TPTP Problem File: SLH0078^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Dedekind_Real/0000_Dedekind_Real/prob_01418_046305__5763236_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1338 ( 543 unt; 71 typ; 0 def)
% Number of atoms : 3828 (1126 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10475 ( 365 ~; 142 |; 176 &;8137 @)
% ( 0 <=>;1655 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 311 ( 311 >; 0 *; 0 +; 0 <<)
% Number of symbols : 64 ( 63 usr; 13 con; 0-2 aty)
% Number of variables : 3459 ( 188 ^;3148 !; 123 ?;3459 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:30:28.228
%------------------------------------------------------------------------------
% 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__Dedekind____Real__Oreal_J,type,
set_Dedekind_real: $tType ).
thf(ty_n_t__Dedekind____Real__Opreal,type,
dedekind_preal: $tType ).
thf(ty_n_t__Dedekind____Real__Oreal,type,
dedekind_real: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (63)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
bit_se4203085406695923979it_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_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_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_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_Oneg__numeral__class_Odbl_001t__Dedekind____Real__Oreal,type,
neg_nu8665065731627423525d_real: dedekind_real > dedekind_real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
neg_numeral_dbl_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Dedekind____Real__Oreal,type,
neg_nu2849763295382964967d_real: dedekind_real > dedekind_real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Dedekind____Real__Opreal,type,
ord_le5708704896291381698_preal: dedekind_preal > dedekind_preal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Dedekind____Real__Oreal,type,
ord_le2991122432403439658d_real: dedekind_real > dedekind_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Dedekind____Real__Opreal,type,
ord_le5604041210740703414_preal: dedekind_preal > dedekind_preal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Dedekind____Real__Oreal,type,
ord_le2716243287969276982d_real: dedekind_real > dedekind_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Dedekind____Real__Oreal,type,
divide9119111104558704680d_real: dedekind_real > dedekind_real > dedekind_real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_001t__Dedekind____Real__Opreal,type,
collec1132657498972982273_preal: ( dedekind_preal > $o ) > set_Dedekind_preal ).
thf(sy_c_Set_OCollect_001t__Dedekind____Real__Oreal,type,
collec1873404778653292075d_real: ( dedekind_real > $o ) > set_Dedekind_real ).
thf(sy_c_member_001t__Dedekind____Real__Opreal,type,
member6871284927547481791_preal: dedekind_preal > set_Dedekind_preal > $o ).
thf(sy_c_member_001t__Dedekind____Real__Oreal,type,
member_Dedekind_real: dedekind_real > set_Dedekind_real > $o ).
thf(sy_v_P,type,
p: set_Dedekind_real ).
thf(sy_v_x____,type,
x: dedekind_real ).
thf(sy_v_y____,type,
y: dedekind_real ).
% Relevant facts (1266)
thf(fact_0_x__in__P,axiom,
member_Dedekind_real @ x @ p ).
% x_in_P
thf(fact_1_not__empty__P,axiom,
? [X: dedekind_real] : ( member_Dedekind_real @ X @ p ) ).
% not_empty_P
thf(fact_2_positive__P,axiom,
! [X2: dedekind_real] :
( ( member_Dedekind_real @ X2 @ p )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ X2 ) ) ).
% positive_P
thf(fact_3_neg__y,axiom,
~ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ y ) ).
% neg_y
thf(fact_4_upper__bound__Ex,axiom,
? [Y: dedekind_real] :
! [X2: dedekind_real] :
( ( member_Dedekind_real @ X2 @ p )
=> ( ord_le2991122432403439658d_real @ X2 @ Y ) ) ).
% upper_bound_Ex
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_Ax_A_092_060in_062_AP_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X: dedekind_real] :
~ ( member_Dedekind_real @ X @ p ) ).
% \<open>\<And>thesis. (\<And>x. x \<in> P \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6_minf_I7_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z )
=> ~ ( ord_le2991122432403439658d_real @ T @ X2 ) ) ).
% minf(7)
thf(fact_7_minf_I7_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z )
=> ~ ( ord_le5708704896291381698_preal @ T @ X2 ) ) ).
% minf(7)
thf(fact_8_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ~ ( ord_less_nat @ T @ X2 ) ) ).
% minf(7)
thf(fact_9_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ~ ( ord_less_int @ T @ X2 ) ) ).
% minf(7)
thf(fact_10_minf_I5_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z )
=> ( ord_le2991122432403439658d_real @ X2 @ T ) ) ).
% minf(5)
thf(fact_11_minf_I5_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z )
=> ( ord_le5708704896291381698_preal @ X2 @ T ) ) ).
% minf(5)
thf(fact_12_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ord_less_nat @ X2 @ T ) ) ).
% minf(5)
thf(fact_13_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ord_less_int @ X2 @ T ) ) ).
% minf(5)
thf(fact_14_minf_I4_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_15_minf_I4_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_16_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_17_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_18_minf_I3_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_19_minf_I3_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_20_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_21_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_22_minf_I2_J,axiom,
! [P: dedekind_real > $o,P2: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z2: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_23_minf_I2_J,axiom,
! [P: dedekind_preal > $o,P2: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z2: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_24_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_25_minf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X: int] :
( ( ord_less_int @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: int] :
! [X: int] :
( ( ord_less_int @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_26_minf_I1_J,axiom,
! [P: dedekind_real > $o,P2: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z2: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_27_minf_I1_J,axiom,
! [P: dedekind_preal > $o,P2: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z2: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_28_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_29_minf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X: int] :
( ( ord_less_int @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: int] :
! [X: int] :
( ( ord_less_int @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_30_pinf_I7_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X2 )
=> ( ord_le2991122432403439658d_real @ T @ X2 ) ) ).
% pinf(7)
thf(fact_31_pinf_I7_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z @ X2 )
=> ( ord_le5708704896291381698_preal @ T @ X2 ) ) ).
% pinf(7)
thf(fact_32_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ord_less_nat @ T @ X2 ) ) ).
% pinf(7)
thf(fact_33_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ord_less_int @ T @ X2 ) ) ).
% pinf(7)
thf(fact_34_pinf_I5_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X2 )
=> ~ ( ord_le2991122432403439658d_real @ X2 @ T ) ) ).
% pinf(5)
thf(fact_35_pinf_I5_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z @ X2 )
=> ~ ( ord_le5708704896291381698_preal @ X2 @ T ) ) ).
% pinf(5)
thf(fact_36_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ~ ( ord_less_nat @ X2 @ T ) ) ).
% pinf(5)
thf(fact_37_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ~ ( ord_less_int @ X2 @ T ) ) ).
% pinf(5)
thf(fact_38_pinf_I4_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_39_pinf_I4_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_40_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_41_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_42_pinf_I3_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_43_pinf_I3_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_44_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_45_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_46_pinf_I1_J,axiom,
! [P: dedekind_real > $o,P2: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z2: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_47_pinf_I1_J,axiom,
! [P: dedekind_preal > $o,P2: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z2: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_48_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_49_pinf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X: int] :
( ( ord_less_int @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: int] :
! [X: int] :
( ( ord_less_int @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_50_pinf_I2_J,axiom,
! [P: dedekind_real > $o,P2: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z2: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_51_pinf_I2_J,axiom,
! [P: dedekind_preal > $o,P2: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z2: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_52_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_53_pinf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X: int] :
( ( ord_less_int @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: int] :
! [X: int] :
( ( ord_less_int @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_54_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_55_field__lbound__gt__zero,axiom,
! [D1: dedekind_real,D2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ D1 )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ D2 )
=> ? [E: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ E )
& ( ord_le2991122432403439658d_real @ E @ D1 )
& ( ord_le2991122432403439658d_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_56_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_57_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_58_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_59_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_60_less__numeral__extra_I3_J,axiom,
~ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ zero_z580800474297136991d_real ) ).
% less_numeral_extra(3)
thf(fact_61_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_62_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_63_real__of__preal__zero__less,axiom,
! [M: dedekind_preal] : ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( dedeki829075424511776921_preal @ M ) ) ).
% real_of_preal_zero_less
thf(fact_64_real__gt__zero__preal__Ex,axiom,
! [X3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ X3 )
= ( ? [Y2: dedekind_preal] :
( X3
= ( dedeki829075424511776921_preal @ Y2 ) ) ) ) ).
% real_gt_zero_preal_Ex
thf(fact_65_order__less__imp__not__less,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ~ ( ord_le2991122432403439658d_real @ Y3 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_66_order__less__imp__not__less,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y3 )
=> ~ ( ord_le5708704896291381698_preal @ Y3 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_67_order__less__imp__not__less,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_68_order__less__imp__not__less,axiom,
! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ~ ( ord_less_int @ Y3 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_69_order__less__imp__not__eq2,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ( Y3 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_70_order__less__imp__not__eq2,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y3 )
=> ( Y3 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_71_order__less__imp__not__eq2,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( Y3 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_72_order__less__imp__not__eq2,axiom,
! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( Y3 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_73_not__less__iff__gr__or__eq,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ~ ( ord_le2991122432403439658d_real @ X3 @ Y3 ) )
= ( ( ord_le2991122432403439658d_real @ Y3 @ X3 )
| ( X3 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_74_not__less__iff__gr__or__eq,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ X3 @ Y3 ) )
= ( ( ord_le5708704896291381698_preal @ Y3 @ X3 )
| ( X3 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_75_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X3 )
| ( X3 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_76_not__less__iff__gr__or__eq,axiom,
! [X3: int,Y3: int] :
( ( ~ ( ord_less_int @ X3 @ Y3 ) )
= ( ( ord_less_int @ Y3 @ X3 )
| ( X3 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_77_real__of__preal__less__iff,axiom,
! [M1: dedekind_preal,M2: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ ( dedeki829075424511776921_preal @ M1 ) @ ( dedeki829075424511776921_preal @ M2 ) )
= ( ord_le5708704896291381698_preal @ M1 @ M2 ) ) ).
% real_of_preal_less_iff
thf(fact_78_zero__reorient,axiom,
! [X3: dedekind_real] :
( ( zero_z580800474297136991d_real = X3 )
= ( X3 = zero_z580800474297136991d_real ) ) ).
% zero_reorient
thf(fact_79_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_80_zero__reorient,axiom,
! [X3: int] :
( ( zero_zero_int = X3 )
= ( X3 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_81_lt__ex,axiom,
! [X3: dedekind_real] :
? [Y: dedekind_real] : ( ord_le2991122432403439658d_real @ Y @ X3 ) ).
% lt_ex
thf(fact_82_lt__ex,axiom,
! [X3: int] :
? [Y: int] : ( ord_less_int @ Y @ X3 ) ).
% lt_ex
thf(fact_83_gt__ex,axiom,
! [X3: dedekind_real] :
? [X_1: dedekind_real] : ( ord_le2991122432403439658d_real @ X3 @ X_1 ) ).
% gt_ex
thf(fact_84_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_85_gt__ex,axiom,
! [X3: int] :
? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% gt_ex
thf(fact_86_dense,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ? [Z: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Z )
& ( ord_le2991122432403439658d_real @ Z @ Y3 ) ) ) ).
% dense
thf(fact_87_less__imp__neq,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ).
% less_imp_neq
thf(fact_88_less__imp__neq,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ).
% less_imp_neq
thf(fact_89_less__imp__neq,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ).
% less_imp_neq
thf(fact_90_less__imp__neq,axiom,
! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ).
% less_imp_neq
thf(fact_91_order_Oasym,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ~ ( ord_le2991122432403439658d_real @ B @ A ) ) ).
% order.asym
thf(fact_92_order_Oasym,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ~ ( ord_le5708704896291381698_preal @ B @ A ) ) ).
% order.asym
thf(fact_93_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_94_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_95_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_96_ord__eq__less__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A = B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_97_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_98_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_99_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_100_ord__less__eq__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( B = C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_101_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_102_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_103_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X )
=> ( P @ Y4 ) )
=> ( P @ X ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_104_antisym__conv3,axiom,
! [Y3: dedekind_real,X3: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ Y3 @ X3 )
=> ( ( ~ ( ord_le2991122432403439658d_real @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_105_antisym__conv3,axiom,
! [Y3: dedekind_preal,X3: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ Y3 @ X3 )
=> ( ( ~ ( ord_le5708704896291381698_preal @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_106_antisym__conv3,axiom,
! [Y3: nat,X3: nat] :
( ~ ( ord_less_nat @ Y3 @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_107_antisym__conv3,axiom,
! [Y3: int,X3: int] :
( ~ ( ord_less_int @ Y3 @ X3 )
=> ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_108_linorder__cases,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ( ( X3 != Y3 )
=> ( ord_le2991122432403439658d_real @ Y3 @ X3 ) ) ) ).
% linorder_cases
thf(fact_109_linorder__cases,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X3 @ Y3 )
=> ( ( X3 != Y3 )
=> ( ord_le5708704896291381698_preal @ Y3 @ X3 ) ) ) ).
% linorder_cases
thf(fact_110_linorder__cases,axiom,
! [X3: nat,Y3: nat] :
( ~ ( ord_less_nat @ X3 @ Y3 )
=> ( ( X3 != Y3 )
=> ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% linorder_cases
thf(fact_111_linorder__cases,axiom,
! [X3: int,Y3: int] :
( ~ ( ord_less_int @ X3 @ Y3 )
=> ( ( X3 != Y3 )
=> ( ord_less_int @ Y3 @ X3 ) ) ) ).
% linorder_cases
thf(fact_112_mem__Collect__eq,axiom,
! [A: dedekind_real,P: dedekind_real > $o] :
( ( member_Dedekind_real @ A @ ( collec1873404778653292075d_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_113_mem__Collect__eq,axiom,
! [A: dedekind_preal,P: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ A @ ( collec1132657498972982273_preal @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_114_Collect__mem__eq,axiom,
! [A2: set_Dedekind_real] :
( ( collec1873404778653292075d_real
@ ^ [X4: dedekind_real] : ( member_Dedekind_real @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_115_Collect__mem__eq,axiom,
! [A2: set_Dedekind_preal] :
( ( collec1132657498972982273_preal
@ ^ [X4: dedekind_preal] : ( member6871284927547481791_preal @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_116_Collect__cong,axiom,
! [P: dedekind_preal > $o,Q: dedekind_preal > $o] :
( ! [X: dedekind_preal] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec1132657498972982273_preal @ P )
= ( collec1132657498972982273_preal @ Q ) ) ) ).
% Collect_cong
thf(fact_117_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_118_dual__order_Oasym,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ~ ( ord_le5708704896291381698_preal @ A @ B ) ) ).
% dual_order.asym
thf(fact_119_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_120_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_121_dual__order_Oirrefl,axiom,
! [A: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_122_dual__order_Oirrefl,axiom,
! [A: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ A @ A ) ).
% dual_order.irrefl
thf(fact_123_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_124_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_125_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ~ ( P4 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_126_linorder__less__wlog,axiom,
! [P: dedekind_real > dedekind_real > $o,A: dedekind_real,B: dedekind_real] :
( ! [A3: dedekind_real,B2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: dedekind_real] : ( P @ A3 @ A3 )
=> ( ! [A3: dedekind_real,B2: dedekind_real] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_127_linorder__less__wlog,axiom,
! [P: dedekind_preal > dedekind_preal > $o,A: dedekind_preal,B: dedekind_preal] :
( ! [A3: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: dedekind_preal] : ( P @ A3 @ A3 )
=> ( ! [A3: dedekind_preal,B2: dedekind_preal] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_128_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_129_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_130_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_131_order_Ostrict__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_132_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_133_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_134_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_135_dual__order_Ostrict__trans,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( ( ord_le5708704896291381698_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_136_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_137_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_138_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_139_order_Ostrict__implies__not__eq,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_140_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_141_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_142_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_143_dual__order_Ostrict__implies__not__eq,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_144_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_145_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_146_linorder__neqE,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( X3 != Y3 )
=> ( ~ ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ( ord_le2991122432403439658d_real @ Y3 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_147_linorder__neqE,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( X3 != Y3 )
=> ( ~ ( ord_le5708704896291381698_preal @ X3 @ Y3 )
=> ( ord_le5708704896291381698_preal @ Y3 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_148_linorder__neqE,axiom,
! [X3: nat,Y3: nat] :
( ( X3 != Y3 )
=> ( ~ ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_149_linorder__neqE,axiom,
! [X3: int,Y3: int] :
( ( X3 != Y3 )
=> ( ~ ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ Y3 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_150_order__less__asym,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ~ ( ord_le2991122432403439658d_real @ Y3 @ X3 ) ) ).
% order_less_asym
thf(fact_151_order__less__asym,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y3 )
=> ~ ( ord_le5708704896291381698_preal @ Y3 @ X3 ) ) ).
% order_less_asym
thf(fact_152_order__less__asym,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% order_less_asym
thf(fact_153_order__less__asym,axiom,
! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ~ ( ord_less_int @ Y3 @ X3 ) ) ).
% order_less_asym
thf(fact_154_linorder__neq__iff,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( X3 != Y3 )
= ( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
| ( ord_le2991122432403439658d_real @ Y3 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_155_linorder__neq__iff,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( X3 != Y3 )
= ( ( ord_le5708704896291381698_preal @ X3 @ Y3 )
| ( ord_le5708704896291381698_preal @ Y3 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_156_linorder__neq__iff,axiom,
! [X3: nat,Y3: nat] :
( ( X3 != Y3 )
= ( ( ord_less_nat @ X3 @ Y3 )
| ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_157_linorder__neq__iff,axiom,
! [X3: int,Y3: int] :
( ( X3 != Y3 )
= ( ( ord_less_int @ X3 @ Y3 )
| ( ord_less_int @ Y3 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_158_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_159_order__less__asym_H,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ~ ( ord_le5708704896291381698_preal @ B @ A ) ) ).
% order_less_asym'
thf(fact_160_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_161_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_162_order__less__trans,axiom,
! [X3: dedekind_real,Y3: dedekind_real,Z3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ( ( ord_le2991122432403439658d_real @ Y3 @ Z3 )
=> ( ord_le2991122432403439658d_real @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_163_order__less__trans,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal,Z3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y3 )
=> ( ( ord_le5708704896291381698_preal @ Y3 @ Z3 )
=> ( ord_le5708704896291381698_preal @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_164_order__less__trans,axiom,
! [X3: nat,Y3: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_165_order__less__trans,axiom,
! [X3: int,Y3: int,Z3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ( ord_less_int @ Y3 @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_166_ord__eq__less__subst,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_167_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_168_ord__eq__less__subst,axiom,
! [A: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_169_ord__eq__less__subst,axiom,
! [A: int,F: dedekind_real > int,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_170_ord__eq__less__subst,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_171_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_172_ord__eq__less__subst,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_173_ord__eq__less__subst,axiom,
! [A: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_174_ord__eq__less__subst,axiom,
! [A: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_175_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_176_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_177_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_178_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_179_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_180_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_181_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_182_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_183_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_184_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_185_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_186_order__less__irrefl,axiom,
! [X3: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_187_order__less__irrefl,axiom,
! [X3: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_188_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_189_order__less__irrefl,axiom,
! [X3: int] :
~ ( ord_less_int @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_190_order__less__subst1,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_191_order__less__subst1,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_192_order__less__subst1,axiom,
! [A: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_193_order__less__subst1,axiom,
! [A: dedekind_real,F: int > dedekind_real,B: int,C: int] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_194_order__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_195_order__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_196_order__less__subst1,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_197_order__less__subst1,axiom,
! [A: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_198_order__less__subst1,axiom,
! [A: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_199_order__less__subst1,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_200_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_201_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_202_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_203_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_204_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_205_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_206_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_207_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_208_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_209_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_210_order__less__not__sym,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ~ ( ord_le2991122432403439658d_real @ Y3 @ X3 ) ) ).
% order_less_not_sym
thf(fact_211_order__less__not__sym,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y3 )
=> ~ ( ord_le5708704896291381698_preal @ Y3 @ X3 ) ) ).
% order_less_not_sym
thf(fact_212_order__less__not__sym,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% order_less_not_sym
thf(fact_213_order__less__not__sym,axiom,
! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ~ ( ord_less_int @ Y3 @ X3 ) ) ).
% order_less_not_sym
thf(fact_214_order__less__imp__triv,axiom,
! [X3: dedekind_real,Y3: dedekind_real,P: $o] :
( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ( ( ord_le2991122432403439658d_real @ Y3 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_215_order__less__imp__triv,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal,P: $o] :
( ( ord_le5708704896291381698_preal @ X3 @ Y3 )
=> ( ( ord_le5708704896291381698_preal @ Y3 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_216_order__less__imp__triv,axiom,
! [X3: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_217_order__less__imp__triv,axiom,
! [X3: int,Y3: int,P: $o] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ( ord_less_int @ Y3 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_218_linorder__less__linear,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
| ( X3 = Y3 )
| ( ord_le2991122432403439658d_real @ Y3 @ X3 ) ) ).
% linorder_less_linear
thf(fact_219_linorder__less__linear,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y3 )
| ( X3 = Y3 )
| ( ord_le5708704896291381698_preal @ Y3 @ X3 ) ) ).
% linorder_less_linear
thf(fact_220_linorder__less__linear,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
| ( X3 = Y3 )
| ( ord_less_nat @ Y3 @ X3 ) ) ).
% linorder_less_linear
thf(fact_221_linorder__less__linear,axiom,
! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
| ( X3 = Y3 )
| ( ord_less_int @ Y3 @ X3 ) ) ).
% linorder_less_linear
thf(fact_222_order__less__imp__not__eq,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_223_order__less__imp__not__eq,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_224_order__less__imp__not__eq,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_225_order__less__imp__not__eq,axiom,
! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_226__092_060open_062y_A_060_Areal__of__preal_A_Ipsup_A_123w_O_Areal__of__preal_Aw_A_092_060in_062_AP_125_J_092_060close_062,axiom,
( ord_le2991122432403439658d_real @ y
@ ( dedeki829075424511776921_preal
@ ( dedekind_psup
@ ( collec1132657498972982273_preal
@ ^ [W: dedekind_preal] : ( member_Dedekind_real @ ( dedeki829075424511776921_preal @ W ) @ p ) ) ) ) ) ).
% \<open>y < real_of_preal (psup {w. real_of_preal w \<in> P})\<close>
thf(fact_227_verit__comp__simplify1_I1_J,axiom,
! [A: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_228_verit__comp__simplify1_I1_J,axiom,
! [A: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_229_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_230_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_231_linordered__field__no__lb,axiom,
! [X2: dedekind_real] :
? [Y: dedekind_real] : ( ord_le2991122432403439658d_real @ Y @ X2 ) ).
% linordered_field_no_lb
thf(fact_232_linordered__field__no__ub,axiom,
! [X2: dedekind_real] :
? [X_1: dedekind_real] : ( ord_le2991122432403439658d_real @ X2 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_233_linorder__neqE__linordered__idom,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( X3 != Y3 )
=> ( ~ ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ( ord_le2991122432403439658d_real @ Y3 @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_234_linorder__neqE__linordered__idom,axiom,
! [X3: int,Y3: int] :
( ( X3 != Y3 )
=> ( ~ ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ Y3 @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_235_dbl__simps_I2_J,axiom,
( ( neg_nu8665065731627423525d_real @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% dbl_simps(2)
thf(fact_236_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_237_real__mult__order,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ X3 )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Y3 )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ X3 @ Y3 ) ) ) ) ).
% real_mult_order
thf(fact_238_mult__zero__left,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ zero_z580800474297136991d_real @ A )
= zero_z580800474297136991d_real ) ).
% mult_zero_left
thf(fact_239_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_240_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_241_mult__zero__right,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% mult_zero_right
thf(fact_242_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_243_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_244_mult__eq__0__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ B )
= zero_z580800474297136991d_real )
= ( ( A = zero_z580800474297136991d_real )
| ( B = zero_z580800474297136991d_real ) ) ) ).
% mult_eq_0_iff
thf(fact_245_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_246_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_247_mult__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ C @ A )
= ( times_2157731159493324635d_real @ C @ B ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_248_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_249_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_250_mult__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ C )
= ( times_2157731159493324635d_real @ B @ C ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_251_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_252_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_253_lambda__zero,axiom,
( ( ^ [H: dedekind_real] : zero_z580800474297136991d_real )
= ( times_2157731159493324635d_real @ zero_z580800474297136991d_real ) ) ).
% lambda_zero
thf(fact_254_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_255_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_256_mult_Oleft__commute,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ B @ ( times_2157731159493324635d_real @ A @ C ) )
= ( times_2157731159493324635d_real @ A @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_257_mult_Oleft__commute,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ B @ ( times_3000655703912201937_preal @ A @ C ) )
= ( times_3000655703912201937_preal @ A @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_258_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_259_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_260_mult_Ocommute,axiom,
( times_2157731159493324635d_real
= ( ^ [A4: dedekind_real,B3: dedekind_real] : ( times_2157731159493324635d_real @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_261_mult_Ocommute,axiom,
( times_3000655703912201937_preal
= ( ^ [A4: dedekind_preal,B3: dedekind_preal] : ( times_3000655703912201937_preal @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_262_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_263_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B3: int] : ( times_times_int @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_264_mult_Oassoc,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ A @ B ) @ C )
= ( times_2157731159493324635d_real @ A @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_265_mult_Oassoc,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ A @ B ) @ C )
= ( times_3000655703912201937_preal @ A @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% mult.assoc
thf(fact_266_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_267_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_268_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ A @ B ) @ C )
= ( times_2157731159493324635d_real @ A @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_269_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ A @ B ) @ C )
= ( times_3000655703912201937_preal @ A @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_270_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_271_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_272_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_273_real__mult__commute,axiom,
( times_2157731159493324635d_real
= ( ^ [Z4: dedekind_real,W: dedekind_real] : ( times_2157731159493324635d_real @ W @ Z4 ) ) ) ).
% real_mult_commute
thf(fact_274_mult__not__zero,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ B )
!= zero_z580800474297136991d_real )
=> ( ( A != zero_z580800474297136991d_real )
& ( B != zero_z580800474297136991d_real ) ) ) ).
% mult_not_zero
thf(fact_275_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_276_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_277_divisors__zero,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ B )
= zero_z580800474297136991d_real )
=> ( ( A = zero_z580800474297136991d_real )
| ( B = zero_z580800474297136991d_real ) ) ) ).
% divisors_zero
thf(fact_278_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_279_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_280_no__zero__divisors,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( B != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ A @ B )
!= zero_z580800474297136991d_real ) ) ) ).
% no_zero_divisors
thf(fact_281_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_282_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_283_mult__left__cancel,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( ( times_2157731159493324635d_real @ C @ A )
= ( times_2157731159493324635d_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_284_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_285_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_286_mult__right__cancel,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( ( times_2157731159493324635d_real @ A @ C )
= ( times_2157731159493324635d_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_287_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_288_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_289_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_290_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_291_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_292_mult__less__cancel__right__disj,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
& ( ord_le2991122432403439658d_real @ A @ B ) )
| ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
& ( ord_le2991122432403439658d_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_293_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_294_mult__strict__right__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_295_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_296_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_297_mult__strict__right__mono__neg,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_298_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_299_mult__less__cancel__left__disj,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
& ( ord_le2991122432403439658d_real @ A @ B ) )
| ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
& ( ord_le2991122432403439658d_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_300_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_301_mult__strict__left__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_302_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_303_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_304_mult__strict__left__mono__neg,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_305_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_306_mult__less__cancel__left__pos,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ord_le2991122432403439658d_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_307_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_308_mult__less__cancel__left__neg,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ord_le2991122432403439658d_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_309_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_310_zero__less__mult__pos2,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ B @ A ) )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_311_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_312_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_313_zero__less__mult__pos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_314_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_315_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_316_zero__less__mult__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
& ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) )
| ( ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real )
& ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_317_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_318_mult__pos__neg2,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ B @ A ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_pos_neg2
thf(fact_319_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_320_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_321_mult__pos__pos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_322_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_323_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_324_mult__pos__neg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_pos_neg
thf(fact_325_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_326_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_327_mult__neg__pos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_neg_pos
thf(fact_328_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_329_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_330_mult__less__0__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ B ) @ zero_z580800474297136991d_real )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
& ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real ) )
| ( ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real )
& ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_331_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_332_not__square__less__zero,axiom,
! [A: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ A ) @ zero_z580800474297136991d_real ) ).
% not_square_less_zero
thf(fact_333_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_334_mult__neg__neg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_335_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_336_real__mult__less__mono2,axiom,
! [Z3: dedekind_real,X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Z3 )
=> ( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ Z3 @ X3 ) @ ( times_2157731159493324635d_real @ Z3 @ Y3 ) ) ) ) ).
% real_mult_less_mono2
thf(fact_337_mult__less__iff1,axiom,
! [Z3: dedekind_real,X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Z3 )
=> ( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ X3 @ Z3 ) @ ( times_2157731159493324635d_real @ Y3 @ Z3 ) )
= ( ord_le2991122432403439658d_real @ X3 @ Y3 ) ) ) ).
% mult_less_iff1
thf(fact_338_mult__less__iff1,axiom,
! [Z3: int,X3: int,Y3: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_int @ ( times_times_int @ X3 @ Z3 ) @ ( times_times_int @ Y3 @ Z3 ) )
= ( ord_less_int @ X3 @ Y3 ) ) ) ).
% mult_less_iff1
thf(fact_339_mult__commute__abs,axiom,
! [C: dedekind_real] :
( ( ^ [X4: dedekind_real] : ( times_2157731159493324635d_real @ X4 @ C ) )
= ( times_2157731159493324635d_real @ C ) ) ).
% mult_commute_abs
thf(fact_340_mult__commute__abs,axiom,
! [C: dedekind_preal] :
( ( ^ [X4: dedekind_preal] : ( times_3000655703912201937_preal @ X4 @ C ) )
= ( times_3000655703912201937_preal @ C ) ) ).
% mult_commute_abs
thf(fact_341_mult__commute__abs,axiom,
! [C: nat] :
( ( ^ [X4: nat] : ( times_times_nat @ X4 @ C ) )
= ( times_times_nat @ C ) ) ).
% mult_commute_abs
thf(fact_342_mult__commute__abs,axiom,
! [C: int] :
( ( ^ [X4: int] : ( times_times_int @ X4 @ C ) )
= ( times_times_int @ C ) ) ).
% mult_commute_abs
thf(fact_343_mult__le__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ A @ B ) )
& ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_344_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_345_mult__le__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ A @ B ) )
& ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_346_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_347_mult__left__less__imp__less,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_348_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_349_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_350_mult__strict__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ C @ D )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_351_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_352_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_353_mult__less__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ A @ B ) )
& ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_354_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_355_mult__right__less__imp__less,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_356_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_357_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_358_mult__strict__mono_H,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ C @ D )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_359_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_360_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_361_order__refl,axiom,
! [X3: dedekind_real] : ( ord_le2716243287969276982d_real @ X3 @ X3 ) ).
% order_refl
thf(fact_362_order__refl,axiom,
! [X3: dedekind_preal] : ( ord_le5604041210740703414_preal @ X3 @ X3 ) ).
% order_refl
thf(fact_363_order__refl,axiom,
! [X3: num] : ( ord_less_eq_num @ X3 @ X3 ) ).
% order_refl
thf(fact_364_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_365_order__refl,axiom,
! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% order_refl
thf(fact_366_dual__order_Orefl,axiom,
! [A: dedekind_real] : ( ord_le2716243287969276982d_real @ A @ A ) ).
% dual_order.refl
thf(fact_367_dual__order_Orefl,axiom,
! [A: dedekind_preal] : ( ord_le5604041210740703414_preal @ A @ A ) ).
% dual_order.refl
thf(fact_368_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_369_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_370_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_371_real__of__preal__le__iff,axiom,
! [M1: dedekind_preal,M2: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ ( dedeki829075424511776921_preal @ M1 ) @ ( dedeki829075424511776921_preal @ M2 ) )
= ( ord_le5604041210740703414_preal @ M1 @ M2 ) ) ).
% real_of_preal_le_iff
thf(fact_372_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_373_verit__comp__simplify1_I2_J,axiom,
! [A: dedekind_real] : ( ord_le2716243287969276982d_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_374_verit__comp__simplify1_I2_J,axiom,
! [A: dedekind_preal] : ( ord_le5604041210740703414_preal @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_375_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_376_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_377_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_378_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_379_verit__la__disequality,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( A = B )
| ~ ( ord_le5604041210740703414_preal @ A @ B )
| ~ ( ord_le5604041210740703414_preal @ B @ A ) ) ).
% verit_la_disequality
thf(fact_380_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_381_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_382_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_383_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_384_nle__le,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ A @ B ) )
= ( ( ord_le5604041210740703414_preal @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_385_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_386_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_387_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_388_le__cases3,axiom,
! [X3: dedekind_real,Y3: dedekind_real,Z3: dedekind_real] :
( ( ( ord_le2716243287969276982d_real @ X3 @ Y3 )
=> ~ ( ord_le2716243287969276982d_real @ Y3 @ Z3 ) )
=> ( ( ( ord_le2716243287969276982d_real @ Y3 @ X3 )
=> ~ ( ord_le2716243287969276982d_real @ X3 @ Z3 ) )
=> ( ( ( ord_le2716243287969276982d_real @ X3 @ Z3 )
=> ~ ( ord_le2716243287969276982d_real @ Z3 @ Y3 ) )
=> ( ( ( ord_le2716243287969276982d_real @ Z3 @ Y3 )
=> ~ ( ord_le2716243287969276982d_real @ Y3 @ X3 ) )
=> ( ( ( ord_le2716243287969276982d_real @ Y3 @ Z3 )
=> ~ ( ord_le2716243287969276982d_real @ Z3 @ X3 ) )
=> ~ ( ( ord_le2716243287969276982d_real @ Z3 @ X3 )
=> ~ ( ord_le2716243287969276982d_real @ X3 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_389_le__cases3,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal,Z3: dedekind_preal] :
( ( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
=> ~ ( ord_le5604041210740703414_preal @ Y3 @ Z3 ) )
=> ( ( ( ord_le5604041210740703414_preal @ Y3 @ X3 )
=> ~ ( ord_le5604041210740703414_preal @ X3 @ Z3 ) )
=> ( ( ( ord_le5604041210740703414_preal @ X3 @ Z3 )
=> ~ ( ord_le5604041210740703414_preal @ Z3 @ Y3 ) )
=> ( ( ( ord_le5604041210740703414_preal @ Z3 @ Y3 )
=> ~ ( ord_le5604041210740703414_preal @ Y3 @ X3 ) )
=> ( ( ( ord_le5604041210740703414_preal @ Y3 @ Z3 )
=> ~ ( ord_le5604041210740703414_preal @ Z3 @ X3 ) )
=> ~ ( ( ord_le5604041210740703414_preal @ Z3 @ X3 )
=> ~ ( ord_le5604041210740703414_preal @ X3 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_390_le__cases3,axiom,
! [X3: num,Y3: num,Z3: num] :
( ( ( ord_less_eq_num @ X3 @ Y3 )
=> ~ ( ord_less_eq_num @ Y3 @ Z3 ) )
=> ( ( ( ord_less_eq_num @ Y3 @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_num @ X3 @ Z3 )
=> ~ ( ord_less_eq_num @ Z3 @ Y3 ) )
=> ( ( ( ord_less_eq_num @ Z3 @ Y3 )
=> ~ ( ord_less_eq_num @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq_num @ Y3 @ Z3 )
=> ~ ( ord_less_eq_num @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_num @ Z3 @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_391_le__cases3,axiom,
! [X3: nat,Y3: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_392_le__cases3,axiom,
! [X3: int,Y3: int,Z3: int] :
( ( ( ord_less_eq_int @ X3 @ Y3 )
=> ~ ( ord_less_eq_int @ Y3 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y3 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X3 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y3 ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y3 )
=> ~ ( ord_less_eq_int @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq_int @ Y3 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_393_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: dedekind_real,Z5: dedekind_real] : ( Y5 = Z5 ) )
= ( ^ [X4: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X4 @ Y2 )
& ( ord_le2716243287969276982d_real @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_394_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: dedekind_preal,Z5: dedekind_preal] : ( Y5 = Z5 ) )
= ( ^ [X4: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X4 @ Y2 )
& ( ord_le5604041210740703414_preal @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_395_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z5: num] : ( Y5 = Z5 ) )
= ( ^ [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
& ( ord_less_eq_num @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_396_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 ) )
= ( ^ [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_397_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z5: int] : ( Y5 = Z5 ) )
= ( ^ [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
& ( ord_less_eq_int @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_398_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_399_ord__eq__le__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A = B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5604041210740703414_preal @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_400_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_401_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_402_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_403_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_404_ord__le__eq__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( B = C )
=> ( ord_le5604041210740703414_preal @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_405_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_406_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_407_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_408_order__antisym,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y3 )
=> ( ( ord_le2716243287969276982d_real @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_409_order__antisym,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
=> ( ( ord_le5604041210740703414_preal @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_410_order__antisym,axiom,
! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ( ord_less_eq_num @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_411_order__antisym,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_412_order__antisym,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ( ord_less_eq_int @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_413_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_414_order_Otrans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5604041210740703414_preal @ A @ C ) ) ) ).
% order.trans
thf(fact_415_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_416_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_417_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_418_order__trans,axiom,
! [X3: dedekind_real,Y3: dedekind_real,Z3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y3 )
=> ( ( ord_le2716243287969276982d_real @ Y3 @ Z3 )
=> ( ord_le2716243287969276982d_real @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_419_order__trans,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal,Z3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
=> ( ( ord_le5604041210740703414_preal @ Y3 @ Z3 )
=> ( ord_le5604041210740703414_preal @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_420_order__trans,axiom,
! [X3: num,Y3: num,Z3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ( ord_less_eq_num @ Y3 @ Z3 )
=> ( ord_less_eq_num @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_421_order__trans,axiom,
! [X3: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_422_order__trans,axiom,
! [X3: int,Y3: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ( ord_less_eq_int @ Y3 @ Z3 )
=> ( ord_less_eq_int @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_423_linorder__wlog,axiom,
! [P: dedekind_real > dedekind_real > $o,A: dedekind_real,B: dedekind_real] :
( ! [A3: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: dedekind_real,B2: dedekind_real] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_424_linorder__wlog,axiom,
! [P: dedekind_preal > dedekind_preal > $o,A: dedekind_preal,B: dedekind_preal] :
( ! [A3: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: dedekind_preal,B2: dedekind_preal] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_425_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A3: num,B2: num] :
( ( ord_less_eq_num @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: num,B2: num] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_426_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_427_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_428_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: dedekind_real,Z5: dedekind_real] : ( Y5 = Z5 ) )
= ( ^ [A4: dedekind_real,B3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B3 @ A4 )
& ( ord_le2716243287969276982d_real @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_429_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: dedekind_preal,Z5: dedekind_preal] : ( Y5 = Z5 ) )
= ( ^ [A4: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B3 @ A4 )
& ( ord_le5604041210740703414_preal @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_430_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: num,Z5: num] : ( Y5 = Z5 ) )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_431_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_432_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z5: int] : ( Y5 = Z5 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_433_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_434_dual__order_Oantisym,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A )
=> ( ( ord_le5604041210740703414_preal @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_435_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_436_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_437_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_438_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_439_dual__order_Otrans,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A )
=> ( ( ord_le5604041210740703414_preal @ C @ B )
=> ( ord_le5604041210740703414_preal @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_440_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_441_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_442_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_443_antisym,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_444_antisym,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_445_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_446_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_447_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_448_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: dedekind_real,Z5: dedekind_real] : ( Y5 = Z5 ) )
= ( ^ [A4: dedekind_real,B3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A4 @ B3 )
& ( ord_le2716243287969276982d_real @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_449_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: dedekind_preal,Z5: dedekind_preal] : ( Y5 = Z5 ) )
= ( ^ [A4: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A4 @ B3 )
& ( ord_le5604041210740703414_preal @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_450_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z5: num] : ( Y5 = Z5 ) )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_451_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_452_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z5: int] : ( Y5 = Z5 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_453_order__subst1,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_454_order__subst1,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_455_order__subst1,axiom,
! [A: dedekind_real,F: num > dedekind_real,B: num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_456_order__subst1,axiom,
! [A: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_457_order__subst1,axiom,
! [A: dedekind_real,F: int > dedekind_real,B: int,C: int] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_458_order__subst1,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_459_order__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_460_order__subst1,axiom,
! [A: dedekind_preal,F: num > dedekind_preal,B: num,C: num] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_461_order__subst1,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_462_order__subst1,axiom,
! [A: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_463_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_464_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_465_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_466_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_467_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_468_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_469_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_470_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_471_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_472_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_473_order__eq__refl,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( X3 = Y3 )
=> ( ord_le2716243287969276982d_real @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_474_order__eq__refl,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( X3 = Y3 )
=> ( ord_le5604041210740703414_preal @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_475_order__eq__refl,axiom,
! [X3: num,Y3: num] :
( ( X3 = Y3 )
=> ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_476_order__eq__refl,axiom,
! [X3: nat,Y3: nat] :
( ( X3 = Y3 )
=> ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_477_order__eq__refl,axiom,
! [X3: int,Y3: int] :
( ( X3 = Y3 )
=> ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_478_linorder__linear,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y3 )
| ( ord_le2716243287969276982d_real @ Y3 @ X3 ) ) ).
% linorder_linear
thf(fact_479_linorder__linear,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
| ( ord_le5604041210740703414_preal @ Y3 @ X3 ) ) ).
% linorder_linear
thf(fact_480_linorder__linear,axiom,
! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
| ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% linorder_linear
thf(fact_481_linorder__linear,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% linorder_linear
thf(fact_482_linorder__linear,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
| ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% linorder_linear
thf(fact_483_ord__eq__le__subst,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_484_ord__eq__le__subst,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_485_ord__eq__le__subst,axiom,
! [A: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_486_ord__eq__le__subst,axiom,
! [A: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_487_ord__eq__le__subst,axiom,
! [A: int,F: dedekind_real > int,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_488_ord__eq__le__subst,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_489_ord__eq__le__subst,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_490_ord__eq__le__subst,axiom,
! [A: num,F: dedekind_preal > num,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_491_ord__eq__le__subst,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_492_ord__eq__le__subst,axiom,
! [A: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_493_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_494_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_495_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_496_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_497_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_498_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_499_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_500_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_501_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_502_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_503_linorder__le__cases,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ~ ( ord_le2716243287969276982d_real @ X3 @ Y3 )
=> ( ord_le2716243287969276982d_real @ Y3 @ X3 ) ) ).
% linorder_le_cases
thf(fact_504_linorder__le__cases,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ~ ( ord_le5604041210740703414_preal @ X3 @ Y3 )
=> ( ord_le5604041210740703414_preal @ Y3 @ X3 ) ) ).
% linorder_le_cases
thf(fact_505_linorder__le__cases,axiom,
! [X3: num,Y3: num] :
( ~ ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% linorder_le_cases
thf(fact_506_linorder__le__cases,axiom,
! [X3: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% linorder_le_cases
thf(fact_507_linorder__le__cases,axiom,
! [X3: int,Y3: int] :
( ~ ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% linorder_le_cases
thf(fact_508_order__antisym__conv,axiom,
! [Y3: dedekind_real,X3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Y3 @ X3 )
=> ( ( ord_le2716243287969276982d_real @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_509_order__antisym__conv,axiom,
! [Y3: dedekind_preal,X3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ Y3 @ X3 )
=> ( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_510_order__antisym__conv,axiom,
! [Y3: num,X3: num] :
( ( ord_less_eq_num @ Y3 @ X3 )
=> ( ( ord_less_eq_num @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_511_order__antisym__conv,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_512_order__antisym__conv,axiom,
! [Y3: int,X3: int] :
( ( ord_less_eq_int @ Y3 @ X3 )
=> ( ( ord_less_eq_int @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_513_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_514_le__numeral__extra_I3_J,axiom,
ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ zero_z580800474297136991d_real ).
% le_numeral_extra(3)
thf(fact_515_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_516_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_517_leD,axiom,
! [Y3: dedekind_real,X3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Y3 @ X3 )
=> ~ ( ord_le2991122432403439658d_real @ X3 @ Y3 ) ) ).
% leD
thf(fact_518_leD,axiom,
! [Y3: dedekind_preal,X3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ Y3 @ X3 )
=> ~ ( ord_le5708704896291381698_preal @ X3 @ Y3 ) ) ).
% leD
thf(fact_519_leD,axiom,
! [Y3: num,X3: num] :
( ( ord_less_eq_num @ Y3 @ X3 )
=> ~ ( ord_less_num @ X3 @ Y3 ) ) ).
% leD
thf(fact_520_leD,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y3 ) ) ).
% leD
thf(fact_521_leD,axiom,
! [Y3: int,X3: int] :
( ( ord_less_eq_int @ Y3 @ X3 )
=> ~ ( ord_less_int @ X3 @ Y3 ) ) ).
% leD
thf(fact_522_leI,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ( ord_le2716243287969276982d_real @ Y3 @ X3 ) ) ).
% leI
thf(fact_523_leI,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X3 @ Y3 )
=> ( ord_le5604041210740703414_preal @ Y3 @ X3 ) ) ).
% leI
thf(fact_524_leI,axiom,
! [X3: num,Y3: num] :
( ~ ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% leI
thf(fact_525_leI,axiom,
! [X3: nat,Y3: nat] :
( ~ ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% leI
thf(fact_526_leI,axiom,
! [X3: int,Y3: int] :
( ~ ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% leI
thf(fact_527_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_528_nless__le,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ A @ B ) )
= ( ~ ( ord_le5604041210740703414_preal @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_529_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_530_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_531_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_532_antisym__conv1,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ( ( ord_le2716243287969276982d_real @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_533_antisym__conv1,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X3 @ Y3 )
=> ( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_534_antisym__conv1,axiom,
! [X3: num,Y3: num] :
( ~ ( ord_less_num @ X3 @ Y3 )
=> ( ( ord_less_eq_num @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_535_antisym__conv1,axiom,
! [X3: nat,Y3: nat] :
( ~ ( ord_less_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_536_antisym__conv1,axiom,
! [X3: int,Y3: int] :
( ~ ( ord_less_int @ X3 @ Y3 )
=> ( ( ord_less_eq_int @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_537_antisym__conv2,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y3 )
=> ( ( ~ ( ord_le2991122432403439658d_real @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_538_antisym__conv2,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
=> ( ( ~ ( ord_le5708704896291381698_preal @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_539_antisym__conv2,axiom,
! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_540_antisym__conv2,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_541_antisym__conv2,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_542_dense__ge,axiom,
! [Z3: dedekind_real,Y3: dedekind_real] :
( ! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ X )
=> ( ord_le2716243287969276982d_real @ Y3 @ X ) )
=> ( ord_le2716243287969276982d_real @ Y3 @ Z3 ) ) ).
% dense_ge
thf(fact_543_dense__le,axiom,
! [Y3: dedekind_real,Z3: dedekind_real] :
( ! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ X @ Z3 ) )
=> ( ord_le2716243287969276982d_real @ Y3 @ Z3 ) ) ).
% dense_le
thf(fact_544_less__le__not__le,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X4: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X4 @ Y2 )
& ~ ( ord_le2716243287969276982d_real @ Y2 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_545_less__le__not__le,axiom,
( ord_le5708704896291381698_preal
= ( ^ [X4: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X4 @ Y2 )
& ~ ( ord_le5604041210740703414_preal @ Y2 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_546_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
& ~ ( ord_less_eq_num @ Y2 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_547_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_548_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
& ~ ( ord_less_eq_int @ Y2 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_549_not__le__imp__less,axiom,
! [Y3: dedekind_real,X3: dedekind_real] :
( ~ ( ord_le2716243287969276982d_real @ Y3 @ X3 )
=> ( ord_le2991122432403439658d_real @ X3 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_550_not__le__imp__less,axiom,
! [Y3: dedekind_preal,X3: dedekind_preal] :
( ~ ( ord_le5604041210740703414_preal @ Y3 @ X3 )
=> ( ord_le5708704896291381698_preal @ X3 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_551_not__le__imp__less,axiom,
! [Y3: num,X3: num] :
( ~ ( ord_less_eq_num @ Y3 @ X3 )
=> ( ord_less_num @ X3 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_552_not__le__imp__less,axiom,
! [Y3: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ord_less_nat @ X3 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_553_not__le__imp__less,axiom,
! [Y3: int,X3: int] :
( ~ ( ord_less_eq_int @ Y3 @ X3 )
=> ( ord_less_int @ X3 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_554_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_555_order_Oorder__iff__strict,axiom,
( ord_le5604041210740703414_preal
= ( ^ [A4: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_556_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_557_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_558_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_559_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_560_order_Ostrict__iff__order,axiom,
( ord_le5708704896291381698_preal
= ( ^ [A4: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_561_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_562_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_563_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_564_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_565_order_Ostrict__trans1,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_566_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_567_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_568_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_569_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_570_order_Ostrict__trans2,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_571_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_572_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_573_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_574_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_575_order_Ostrict__iff__not,axiom,
( ord_le5708704896291381698_preal
= ( ^ [A4: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A4 @ B3 )
& ~ ( ord_le5604041210740703414_preal @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_576_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_577_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_578_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_579_dense__ge__bounded,axiom,
! [Z3: dedekind_real,X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ X3 )
=> ( ! [W2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ W2 )
=> ( ( ord_le2991122432403439658d_real @ W2 @ X3 )
=> ( ord_le2716243287969276982d_real @ Y3 @ W2 ) ) )
=> ( ord_le2716243287969276982d_real @ Y3 @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_580_dense__le__bounded,axiom,
! [X3: dedekind_real,Y3: dedekind_real,Z3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ( ! [W2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ W2 )
=> ( ( ord_le2991122432403439658d_real @ W2 @ Y3 )
=> ( ord_le2716243287969276982d_real @ W2 @ Z3 ) ) )
=> ( ord_le2716243287969276982d_real @ Y3 @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_581_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_582_dual__order_Oorder__iff__strict,axiom,
( ord_le5604041210740703414_preal
= ( ^ [B3: dedekind_preal,A4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_583_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_584_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_585_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_586_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_587_dual__order_Ostrict__iff__order,axiom,
( ord_le5708704896291381698_preal
= ( ^ [B3: dedekind_preal,A4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_588_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_589_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_590_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_591_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_592_dual__order_Ostrict__trans1,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A )
=> ( ( ord_le5708704896291381698_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_593_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_594_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_595_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_596_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_597_dual__order_Ostrict__trans2,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( ( ord_le5604041210740703414_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_598_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_599_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_600_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_601_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_602_dual__order_Ostrict__iff__not,axiom,
( ord_le5708704896291381698_preal
= ( ^ [B3: dedekind_preal,A4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B3 @ A4 )
& ~ ( ord_le5604041210740703414_preal @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_603_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_604_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_605_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_606_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_607_order_Ostrict__implies__order,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ord_le5604041210740703414_preal @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_608_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_609_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_610_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_611_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_612_dual__order_Ostrict__implies__order,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( ord_le5604041210740703414_preal @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_613_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_614_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_615_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_616_order__le__less,axiom,
( ord_le2716243287969276982d_real
= ( ^ [X4: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Y2 )
| ( X4 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_617_order__le__less,axiom,
( ord_le5604041210740703414_preal
= ( ^ [X4: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Y2 )
| ( X4 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_618_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X4: num,Y2: num] :
( ( ord_less_num @ X4 @ Y2 )
| ( X4 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_619_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
| ( X4 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_620_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
| ( X4 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_621_order__less__le,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X4: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X4 @ Y2 )
& ( X4 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_622_order__less__le,axiom,
( ord_le5708704896291381698_preal
= ( ^ [X4: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X4 @ Y2 )
& ( X4 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_623_order__less__le,axiom,
( ord_less_num
= ( ^ [X4: num,Y2: num] :
( ( ord_less_eq_num @ X4 @ Y2 )
& ( X4 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_624_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
& ( X4 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_625_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
& ( X4 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_626_linorder__not__le,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ~ ( ord_le2716243287969276982d_real @ X3 @ Y3 ) )
= ( ord_le2991122432403439658d_real @ Y3 @ X3 ) ) ).
% linorder_not_le
thf(fact_627_linorder__not__le,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ X3 @ Y3 ) )
= ( ord_le5708704896291381698_preal @ Y3 @ X3 ) ) ).
% linorder_not_le
thf(fact_628_linorder__not__le,axiom,
! [X3: num,Y3: num] :
( ( ~ ( ord_less_eq_num @ X3 @ Y3 ) )
= ( ord_less_num @ Y3 @ X3 ) ) ).
% linorder_not_le
thf(fact_629_linorder__not__le,axiom,
! [X3: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y3 ) )
= ( ord_less_nat @ Y3 @ X3 ) ) ).
% linorder_not_le
thf(fact_630_linorder__not__le,axiom,
! [X3: int,Y3: int] :
( ( ~ ( ord_less_eq_int @ X3 @ Y3 ) )
= ( ord_less_int @ Y3 @ X3 ) ) ).
% linorder_not_le
thf(fact_631_linorder__not__less,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ~ ( ord_le2991122432403439658d_real @ X3 @ Y3 ) )
= ( ord_le2716243287969276982d_real @ Y3 @ X3 ) ) ).
% linorder_not_less
thf(fact_632_linorder__not__less,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ X3 @ Y3 ) )
= ( ord_le5604041210740703414_preal @ Y3 @ X3 ) ) ).
% linorder_not_less
thf(fact_633_linorder__not__less,axiom,
! [X3: num,Y3: num] :
( ( ~ ( ord_less_num @ X3 @ Y3 ) )
= ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% linorder_not_less
thf(fact_634_linorder__not__less,axiom,
! [X3: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% linorder_not_less
thf(fact_635_linorder__not__less,axiom,
! [X3: int,Y3: int] :
( ( ~ ( ord_less_int @ X3 @ Y3 ) )
= ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% linorder_not_less
thf(fact_636_order__less__imp__le,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ( ord_le2716243287969276982d_real @ X3 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_637_order__less__imp__le,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y3 )
=> ( ord_le5604041210740703414_preal @ X3 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_638_order__less__imp__le,axiom,
! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_639_order__less__imp__le,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_640_order__less__imp__le,axiom,
! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_641_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_642_order__le__neq__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( A != B )
=> ( ord_le5708704896291381698_preal @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_643_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_644_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_645_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_646_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_647_order__neq__le__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( A != B )
=> ( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ord_le5708704896291381698_preal @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_648_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_649_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_650_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_651_order__le__less__trans,axiom,
! [X3: dedekind_real,Y3: dedekind_real,Z3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y3 )
=> ( ( ord_le2991122432403439658d_real @ Y3 @ Z3 )
=> ( ord_le2991122432403439658d_real @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_652_order__le__less__trans,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal,Z3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
=> ( ( ord_le5708704896291381698_preal @ Y3 @ Z3 )
=> ( ord_le5708704896291381698_preal @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_653_order__le__less__trans,axiom,
! [X3: num,Y3: num,Z3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ( ord_less_num @ Y3 @ Z3 )
=> ( ord_less_num @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_654_order__le__less__trans,axiom,
! [X3: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_655_order__le__less__trans,axiom,
! [X3: int,Y3: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ( ord_less_int @ Y3 @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_656_order__less__le__trans,axiom,
! [X3: dedekind_real,Y3: dedekind_real,Z3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
=> ( ( ord_le2716243287969276982d_real @ Y3 @ Z3 )
=> ( ord_le2991122432403439658d_real @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_657_order__less__le__trans,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal,Z3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y3 )
=> ( ( ord_le5604041210740703414_preal @ Y3 @ Z3 )
=> ( ord_le5708704896291381698_preal @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_658_order__less__le__trans,axiom,
! [X3: num,Y3: num,Z3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ( ord_less_eq_num @ Y3 @ Z3 )
=> ( ord_less_num @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_659_order__less__le__trans,axiom,
! [X3: nat,Y3: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_660_order__less__le__trans,axiom,
! [X3: int,Y3: int,Z3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ( ord_less_eq_int @ Y3 @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_661_order__le__less__subst1,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_662_order__le__less__subst1,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_663_order__le__less__subst1,axiom,
! [A: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_664_order__le__less__subst1,axiom,
! [A: dedekind_real,F: int > dedekind_real,B: int,C: int] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_665_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_666_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_667_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_668_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_669_order__le__less__subst1,axiom,
! [A: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_670_order__le__less__subst1,axiom,
! [A: num,F: dedekind_preal > num,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_671_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_672_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_673_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_674_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_675_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_676_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_677_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_678_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_679_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_680_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_681_order__less__le__subst1,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_682_order__less__le__subst1,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_683_order__less__le__subst1,axiom,
! [A: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_684_order__less__le__subst1,axiom,
! [A: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_685_order__less__le__subst1,axiom,
! [A: int,F: dedekind_real > int,B: dedekind_real,C: dedekind_real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_686_order__less__le__subst1,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_687_order__less__le__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_688_order__less__le__subst1,axiom,
! [A: num,F: dedekind_preal > num,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_689_order__less__le__subst1,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_690_order__less__le__subst1,axiom,
! [A: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_691_order__less__le__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_692_order__less__le__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_693_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_694_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > dedekind_real,C: dedekind_real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_695_order__less__le__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_696_order__less__le__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_697_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_698_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > dedekind_preal,C: dedekind_preal] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_699_order__less__le__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_700_order__less__le__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_701_linorder__le__less__linear,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y3 )
| ( ord_le2991122432403439658d_real @ Y3 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_702_linorder__le__less__linear,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
| ( ord_le5708704896291381698_preal @ Y3 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_703_linorder__le__less__linear,axiom,
! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
| ( ord_less_num @ Y3 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_704_linorder__le__less__linear,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
| ( ord_less_nat @ Y3 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_705_linorder__le__less__linear,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
| ( ord_less_int @ Y3 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_706_order__le__imp__less__or__eq,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y3 )
=> ( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_707_order__le__imp__less__or__eq,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
=> ( ( ord_le5708704896291381698_preal @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_708_order__le__imp__less__or__eq,axiom,
! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ( ord_less_num @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_709_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_710_order__le__imp__less__or__eq,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ( ord_less_int @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_711_minf_I8_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z )
=> ~ ( ord_le2716243287969276982d_real @ T @ X2 ) ) ).
% minf(8)
thf(fact_712_minf_I8_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z )
=> ~ ( ord_le5604041210740703414_preal @ T @ X2 ) ) ).
% minf(8)
thf(fact_713_minf_I8_J,axiom,
! [T: num] :
? [Z: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z )
=> ~ ( ord_less_eq_num @ T @ X2 ) ) ).
% minf(8)
thf(fact_714_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X2 ) ) ).
% minf(8)
thf(fact_715_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% minf(8)
thf(fact_716_minf_I6_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z )
=> ( ord_le2716243287969276982d_real @ X2 @ T ) ) ).
% minf(6)
thf(fact_717_minf_I6_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z )
=> ( ord_le5604041210740703414_preal @ X2 @ T ) ) ).
% minf(6)
thf(fact_718_minf_I6_J,axiom,
! [T: num] :
? [Z: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z )
=> ( ord_less_eq_num @ X2 @ T ) ) ).
% minf(6)
thf(fact_719_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ord_less_eq_nat @ X2 @ T ) ) ).
% minf(6)
thf(fact_720_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ord_less_eq_int @ X2 @ T ) ) ).
% minf(6)
thf(fact_721_pinf_I8_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X2 )
=> ( ord_le2716243287969276982d_real @ T @ X2 ) ) ).
% pinf(8)
thf(fact_722_pinf_I8_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z @ X2 )
=> ( ord_le5604041210740703414_preal @ T @ X2 ) ) ).
% pinf(8)
thf(fact_723_pinf_I8_J,axiom,
! [T: num] :
? [Z: num] :
! [X2: num] :
( ( ord_less_num @ Z @ X2 )
=> ( ord_less_eq_num @ T @ X2 ) ) ).
% pinf(8)
thf(fact_724_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ord_less_eq_nat @ T @ X2 ) ) ).
% pinf(8)
thf(fact_725_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ord_less_eq_int @ T @ X2 ) ) ).
% pinf(8)
thf(fact_726_pinf_I6_J,axiom,
! [T: dedekind_real] :
? [Z: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z @ X2 )
=> ~ ( ord_le2716243287969276982d_real @ X2 @ T ) ) ).
% pinf(6)
thf(fact_727_pinf_I6_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z @ X2 )
=> ~ ( ord_le5604041210740703414_preal @ X2 @ T ) ) ).
% pinf(6)
thf(fact_728_pinf_I6_J,axiom,
! [T: num] :
? [Z: num] :
! [X2: num] :
( ( ord_less_num @ Z @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ T ) ) ).
% pinf(6)
thf(fact_729_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ T ) ) ).
% pinf(6)
thf(fact_730_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% pinf(6)
thf(fact_731_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_732_verit__comp__simplify1_I3_J,axiom,
! [B4: dedekind_preal,A5: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ B4 @ A5 ) )
= ( ord_le5708704896291381698_preal @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_733_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_734_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_735_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_736_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D3: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A @ X )
& ( ord_less_nat @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_737_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X2: int] :
( ( ( ord_less_eq_int @ A @ X2 )
& ( ord_less_int @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D3: int] :
( ! [X: int] :
( ( ( ord_less_eq_int @ A @ X )
& ( ord_less_int @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_int @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_738_real__less__def,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X4: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X4 @ Y2 )
& ( X4 != Y2 ) ) ) ) ).
% real_less_def
thf(fact_739_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_740_mult__le__cancel__iff2,axiom,
! [Z3: dedekind_real,X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Z3 )
=> ( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ Z3 @ X3 ) @ ( times_2157731159493324635d_real @ Z3 @ Y3 ) )
= ( ord_le2716243287969276982d_real @ X3 @ Y3 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_741_mult__le__cancel__iff2,axiom,
! [Z3: int,X3: int,Y3: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z3 @ X3 ) @ ( times_times_int @ Z3 @ Y3 ) )
= ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_742_mult__le__cancel__iff1,axiom,
! [Z3: dedekind_real,X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Z3 )
=> ( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ X3 @ Z3 ) @ ( times_2157731159493324635d_real @ Y3 @ Z3 ) )
= ( ord_le2716243287969276982d_real @ X3 @ Y3 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_743_mult__le__cancel__iff1,axiom,
! [Z3: int,X3: int,Y3: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ ( times_times_int @ X3 @ Z3 ) @ ( times_times_int @ Y3 @ Z3 ) )
= ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_744_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_745_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_746_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_747_zero__le__mult__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
& ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) )
| ( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
& ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_748_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_749_mult__nonneg__nonpos2,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ B @ A ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_750_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_751_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_752_mult__nonpos__nonneg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_753_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_754_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_755_mult__nonneg__nonpos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_756_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_757_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_758_mult__nonneg__nonneg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_759_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_760_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_761_split__mult__neg__le,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
& ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real ) )
| ( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
& ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) ) )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ).
% split_mult_neg_le
thf(fact_762_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_763_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_764_mult__le__0__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ B ) @ zero_z580800474297136991d_real )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
& ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real ) )
| ( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
& ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_765_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_766_mult__right__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_767_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_768_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_769_mult__right__mono__neg,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A )
=> ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_770_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_771_mult__left__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_772_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_773_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_774_mult__nonpos__nonpos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_775_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_776_mult__left__mono__neg,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A )
=> ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_777_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_778_split__mult__pos__le,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
& ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) )
| ( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
& ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real ) ) )
=> ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_779_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_780_zero__le__square,axiom,
! [A: dedekind_real] : ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A @ A ) ) ).
% zero_le_square
thf(fact_781_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_782_mult__mono_H,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ C @ D )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_783_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_784_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_785_mult__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ C @ D )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_786_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_787_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_788_real__of__preal__mult,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( dedeki829075424511776921_preal @ ( times_3000655703912201937_preal @ X3 @ Y3 ) )
= ( times_2157731159493324635d_real @ ( dedeki829075424511776921_preal @ X3 ) @ ( dedeki829075424511776921_preal @ Y3 ) ) ) ).
% real_of_preal_mult
thf(fact_789_mult__less__le__imp__less,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ C @ D )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_790_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_791_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_792_mult__le__less__imp__less,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ C @ D )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_793_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_794_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_795_mult__right__le__imp__le,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_796_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_797_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_798_mult__left__le__imp__le,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_799_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_800_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_801_mult__le__cancel__left__pos,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ord_le2716243287969276982d_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_802_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_803_mult__le__cancel__left__neg,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ord_le2716243287969276982d_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_804_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_805_mult__less__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ A @ B ) )
& ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_806_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_807_mult__le__cancel__left1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ C @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ B ) )
& ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ B @ one_on6069100329679821595d_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_808_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_809_mult__le__cancel__left2,axiom,
! [C: dedekind_real,A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A ) @ C )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ A @ one_on6069100329679821595d_real ) )
& ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_810_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_811_mult__le__cancel__right1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ C @ ( times_2157731159493324635d_real @ B @ C ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ B ) )
& ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ B @ one_on6069100329679821595d_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_812_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_813_mult__le__cancel__right2,axiom,
! [A: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ C ) @ C )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ A @ one_on6069100329679821595d_real ) )
& ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_814_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_815_mult__less__cancel__left1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ C @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ B ) )
& ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ B @ one_on6069100329679821595d_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_816_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_817_mult__less__cancel__left2,axiom,
! [C: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A ) @ C )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ A @ one_on6069100329679821595d_real ) )
& ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_818_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_819_mult__less__cancel__right1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ C @ ( times_2157731159493324635d_real @ B @ C ) )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ B ) )
& ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ B @ one_on6069100329679821595d_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_820_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_821_mult__less__cancel__right2,axiom,
! [A: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A @ C ) @ C )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ A @ one_on6069100329679821595d_real ) )
& ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_822_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_823_field__le__mult__one__interval,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ! [Z: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Z )
=> ( ( ord_le2991122432403439658d_real @ Z @ one_on6069100329679821595d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ Z @ X3 ) @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ X3 @ Y3 ) ) ).
% field_le_mult_one_interval
thf(fact_824_mult__1,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ A )
= A ) ).
% mult_1
thf(fact_825_mult__1,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ A )
= A ) ).
% mult_1
thf(fact_826_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_827_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_828_mult_Oright__neutral,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ one_on6069100329679821595d_real )
= A ) ).
% mult.right_neutral
thf(fact_829_mult_Oright__neutral,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ A @ one_on9143529541772854033_preal )
= A ) ).
% mult.right_neutral
thf(fact_830_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_831_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_832_mult__cancel__left1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( C
= ( times_2157731159493324635d_real @ C @ B ) )
= ( ( C = zero_z580800474297136991d_real )
| ( B = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_left1
thf(fact_833_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_834_mult__cancel__left2,axiom,
! [C: dedekind_real,A: dedekind_real] :
( ( ( times_2157731159493324635d_real @ C @ A )
= C )
= ( ( C = zero_z580800474297136991d_real )
| ( A = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_left2
thf(fact_835_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_836_mult__cancel__right1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( C
= ( times_2157731159493324635d_real @ B @ C ) )
= ( ( C = zero_z580800474297136991d_real )
| ( B = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_right1
thf(fact_837_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_838_mult__cancel__right2,axiom,
! [A: dedekind_real,C: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ C )
= C )
= ( ( C = zero_z580800474297136991d_real )
| ( A = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_right2
thf(fact_839_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_840_preal__mult__1,axiom,
! [Z3: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ Z3 )
= Z3 ) ).
% preal_mult_1
thf(fact_841_preal__mult__assoc,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal,Z3: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ X3 @ Y3 ) @ Z3 )
= ( times_3000655703912201937_preal @ X3 @ ( times_3000655703912201937_preal @ Y3 @ Z3 ) ) ) ).
% preal_mult_assoc
thf(fact_842_preal__mult__commute,axiom,
( times_3000655703912201937_preal
= ( ^ [X4: dedekind_preal,Y2: dedekind_preal] : ( times_3000655703912201937_preal @ Y2 @ X4 ) ) ) ).
% preal_mult_commute
thf(fact_843_real__le__antisym,axiom,
! [Z3: dedekind_real,W3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Z3 @ W3 )
=> ( ( ord_le2716243287969276982d_real @ W3 @ Z3 )
=> ( Z3 = W3 ) ) ) ).
% real_le_antisym
thf(fact_844_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_845_real__le__refl,axiom,
! [W3: dedekind_real] : ( ord_le2716243287969276982d_real @ W3 @ W3 ) ).
% real_le_refl
thf(fact_846_one__reorient,axiom,
! [X3: dedekind_preal] :
( ( one_on9143529541772854033_preal = X3 )
= ( X3 = one_on9143529541772854033_preal ) ) ).
% one_reorient
thf(fact_847_one__reorient,axiom,
! [X3: dedekind_real] :
( ( one_on6069100329679821595d_real = X3 )
= ( X3 = one_on6069100329679821595d_real ) ) ).
% one_reorient
thf(fact_848_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_849_one__reorient,axiom,
! [X3: int] :
( ( one_one_int = X3 )
= ( X3 = one_one_int ) ) ).
% one_reorient
thf(fact_850_le__numeral__extra_I4_J,axiom,
ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ one_on6069100329679821595d_real ).
% le_numeral_extra(4)
thf(fact_851_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_852_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_853_zero__neq__one,axiom,
zero_z580800474297136991d_real != one_on6069100329679821595d_real ).
% zero_neq_one
thf(fact_854_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_855_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_856_less__numeral__extra_I4_J,axiom,
~ ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ one_on6069100329679821595d_real ) ).
% less_numeral_extra(4)
thf(fact_857_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_858_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_859_mult_Ocomm__neutral,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ one_on6069100329679821595d_real )
= A ) ).
% mult.comm_neutral
thf(fact_860_mult_Ocomm__neutral,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ A @ one_on9143529541772854033_preal )
= A ) ).
% mult.comm_neutral
thf(fact_861_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_862_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_863_comm__monoid__mult__class_Omult__1,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_864_comm__monoid__mult__class_Omult__1,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_865_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_866_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_867_real__zero__not__eq__one,axiom,
zero_z580800474297136991d_real != one_on6069100329679821595d_real ).
% real_zero_not_eq_one
thf(fact_868_real__mult__1,axiom,
! [Z3: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ Z3 )
= Z3 ) ).
% real_mult_1
thf(fact_869_lambda__one,axiom,
( ( ^ [X4: dedekind_real] : X4 )
= ( times_2157731159493324635d_real @ one_on6069100329679821595d_real ) ) ).
% lambda_one
thf(fact_870_lambda__one,axiom,
( ( ^ [X4: dedekind_preal] : X4 )
= ( times_3000655703912201937_preal @ one_on9143529541772854033_preal ) ) ).
% lambda_one
thf(fact_871_lambda__one,axiom,
( ( ^ [X4: nat] : X4 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_872_lambda__one,axiom,
( ( ^ [X4: int] : X4 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_873_not__one__le__zero,axiom,
~ ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ zero_z580800474297136991d_real ) ).
% not_one_le_zero
thf(fact_874_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_875_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_876_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ one_on6069100329679821595d_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_877_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_878_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_879_zero__less__one__class_Ozero__le__one,axiom,
ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ one_on6069100329679821595d_real ).
% zero_less_one_class.zero_le_one
thf(fact_880_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_881_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_882_not__one__less__zero,axiom,
~ ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ zero_z580800474297136991d_real ) ).
% not_one_less_zero
thf(fact_883_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_884_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_885_zero__less__one,axiom,
ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ one_on6069100329679821595d_real ).
% zero_less_one
thf(fact_886_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_887_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_888_less__numeral__extra_I1_J,axiom,
ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ one_on6069100329679821595d_real ).
% less_numeral_extra(1)
thf(fact_889_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_890_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_891_less__1__mult,axiom,
! [M: dedekind_real,N: dedekind_real] :
( ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ M )
=> ( ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ N )
=> ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ ( times_2157731159493324635d_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_892_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_893_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_894_real__mult__inverse__left__ex,axiom,
! [X3: dedekind_real] :
( ( X3 != zero_z580800474297136991d_real )
=> ~ ! [Y: dedekind_real] :
( ( times_2157731159493324635d_real @ Y @ X3 )
!= one_on6069100329679821595d_real ) ) ).
% real_mult_inverse_left_ex
thf(fact_895_mult__left__le__one__le,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ X3 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ Y3 )
=> ( ( ord_le2716243287969276982d_real @ Y3 @ one_on6069100329679821595d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ Y3 @ X3 ) @ X3 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_896_mult__left__le__one__le,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
=> ( ( ord_less_eq_int @ Y3 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y3 @ X3 ) @ X3 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_897_mult__right__le__one__le,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ X3 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ Y3 )
=> ( ( ord_le2716243287969276982d_real @ Y3 @ one_on6069100329679821595d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ X3 @ Y3 ) @ X3 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_898_mult__right__le__one__le,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
=> ( ( ord_less_eq_int @ Y3 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X3 @ Y3 ) @ X3 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_899_mult__le__one,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ one_on6069100329679821595d_real )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B )
=> ( ( ord_le2716243287969276982d_real @ B @ one_on6069100329679821595d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ B ) @ one_on6069100329679821595d_real ) ) ) ) ).
% mult_le_one
thf(fact_900_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_901_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_902_mult__left__le,axiom,
! [C: dedekind_real,A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ C @ one_on6069100329679821595d_real )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_903_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_904_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_905_dbl__inc__simps_I2_J,axiom,
( ( neg_nu2849763295382964967d_real @ zero_z580800474297136991d_real )
= one_on6069100329679821595d_real ) ).
% dbl_inc_simps(2)
thf(fact_906_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_907_convex__bound__lt,axiom,
! [X3: real,A: real,Y3: real,U: real,V: real] :
( ( ord_less_real @ X3 @ A )
=> ( ( ord_less_real @ Y3 @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_908_convex__bound__lt,axiom,
! [X3: dedekind_real,A: dedekind_real,Y3: dedekind_real,U: dedekind_real,V: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ A )
=> ( ( ord_le2991122432403439658d_real @ Y3 @ A )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ U )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ V )
=> ( ( ( plus_p4060926892116697567d_real @ U @ V )
= one_on6069100329679821595d_real )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ U @ X3 ) @ ( times_2157731159493324635d_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_909_convex__bound__lt,axiom,
! [X3: int,A: int,Y3: int,U: int,V: int] :
( ( ord_less_int @ X3 @ A )
=> ( ( ord_less_int @ Y3 @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_910_divide__le__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_911_divide__le__eq__1__neg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ ( divide9119111104558704680d_real @ B @ A ) @ one_on6069100329679821595d_real )
= ( ord_le2716243287969276982d_real @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_912_divide__le__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_913_divide__le__eq__1__pos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ ( divide9119111104558704680d_real @ B @ A ) @ one_on6069100329679821595d_real )
= ( ord_le2716243287969276982d_real @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_914_le__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_915_le__divide__eq__1__neg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ ( divide9119111104558704680d_real @ B @ A ) )
= ( ord_le2716243287969276982d_real @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_916_le__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_917_le__divide__eq__1__pos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ ( divide9119111104558704680d_real @ B @ A ) )
= ( ord_le2716243287969276982d_real @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_918_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_919_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_920_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_921_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_922_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_923_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_924_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_925_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_926_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_927_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_928_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_929_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_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_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_938_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_939_add__0,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ zero_z580800474297136991d_real @ A )
= A ) ).
% add_0
thf(fact_940_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_941_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_942_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y3 ) )
= ( ( X3 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_943_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y3: nat] :
( ( ( plus_plus_nat @ X3 @ Y3 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_944_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_945_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_946_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_947_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_948_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_949_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_950_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_951_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_952_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_953_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_954_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_955_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_956_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_957_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_958_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_959_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_960_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_961_double__zero__sym,axiom,
! [A: dedekind_real] :
( ( zero_z580800474297136991d_real
= ( plus_p4060926892116697567d_real @ A @ A ) )
= ( A = zero_z580800474297136991d_real ) ) ).
% double_zero_sym
thf(fact_962_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_963_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_964_add_Oright__neutral,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% add.right_neutral
thf(fact_965_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_966_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_967_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_968_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_969_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_970_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_971_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_972_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_973_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_974_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_975_division__ring__divide__zero,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ A @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% division_ring_divide_zero
thf(fact_976_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_977_divide__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ A @ C )
= ( divide9119111104558704680d_real @ B @ C ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_978_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_979_divide__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ C @ A )
= ( divide9119111104558704680d_real @ C @ B ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_980_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_981_div__by__0,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ A @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% div_by_0
thf(fact_982_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_983_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_984_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_985_divide__eq__0__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ A @ B )
= zero_z580800474297136991d_real )
= ( ( A = zero_z580800474297136991d_real )
| ( B = zero_z580800474297136991d_real ) ) ) ).
% divide_eq_0_iff
thf(fact_986_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_987_div__0,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ zero_z580800474297136991d_real @ A )
= zero_z580800474297136991d_real ) ).
% div_0
thf(fact_988_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_989_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_990_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_991_times__divide__eq__right,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ ( divide9119111104558704680d_real @ B @ C ) )
= ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_992_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_993_divide__divide__eq__right,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( divide9119111104558704680d_real @ A @ ( divide9119111104558704680d_real @ B @ C ) )
= ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_994_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_995_divide__divide__eq__left,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( divide9119111104558704680d_real @ ( divide9119111104558704680d_real @ A @ B ) @ C )
= ( divide9119111104558704680d_real @ A @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_996_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_997_times__divide__eq__left,axiom,
! [B: dedekind_real,C: dedekind_real,A: dedekind_real] :
( ( times_2157731159493324635d_real @ ( divide9119111104558704680d_real @ B @ C ) @ A )
= ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_998_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_999_div__by__1,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ A @ one_on6069100329679821595d_real )
= A ) ).
% div_by_1
thf(fact_1000_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_1001_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_1002_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_1003_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_1004_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_1005_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_1006_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_1007_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_1008_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_1009_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_1010_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_1011_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_1012_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_1013_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_1014_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_1015_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_1016_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_1017_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_1018_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_1019_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_1020_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_1021_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_1022_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_1023_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_1024_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_1025_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_1026_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_1027_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_1028_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_1029_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_1030_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_1031_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_1032_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_1033_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_1034_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_1035_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_1036_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_1037_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_1038_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_1039_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_1040_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_1041_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_1042_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_1043_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_1044_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_1045_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_1046_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_1047_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1048_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1049_nonzero__mult__div__cancel__right,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( B != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1050_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1051_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1052_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1053_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1054_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1055_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ B @ C ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1056_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1057_nonzero__mult__div__cancel__left,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1058_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1059_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1060_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1061_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1062_real__of__preal__add,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( dedeki829075424511776921_preal @ ( plus_p3173629198307831117_preal @ X3 @ Y3 ) )
= ( plus_p4060926892116697567d_real @ ( dedeki829075424511776921_preal @ X3 ) @ ( dedeki829075424511776921_preal @ Y3 ) ) ) ).
% real_of_preal_add
thf(fact_1063_real__mult__congruent2__lemma,axiom,
! [X1: dedekind_preal,Y22: dedekind_preal,X22: dedekind_preal,Y1: dedekind_preal,X3: dedekind_preal,Y3: 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 @ Y3 @ Y1 ) ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X3 @ Y22 ) @ ( times_3000655703912201937_preal @ Y3 @ X22 ) ) )
= ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X3 @ X22 ) @ ( times_3000655703912201937_preal @ Y3 @ Y22 ) ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X3 @ Y1 ) @ ( times_3000655703912201937_preal @ Y3 @ X1 ) ) ) ) ) ).
% real_mult_congruent2_lemma
thf(fact_1064_preal__add__mult__distrib2,axiom,
! [W3: dedekind_preal,X3: dedekind_preal,Y3: dedekind_preal] :
( ( times_3000655703912201937_preal @ W3 @ ( plus_p3173629198307831117_preal @ X3 @ Y3 ) )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ W3 @ X3 ) @ ( times_3000655703912201937_preal @ W3 @ Y3 ) ) ) ).
% preal_add_mult_distrib2
thf(fact_1065_preal__add__mult__distrib,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal,W3: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ X3 @ Y3 ) @ W3 )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X3 @ W3 ) @ ( times_3000655703912201937_preal @ Y3 @ W3 ) ) ) ).
% preal_add_mult_distrib
thf(fact_1066_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_1067_preal__eq__le__imp__le,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal,D: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ A @ B )
= ( plus_p3173629198307831117_preal @ C @ D ) )
=> ( ( ord_le5604041210740703414_preal @ C @ A )
=> ( ord_le5604041210740703414_preal @ B @ D ) ) ) ).
% preal_eq_le_imp_le
thf(fact_1068_real__add__mult__distrib,axiom,
! [Z1: dedekind_real,Z22: dedekind_real,W3: dedekind_real] :
( ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ Z1 @ Z22 ) @ W3 )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ Z1 @ W3 ) @ ( times_2157731159493324635d_real @ Z22 @ W3 ) ) ) ).
% real_add_mult_distrib
thf(fact_1069_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_1070_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_1071_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_1072_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_1073_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_1074_real__add__left__mono,axiom,
! [X3: dedekind_real,Y3: dedekind_real,Z3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ Z3 @ X3 ) @ ( plus_p4060926892116697567d_real @ Z3 @ Y3 ) ) ) ).
% real_add_left_mono
thf(fact_1075_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_1076_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_1077_preal__trans__lemma,axiom,
! [X3: dedekind_preal,Y1: dedekind_preal,X1: dedekind_preal,Y3: dedekind_preal,Y22: dedekind_preal,X22: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ X3 @ Y1 )
= ( plus_p3173629198307831117_preal @ X1 @ Y3 ) )
=> ( ( ( plus_p3173629198307831117_preal @ X3 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y3 ) )
=> ( ( plus_p3173629198307831117_preal @ X1 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y1 ) ) ) ) ).
% preal_trans_lemma
thf(fact_1078_preal__add__commute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [X4: dedekind_preal,Y2: dedekind_preal] : ( plus_p3173629198307831117_preal @ Y2 @ X4 ) ) ) ).
% preal_add_commute
thf(fact_1079_preal__add__assoc,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal,Z3: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ X3 @ Y3 ) @ Z3 )
= ( plus_p3173629198307831117_preal @ X3 @ ( plus_p3173629198307831117_preal @ Y3 @ Z3 ) ) ) ).
% preal_add_assoc
thf(fact_1080_real__of__preal__trichotomy,axiom,
! [X3: dedekind_real] :
? [M4: dedekind_preal] :
( ( X3
= ( dedeki829075424511776921_preal @ M4 ) )
| ( X3 = zero_z580800474297136991d_real )
| ( X3
= ( uminus7714077491378687647d_real @ ( dedeki829075424511776921_preal @ M4 ) ) ) ) ).
% real_of_preal_trichotomy
thf(fact_1081_real__less__sum__gt__zero,axiom,
! [W3: dedekind_real,S: dedekind_real] :
( ( ord_le2991122432403439658d_real @ W3 @ S )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ S @ ( uminus7714077491378687647d_real @ W3 ) ) ) ) ).
% real_less_sum_gt_zero
thf(fact_1082_real__sum__gt__zero__less,axiom,
! [S: dedekind_real,W3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ S @ ( uminus7714077491378687647d_real @ W3 ) ) )
=> ( ord_le2991122432403439658d_real @ W3 @ S ) ) ).
% real_sum_gt_zero_less
thf(fact_1083_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_1084_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_1085_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_1086_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1087_num__induct,axiom,
! [P: num > $o,X3: num] :
( ( P @ one )
=> ( ! [X: num] :
( ( P @ X )
=> ( P @ ( inc @ X ) ) )
=> ( P @ X3 ) ) ) ).
% num_induct
thf(fact_1088_add__One,axiom,
! [X3: num] :
( ( plus_plus_num @ X3 @ one )
= ( inc @ X3 ) ) ).
% add_One
thf(fact_1089_add__inc,axiom,
! [X3: num,Y3: num] :
( ( plus_plus_num @ X3 @ ( inc @ Y3 ) )
= ( inc @ ( plus_plus_num @ X3 @ Y3 ) ) ) ).
% add_inc
thf(fact_1090_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1091_le__num__One__iff,axiom,
! [X3: num] :
( ( ord_less_eq_num @ X3 @ one )
= ( X3 = one ) ) ).
% le_num_One_iff
thf(fact_1092_mult__inc,axiom,
! [X3: num,Y3: num] :
( ( times_times_num @ X3 @ ( inc @ Y3 ) )
= ( plus_plus_num @ ( times_times_num @ X3 @ Y3 ) @ X3 ) ) ).
% mult_inc
thf(fact_1093_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1094_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1095_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1096_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1097_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1098_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1099_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1100_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1101_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1102_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1103_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1104_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1105_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1106_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1107_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1108_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1109_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1110_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1111_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1112_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1113_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1114_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1115_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1116_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1117_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1118_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I2: nat,J2: nat] :
( ( ( ord_less_nat @ J2 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_div
thf(fact_1119_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1120_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1121_div__less__iff__less__mult,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1122_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1123_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1124_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1125_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1126_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1127_div__mult2__eq,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q3 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q3 ) ) ).
% div_mult2_eq
thf(fact_1128_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_1129_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1130_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1131_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1132_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1133_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1134_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1135_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1136_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1137_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_1138_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1139_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1140_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1141_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_1142_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1143_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1144_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_1145_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_1146_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1147_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1148_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_1149_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_1150_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1151_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1152_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1153_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1154_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1155_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1156_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1157_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1158_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1159_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1160_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1161_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1162_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1163_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1164_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1165_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1166_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1167_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1168_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_1169_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_1170_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1171_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_1172_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_1173_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1174_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_1175_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_1176_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1177_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_1178_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_1179_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_1180_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_1181_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_1182_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M5: nat] :
( ( P @ X3 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1183_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1184_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1185_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1186_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1187_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1188_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1189_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1190_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1191_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1192_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_1193_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_1194_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_1195_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1196_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1197_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1198_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1199_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1200_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_1201_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_1202_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1203_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_1204_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_1205_zle__add1__eq__le,axiom,
! [W3: int,Z3: int] :
( ( ord_less_int @ W3 @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ord_less_eq_int @ W3 @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_1206_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_1207_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_1208_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_1209_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_1210_zless__imp__add1__zle,axiom,
! [W3: int,Z3: int] :
( ( ord_less_int @ W3 @ Z3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W3 @ one_one_int ) @ Z3 ) ) ).
% zless_imp_add1_zle
thf(fact_1211_add1__zle__eq,axiom,
! [W3: int,Z3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W3 @ one_one_int ) @ Z3 )
= ( ord_less_int @ W3 @ Z3 ) ) ).
% add1_zle_eq
thf(fact_1212_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1213_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_1214_zless__add1__eq,axiom,
! [W3: int,Z3: int] :
( ( ord_less_int @ W3 @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ( ord_less_int @ W3 @ Z3 )
| ( W3 = Z3 ) ) ) ).
% zless_add1_eq
thf(fact_1215_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_1216_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_1217_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1218_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W3: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W3 )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W3 ) @ ( times_times_int @ Z22 @ W3 ) ) ) ).
% int_distrib(1)
thf(fact_1219_int__distrib_I2_J,axiom,
! [W3: int,Z1: int,Z22: int] :
( ( times_times_int @ W3 @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W3 @ Z1 ) @ ( times_times_int @ W3 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1220_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X: int] :
( ( P @ X )
=> ( P @ ( plus_plus_int @ X @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1221_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I2: int,J2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
& ( ord_less_int @ J2 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I2: int,J2: int] :
( ( ( ord_less_int @ K @ J2 )
& ( ord_less_eq_int @ J2 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_zdiv
thf(fact_1222_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_1223_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_1224_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_1225_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_1226_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_1227_zdiv__mono1__neg,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1228_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_1229_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_1230_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_1231_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_1232_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_1233_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_1234_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_1235_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_1236_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_1237_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_1238_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_1239_verit__less__mono__div__int2,axiom,
! [A2: int,B5: int,N: int] :
( ( ord_less_eq_int @ A2 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1240_set__bit__greater__eq,axiom,
! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% set_bit_greater_eq
thf(fact_1241_unset__bit__less__eq,axiom,
! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% unset_bit_less_eq
thf(fact_1242_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_1243_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1244_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1245_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1246_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_1247_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1248_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_1249_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_1250_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1251_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1252_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1253_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1254_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1255_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1256_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_1257_real__average__minus__second,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
= ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% real_average_minus_second
thf(fact_1258_real__average__minus__first,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
= ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% real_average_minus_first
thf(fact_1259_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1260_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1261_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1262_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1263_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1264_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1265_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
% Conjectures (1)
thf(conj_0,conjecture,
? [X2: dedekind_real] :
( ( member_Dedekind_real @ X2 @ p )
& ( ord_le2991122432403439658d_real @ y @ X2 ) ) ).
%------------------------------------------------------------------------------