nonzero,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( invers3762989301784728874d_real @ A )
!= zero_z580800474297136991d_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_1048_nonzero__imp__inverse__nonzero,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_1049_preal__complete,axiom,
! [P: set_Dedekind_preal,Y6: dedekind_preal,Z5: dedekind_preal] :
( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ P )
=> ( ord_le5604041210740703414_preal @ X6 @ Y6 ) )
=> ( ( P != bot_bo4848840305443107682_preal )
=> ( ( ? [X5: dedekind_preal] :
( ( member6871284927547481791_preal @ X5 @ P )
& ( ord_le5708704896291381698_preal @ Z5 @ X5 ) ) )
= ( ord_le5708704896291381698_preal @ Z5 @ ( dedekind_psup @ P ) ) ) ) ) ).
% preal_complete
thf(fact_1050_psup__le__ub,axiom,
! [P: set_Dedekind_preal,Y6: dedekind_preal] :
( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ P )
=> ( ord_le5604041210740703414_preal @ X6 @ Y6 ) )
=> ( ( P != bot_bo4848840305443107682_preal )
=> ( ord_le5604041210740703414_preal @ ( dedekind_psup @ P ) @ Y6 ) ) ) ).
% psup_le_ub
thf(fact_1051_real__of__preal__mult,axiom,
! [X3: dedekind_preal,Y: dedekind_preal] :
( ( dedeki829075424511776921_preal @ ( times_3000655703912201937_preal @ X3 @ Y ) )
= ( times_2157731159493324635d_real @ ( dedeki829075424511776921_preal @ X3 ) @ ( dedeki829075424511776921_preal @ Y ) ) ) ).
% real_of_preal_mult
thf(fact_1052_preal__add__mult__distrib,axiom,
! [X3: dedekind_preal,Y: dedekind_preal,W: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ X3 @ Y ) @ W )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X3 @ W ) @ ( times_3000655703912201937_preal @ Y @ W ) ) ) ).
% preal_add_mult_distrib
thf(fact_1053_preal__add__mult__distrib2,axiom,
! [W: dedekind_preal,X3: dedekind_preal,Y: dedekind_preal] :
( ( times_3000655703912201937_preal @ W @ ( plus_p3173629198307831117_preal @ X3 @ Y ) )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ W @ X3 ) @ ( times_3000655703912201937_preal @ W @ Y ) ) ) ).
% preal_add_mult_distrib2
thf(fact_1054_real__mult__congruent2__lemma,axiom,
! [X1: dedekind_preal,Y22: dedekind_preal,X22: dedekind_preal,Y1: dedekind_preal,X3: dedekind_preal,Y: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ X1 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y1 ) )
=> ( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X3 @ X1 ) @ ( times_3000655703912201937_preal @ Y @ Y1 ) ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X3 @ Y22 ) @ ( times_3000655703912201937_preal @ Y @ X22 ) ) )
= ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X3 @ X22 ) @ ( times_3000655703912201937_preal @ Y @ Y22 ) ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X3 @ Y1 ) @ ( times_3000655703912201937_preal @ Y @ X1 ) ) ) ) ) ).
% real_mult_congruent2_lemma
thf(fact_1055_real__mult__less__mono2,axiom,
! [Z3: dedekind_real,X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Z3 )
=> ( ( ord_le2991122432403439658d_real @ X3 @ Y )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ Z3 @ X3 ) @ ( times_2157731159493324635d_real @ Z3 @ Y ) ) ) ) ).
% real_mult_less_mono2
thf(fact_1056_real__mult__order,axiom,
! [X3: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ X3 )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Y )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ X3 @ Y ) ) ) ) ).
% real_mult_order
thf(fact_1057_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_1058_real__mult__inverse__left,axiom,
! [X3: dedekind_real] :
( ( X3 != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ X3 ) @ X3 )
= one_on6069100329679821595d_real ) ) ).
% real_mult_inverse_left
thf(fact_1059_preal__mult__inverse__right,axiom,
! [R: dedekind_preal] :
( ( times_3000655703912201937_preal @ R @ ( invers3090987106763476162_preal @ R ) )
= one_on9143529541772854033_preal ) ).
% preal_mult_inverse_right
thf(fact_1060_preal__mult__inverse,axiom,
! [R: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( invers3090987106763476162_preal @ R ) @ R )
= one_on9143529541772854033_preal ) ).
% preal_mult_inverse
thf(fact_1061_real__mult__1,axiom,
! [Z3: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ Z3 )
= Z3 ) ).
% real_mult_1
thf(fact_1062_preal__mult__1,axiom,
! [Z3: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ Z3 )
= Z3 ) ).
% preal_mult_1
thf(fact_1063_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_1064_preal__mult__assoc,axiom,
! [X3: dedekind_preal,Y: dedekind_preal,Z3: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ X3 @ Y ) @ Z3 )
= ( times_3000655703912201937_preal @ X3 @ ( times_3000655703912201937_preal @ Y @ Z3 ) ) ) ).
% preal_mult_assoc
thf(fact_1065_real__mult__commute,axiom,
( times_2157731159493324635d_real
= ( ^ [Z6: dedekind_real,W3: dedekind_real] : ( times_2157731159493324635d_real @ W3 @ Z6 ) ) ) ).
% real_mult_commute
thf(fact_1066_preal__mult__commute,axiom,
( times_3000655703912201937_preal
= ( ^ [X5: dedekind_preal,Y4: dedekind_preal] : ( times_3000655703912201937_preal @ Y4 @ X5 ) ) ) ).
% preal_mult_commute
thf(fact_1067_real__mult__inverse__left__ex,axiom,
! [X3: dedekind_real] :
( ( X3 != zero_z580800474297136991d_real )
=> ~ ! [Y2: dedekind_real] :
( ( times_2157731159493324635d_real @ Y2 @ X3 )
!= one_on6069100329679821595d_real ) ) ).
% real_mult_inverse_left_ex
thf(fact_1068_real__zero__not__eq__one,axiom,
zero_z580800474297136991d_real != one_on6069100329679821595d_real ).
% real_zero_not_eq_one
thf(fact_1069_divide__preal__def,axiom,
( divide4190755330972744004_preal
= ( ^ [R2: dedekind_preal,S2: dedekind_preal] : ( times_3000655703912201937_preal @ R2 @ ( invers3090987106763476162_preal @ S2 ) ) ) ) ).
% divide_preal_def
thf(fact_1070_real__divide__def,axiom,
( divide9119111104558704680d_real
= ( ^ [R2: dedekind_real,S2: dedekind_real] : ( times_2157731159493324635d_real @ R2 @ ( invers3762989301784728874d_real @ S2 ) ) ) ) ).
% real_divide_def
thf(fact_1071_real__of__preal__trichotomy,axiom,
! [X3: dedekind_real] :
? [M3: dedekind_preal] :
( ( X3
= ( dedeki829075424511776921_preal @ M3 ) )
| ( X3 = zero_z580800474297136991d_real )
| ( X3
= ( uminus7714077491378687647d_real @ ( dedeki829075424511776921_preal @ M3 ) ) ) ) ).
% real_of_preal_trichotomy
thf(fact_1072_real__sum__gt__zero__less,axiom,
! [S: dedekind_real,W: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ S @ ( uminus7714077491378687647d_real @ W ) ) )
=> ( ord_le2991122432403439658d_real @ W @ S ) ) ).
% real_sum_gt_zero_less
thf(fact_1073_real__less__sum__gt__zero,axiom,
! [W: dedekind_real,S: dedekind_real] :
( ( ord_le2991122432403439658d_real @ W @ S )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ S @ ( uminus7714077491378687647d_real @ W ) ) ) ) ).
% real_less_sum_gt_zero
thf(fact_1074_add__inc,axiom,
! [X3: num,Y: num] :
( ( plus_plus_num @ X3 @ ( inc @ Y ) )
= ( inc @ ( plus_plus_num @ X3 @ Y ) ) ) ).
% add_inc
thf(fact_1075_mult__inc,axiom,
! [X3: num,Y: num] :
( ( times_times_num @ X3 @ ( inc @ Y ) )
= ( plus_plus_num @ ( times_times_num @ X3 @ Y ) @ X3 ) ) ).
% mult_inc
thf(fact_1076_semiring__norm_I75_J,axiom,
! [M2: num] :
~ ( ord_less_num @ M2 @ one ) ).
% semiring_norm(75)
thf(fact_1077_semiring__norm_I68_J,axiom,
! [N2: num] : ( ord_less_eq_num @ one @ N2 ) ).
% semiring_norm(68)
thf(fact_1078_add__One__commute,axiom,
! [N2: num] :
( ( plus_plus_num @ one @ N2 )
= ( plus_plus_num @ N2 @ one ) ) ).
% add_One_commute
thf(fact_1079_le__num__One__iff,axiom,
! [X3: num] :
( ( ord_less_eq_num @ X3 @ one )
= ( X3 = one ) ) ).
% le_num_One_iff
thf(fact_1080_add__One,axiom,
! [X3: num] :
( ( plus_plus_num @ X3 @ one )
= ( inc @ X3 ) ) ).
% add_One
thf(fact_1081_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_1082_add__self__div__2,axiom,
! [M2: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M2 ) ).
% add_self_div_2
thf(fact_1083_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_1084_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_1085_semiring__norm_I6_J,axiom,
! [M2: num,N2: num] :
( ( plus_plus_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) )
= ( bit0 @ ( plus_plus_num @ M2 @ N2 ) ) ) ).
% semiring_norm(6)
thf(fact_1086_semiring__norm_I71_J,axiom,
! [M2: num,N2: num] :
( ( ord_less_eq_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) )
= ( ord_less_eq_num @ M2 @ N2 ) ) ).
% semiring_norm(71)
thf(fact_1087_semiring__norm_I78_J,axiom,
! [M2: num,N2: num] :
( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) )
= ( ord_less_num @ M2 @ N2 ) ) ).
% semiring_norm(78)
thf(fact_1088_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_1089_semiring__norm_I69_J,axiom,
! [M2: num] :
~ ( ord_less_eq_num @ ( bit0 @ M2 ) @ one ) ).
% semiring_norm(69)
thf(fact_1090_semiring__norm_I76_J,axiom,
! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).
% semiring_norm(76)
thf(fact_1091_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_1092_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2 != zero_zero_int )
=> ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_1093_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_1094_plus__inverse__ge__2,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_1095_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_1096_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_1097_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1098_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1099_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1100_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_1101_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1102_nat__0__less__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1103_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1104_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1105_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1106_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1107_mult__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1108_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1109_mult__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1110_mult__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1111_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1112_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1113_nat__1__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1114_nat__mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1115_ln__inj__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X3 )
= ( ln_ln_real @ Y ) )
= ( X3 = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_1116_ln__less__cancel__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X3 @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_1117_ln__less__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
= ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_1118_ln__gt__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
= ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% ln_gt_zero_iff
thf(fact_1119_ln__eq__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ( ln_ln_real @ X3 )
= zero_zero_real )
= ( X3 = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_1120_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1121_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1122_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1123_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1124_set__bit__nonnegative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_1125_real__add__minus__iff,axiom,
! [X3: real,A: real] :
( ( ( plus_plus_real @ X3 @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X3 = A ) ) ).
% real_add_minus_iff
thf(fact_1126_set__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% set_bit_negative_int_iff
thf(fact_1127_div__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1128_div__mult__self1__is__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M2 ) @ N2 )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1129_div__mult__self__is__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N2 ) @ N2 )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1130_ln__le__cancel__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_1131_ln__le__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
= ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_1132_ln__ge__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
= ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% ln_ge_zero_iff
thf(fact_1133_zle__add1__eq__le,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_1134_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_1135_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_1136_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1137_ln__add__one__self__le__self,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) ).
% ln_add_one_self_le_self
thf(fact_1138_ln__ge__zero,axiom,
! [X3: real] :
( ( ord_less_eq_real @ one_one_real @ X3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).
% ln_ge_zero
thf(fact_1139_ln__less__self,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).
% ln_less_self
thf(fact_1140_ln__bound,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).
% ln_bound
thf(fact_1141_ln__ge__zero__imp__ge__one,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_1142_ln__gt__zero__imp__gt__one,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_1143_ln__less__zero,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_1144_ln__gt__zero,axiom,
! [X3: real] :
( ( ord_less_real @ one_one_real @ X3 )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).
% ln_gt_zero
thf(fact_1145_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1146_odd__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 ) @ zero_zero_int )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1147_le__imp__0__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z3 ) ) ) ).
% le_imp_0_less
thf(fact_1148_div__less__dividend,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1149_int__div__less__self,axiom,
! [X3: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X3 )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X3 @ K ) @ X3 ) ) ) ).
% int_div_less_self
thf(fact_1150_div__eq__dividend__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N2 )
= M2 )
= ( N2 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1151_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_1152_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_1153_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_1154_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_1155_div__greater__zero__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ N2 @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_1156_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_1157_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_1158_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_1159_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_1160_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_1161_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_1162_div__le__mono2,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1163_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_1164_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_1165_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ~ ( P @ I2 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1166_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_1167_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_1168_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_1169_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N2 )
| ( N2 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1170_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X5: real,Y4: real] :
( ( ord_less_real @ X5 @ Y4 )
| ( X5 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_1171_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1172_div__le__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ).
% div_le_dividend
thf(fact_1173_div__le__mono,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).
% div_le_mono
thf(fact_1174_add1__zle__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z3 )
= ( ord_less_int @ W @ Z3 ) ) ).
% add1_zle_eq
thf(fact_1175_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_1176_zless__add1__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z3 )
| ( W = Z3 ) ) ) ).
% zless_add1_eq
thf(fact_1177_zless__imp__add1__zle,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ Z3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z3 ) ) ).
% zless_imp_add1_zle
thf(fact_1178_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1179_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1180_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1181_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_1182_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1183_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1184_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_1185_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_1186_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1187_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_1188_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1189_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1190_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_1191_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1192_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_1193_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1194_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1195_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1196_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1197_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1198_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1199_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_1200_add__leE,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1201_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_1202_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_1203_add__leD1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_1204_add__leD2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1205_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_1206_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_1207_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_1208_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1209_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1210_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
? [K3: nat] :
( N
= ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1211_verit__la__generic,axiom,
! [A: int,X3: int] :
( ( ord_less_eq_int @ A @ X3 )
| ( A = X3 )
| ( ord_less_eq_int @ X3 @ A ) ) ).
% verit_la_generic
thf(fact_1212_complete__real,axiom,
! [S3: set_real] :
( ? [X: real] : ( member_real @ X @ S3 )
=> ( ? [Z2: real] :
! [X2: real] :
( ( member_real @ X2 @ S3 )
=> ( ord_less_eq_real @ X2 @ Z2 ) )
=> ? [Y2: real] :
( ! [X: real] :
( ( member_real @ X @ S3 )
=> ( ord_less_eq_real @ X @ Y2 ) )
& ! [Z2: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S3 )
=> ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ord_less_eq_real @ Y2 @ Z2 ) ) ) ) ) ).
% complete_real
thf(fact_1213_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1214_imp__le__cong,axiom,
! [X3: int,X7: int,P: $o,P2: $o] :
( ( X3 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_1215_conj__le__cong,axiom,
! [X3: int,X7: int,P: $o,P2: $o] :
( ( X3 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_1216_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1217_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1218_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1219_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1220_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1221_set__bit__greater__eq,axiom,
! [K: int,N2: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N2 @ K ) ) ).
% set_bit_greater_eq
thf(fact_1222_real__add__le__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X3 @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X3 ) ) ) ).
% real_add_le_0_iff
thf(fact_1223_real__0__le__add__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X3 ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1224_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1225_real__0__less__add__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X3 ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1226_real__add__less__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X3 @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X3 ) ) ) ).
% real_add_less_0_iff
thf(fact_1227_verit__less__mono__div__int2,axiom,
! [A3: int,B5: int,N2: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N2 ) @ ( divide_divide_int @ A3 @ N2 ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1228_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_1229_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_1230_ln__inverse,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ln_ln_real @ ( inverse_inverse_real @ X3 ) )
= ( uminus_uminus_real @ ( ln_ln_real @ X3 ) ) ) ) ).
% ln_inverse
thf(fact_1231_ln__add__one__self__le__self2,axiom,
! [X3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) ).
% ln_add_one_self_le_self2
thf(fact_1232_real__minus__mult__self__le,axiom,
! [U: real,X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X3 @ X3 ) ) ).
% real_minus_mult_self_le
thf(fact_1233_add__mult__distrib,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1234_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1235_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_1236_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1237_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_1238_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_1239_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1240_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1241_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1242_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1243_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1244_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_1245_div__times__less__eq__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_1246_times__div__less__eq__dividend,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M2 @ N2 ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_1247_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N2 @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q3 ) @ N2 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1248_dividend__less__times__div,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M2 @ N2 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1249_dividend__less__div__times,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% dividend_less_div_times
thf(fact_1250_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1251_int__div__pos__eq,axiom,
! [A: int,B: int,Q3: int,R: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ R @ B )
=> ( ( divide_divide_int @ A @ B )
= Q3 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1252_int__div__neg__eq,axiom,
! [A: int,B: int,Q3: int,R: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R ) )
=> ( ( ord_less_eq_int @ R @ zero_zero_int )
=> ( ( ord_less_int @ B @ R )
=> ( ( divide_divide_int @ A @ B )
= Q3 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1253_split__zdiv,axiom,
! [P: int > $o,N2: int,K: int] :
( ( P @ ( divide_divide_int @ N2 @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I4: int,J2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
& ( ord_less_int @ J2 @ K )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J2 ) ) )
=> ( P @ I4 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I4: int,J2: int] :
( ( ( ord_less_int @ K @ J2 )
& ( ord_less_eq_int @ J2 @ zero_zero_int )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J2 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_zdiv
thf(fact_1254_split__div,axiom,
! [P: nat > $o,M2: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N2 != zero_zero_nat )
=> ! [I4: nat,J2: nat] :
( ( ( ord_less_nat @ J2 @ N2 )
& ( M2
= ( plus_plus_nat @ ( times_times_nat @ N2 @ I4 ) @ J2 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_div
thf(fact_1255_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X: int] :
( ( P @ X )
=> ( P @ ( plus_plus_int @ X @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1256_nat__mult__div__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1257_less__mult__imp__div__less,axiom,
! [M2: nat,I: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I @ N2 ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1258_div__less__iff__less__mult,axiom,
! [Q3: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q3 ) @ N2 )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N2 @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1259_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1260_pos__zmult__eq__1__iff,axiom,
! [M2: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N2 )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N2 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1261_ln__mult,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( times_times_real @ X3 @ Y ) )
= ( plus_plus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_1262_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1263_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1264_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1265_mult__eq__self__implies__10,axiom,
! [M2: nat,N2: nat] :
( ( M2
= ( times_times_nat @ M2 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1266_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1267_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1268_nat__induct2,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct2
thf(fact_1269_not__real__square__gt__zero,axiom,
! [X3: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X3 @ X3 ) ) )
= ( X3 = zero_zero_real ) ) ).
% not_real_square_gt_zero
% Conjectures (1)
thf(conj_0,conjecture,
( ( ord_le2991122432403439658d_real @ ( dedeki829075424511776921_preal @ m1 ) @ ( dedeki829075424511776921_preal @ m2 ) )
= ( ord_le5708704896291381698_preal @ m1 @ m2 ) ) ).
%------------------------------------------------------------------------------