TPTP Problem File: SLH0199^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Dedekind_Real/0000_Dedekind_Real/prob_00120_002964__5583446_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1352 ( 518 unt; 83 typ; 0 def)
% Number of atoms : 3668 (1140 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9624 ( 393 ~; 115 |; 163 &;7361 @)
% ( 0 <=>;1592 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 370 ( 370 >; 0 *; 0 +; 0 <<)
% Number of symbols : 74 ( 73 usr; 12 con; 0-3 aty)
% Number of variables : 3465 ( 205 ^;3134 !; 126 ?;3465 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:15:11.544
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
set_set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Dedekind____Real__Opreal,type,
dedekind_preal: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (73)
thf(sy_c_Dedekind__Real_Oadd__set,type,
dedekind_add_set: set_rat > set_rat > set_rat ).
thf(sy_c_Dedekind__Real_Ocut,type,
dedekind_cut: set_rat > $o ).
thf(sy_c_Dedekind__Real_Omult__set,type,
dedekind_mult_set: set_rat > set_rat > set_rat ).
thf(sy_c_Dedekind__Real_Opreal_OAbs__preal,type,
dedekind_Abs_preal: set_rat > dedekind_preal ).
thf(sy_c_Dedekind__Real_Opreal_ORep__preal,type,
dedekind_Rep_preal: dedekind_preal > set_rat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
one_one_rat: rat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Dedekind____Real__Opreal,type,
plus_p3173629198307831117_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
plus_plus_rat: rat > rat > rat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Dedekind____Real__Opreal,type,
times_3000655703912201937_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
times_times_rat: rat > rat > rat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
zero_zero_rat: rat ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
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_001t__Rat__Orat,type,
neg_numeral_dbl_rat: rat > rat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
neg_nu5219082963157363817nc_rat: rat > rat ).
thf(sy_c_Num_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_Orderings_Obot__class_Obot_001_062_It__Rat__Orat_M_Eo_J,type,
bot_bot_rat_o: rat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Rat__Orat_J_M_Eo_J,type,
bot_bot_set_rat_o: set_rat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
bot_bot_set_rat: set_rat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
bot_bot_set_set_rat: set_set_rat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Dedekind____Real__Opreal,type,
ord_le5708704896291381698_preal: dedekind_preal > dedekind_preal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
ord_less_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
ord_less_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
ord_less_set_set_rat: set_set_rat > set_set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Dedekind____Real__Opreal,type,
ord_le5604041210740703414_preal: dedekind_preal > dedekind_preal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
ord_less_eq_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
ord_less_eq_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
ord_le513522071413781156et_rat: set_set_rat > set_set_rat > $o ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Rat__Orat,type,
field_2639924705303425560at_rat: rat > rat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
divide_divide_rat: rat > rat > rat ).
thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
collect_rat: ( rat > $o ) > set_rat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Rat__Orat_J,type,
collect_set_rat: ( set_rat > $o ) > set_set_rat ).
thf(sy_c_Set_Ois__empty_001t__Rat__Orat,type,
is_empty_rat: set_rat > $o ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Int__Oint,type,
set_or1207661135979820486an_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Num__Onum,type,
set_or6990855429499425204an_num: num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Rat__Orat,type,
set_or575021546402375026an_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_It__Rat__Orat_J,type,
set_or6174011595382531368et_rat: set_rat > set_set_rat ).
thf(sy_c_Typedef_Otype__definition_001t__Dedekind____Real__Opreal_001t__Set__Oset_It__Rat__Orat_J,type,
type_d4900610042970207096et_rat: ( dedekind_preal > set_rat ) > ( set_rat > dedekind_preal ) > set_set_rat > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > set_rat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Rat__Orat_J,type,
member_set_rat: set_rat > set_set_rat > $o ).
thf(sy_v_A,type,
a: set_rat ).
thf(sy_v_y,type,
y: rat ).
% Relevant facts (1266)
thf(fact_0_preal__Ex__mem,axiom,
! [A: set_rat] :
( ( dedekind_cut @ A )
=> ? [X: rat] : ( member_rat @ X @ A ) ) ).
% preal_Ex_mem
thf(fact_1_Abs__preal__inject,axiom,
! [X2: set_rat,Y: set_rat] :
( ( member_set_rat @ X2 @ ( collect_set_rat @ dedekind_cut ) )
=> ( ( member_set_rat @ Y @ ( collect_set_rat @ dedekind_cut ) )
=> ( ( ( dedekind_Abs_preal @ X2 )
= ( dedekind_Abs_preal @ Y ) )
= ( X2 = Y ) ) ) ) ).
% Abs_preal_inject
thf(fact_2_minf_I7_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z )
=> ~ ( ord_less_rat @ T @ X3 ) ) ).
% minf(7)
thf(fact_3_minf_I7_J,axiom,
! [T: num] :
? [Z: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z )
=> ~ ( ord_less_num @ T @ X3 ) ) ).
% minf(7)
thf(fact_4_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ~ ( ord_less_nat @ T @ X3 ) ) ).
% minf(7)
thf(fact_5_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z )
=> ~ ( ord_less_int @ T @ X3 ) ) ).
% minf(7)
thf(fact_6_minf_I5_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z )
=> ( ord_less_rat @ X3 @ T ) ) ).
% minf(5)
thf(fact_7_minf_I5_J,axiom,
! [T: num] :
? [Z: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z )
=> ( ord_less_num @ X3 @ T ) ) ).
% minf(5)
thf(fact_8_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( ord_less_nat @ X3 @ T ) ) ).
% minf(5)
thf(fact_9_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z )
=> ( ord_less_int @ X3 @ T ) ) ).
% minf(5)
thf(fact_10_minf_I4_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_11_minf_I4_J,axiom,
! [T: num] :
? [Z: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_12_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_13_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_14_minf_I3_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_15_minf_I3_J,axiom,
! [T: num] :
? [Z: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_16_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_17_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_18_minf_I2_J,axiom,
! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z2: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_19_minf_I2_J,axiom,
! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z2: num] :
! [X: num] :
( ( ord_less_num @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: num] :
! [X: num] :
( ( ord_less_num @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_20_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] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_21_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] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_22_minf_I1_J,axiom,
! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z2: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_23_minf_I1_J,axiom,
! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z2: num] :
! [X: num] :
( ( ord_less_num @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: num] :
! [X: num] :
( ( ord_less_num @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_24_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] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_25_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] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_26_pinf_I7_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ( ord_less_rat @ T @ X3 ) ) ).
% pinf(7)
thf(fact_27_pinf_I7_J,axiom,
! [T: num] :
? [Z: num] :
! [X3: num] :
( ( ord_less_num @ Z @ X3 )
=> ( ord_less_num @ T @ X3 ) ) ).
% pinf(7)
thf(fact_28_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( ord_less_nat @ T @ X3 ) ) ).
% pinf(7)
thf(fact_29_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X3: int] :
( ( ord_less_int @ Z @ X3 )
=> ( ord_less_int @ T @ X3 ) ) ).
% pinf(7)
thf(fact_30_pinf_I5_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ~ ( ord_less_rat @ X3 @ T ) ) ).
% pinf(5)
thf(fact_31_pinf_I5_J,axiom,
! [T: num] :
? [Z: num] :
! [X3: num] :
( ( ord_less_num @ Z @ X3 )
=> ~ ( ord_less_num @ X3 @ T ) ) ).
% pinf(5)
thf(fact_32_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ~ ( ord_less_nat @ X3 @ T ) ) ).
% pinf(5)
thf(fact_33_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X3: int] :
( ( ord_less_int @ Z @ X3 )
=> ~ ( ord_less_int @ X3 @ T ) ) ).
% pinf(5)
thf(fact_34_pinf_I4_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_35_pinf_I4_J,axiom,
! [T: num] :
? [Z: num] :
! [X3: num] :
( ( ord_less_num @ Z @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_36_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_37_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X3: int] :
( ( ord_less_int @ Z @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_38_Dedekind__Real_OAbs__preal__induct,axiom,
! [P: dedekind_preal > $o,X2: dedekind_preal] :
( ! [X: set_rat] :
( ( dedekind_cut @ X )
=> ( P @ ( dedekind_Abs_preal @ X ) ) )
=> ( P @ X2 ) ) ).
% Dedekind_Real.Abs_preal_induct
thf(fact_39_preal_OAbs__preal__induct,axiom,
! [P: dedekind_preal > $o,X2: dedekind_preal] :
( ! [Y2: set_rat] :
( ( member_set_rat @ Y2 @ ( collect_set_rat @ dedekind_cut ) )
=> ( P @ ( dedekind_Abs_preal @ Y2 ) ) )
=> ( P @ X2 ) ) ).
% preal.Abs_preal_induct
thf(fact_40_Abs__preal__cases,axiom,
! [X2: dedekind_preal] :
~ ! [Y2: set_rat] :
( ( X2
= ( dedekind_Abs_preal @ Y2 ) )
=> ~ ( member_set_rat @ Y2 @ ( collect_set_rat @ dedekind_cut ) ) ) ).
% Abs_preal_cases
thf(fact_41_pinf_I1_J,axiom,
! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z2: rat] :
! [X: rat] :
( ( ord_less_rat @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: rat] :
! [X: rat] :
( ( ord_less_rat @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_42_pinf_I1_J,axiom,
! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z2: num] :
! [X: num] :
( ( ord_less_num @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: num] :
! [X: num] :
( ( ord_less_num @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: num] :
! [X3: num] :
( ( ord_less_num @ Z @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_43_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] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_44_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] :
! [X3: int] :
( ( ord_less_int @ Z @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_45_pinf_I2_J,axiom,
! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z2: rat] :
! [X: rat] :
( ( ord_less_rat @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: rat] :
! [X: rat] :
( ( ord_less_rat @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_46_pinf_I2_J,axiom,
! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z2: num] :
! [X: num] :
( ( ord_less_num @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: num] :
! [X: num] :
( ( ord_less_num @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: num] :
! [X3: num] :
( ( ord_less_num @ Z @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_47_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] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_48_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] :
! [X3: int] :
( ( ord_less_int @ Z @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_49_pinf_I3_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_50_pinf_I3_J,axiom,
! [T: num] :
? [Z: num] :
! [X3: num] :
( ( ord_less_num @ Z @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_51_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_52_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X3: int] :
( ( ord_less_int @ Z @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_53_Abs__preal__inverse,axiom,
! [Y: set_rat] :
( ( member_set_rat @ Y @ ( collect_set_rat @ dedekind_cut ) )
=> ( ( dedekind_Rep_preal @ ( dedekind_Abs_preal @ Y ) )
= Y ) ) ).
% Abs_preal_inverse
thf(fact_54_order__less__imp__not__less,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ~ ( ord_less_set_rat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_55_order__less__imp__not__less,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ~ ( ord_less_rat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_56_order__less__imp__not__less,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ~ ( ord_less_num @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_57_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_58_order__less__imp__not__less,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_59_order__less__imp__not__eq2,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_60_order__less__imp__not__eq2,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_61_order__less__imp__not__eq2,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_62_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_63_order__less__imp__not__eq2,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_64_order__less__imp__not__eq,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_65_order__less__imp__not__eq,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_66_order__less__imp__not__eq,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_67_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_68_order__less__imp__not__eq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_69_linorder__less__linear,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_rat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_70_linorder__less__linear,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_num @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_71_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_72_linorder__less__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_int @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_73_order__less__imp__triv,axiom,
! [X2: set_rat,Y: set_rat,P: $o] :
( ( ord_less_set_rat @ X2 @ Y )
=> ( ( ord_less_set_rat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_74_order__less__imp__triv,axiom,
! [X2: rat,Y: rat,P: $o] :
( ( ord_less_rat @ X2 @ Y )
=> ( ( ord_less_rat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_75_order__less__imp__triv,axiom,
! [X2: num,Y: num,P: $o] :
( ( ord_less_num @ X2 @ Y )
=> ( ( ord_less_num @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_76_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_77_order__less__imp__triv,axiom,
! [X2: int,Y: int,P: $o] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_78_order__less__not__sym,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ~ ( ord_less_set_rat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_79_order__less__not__sym,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ~ ( ord_less_rat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_80_order__less__not__sym,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ~ ( ord_less_num @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_81_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_82_order__less__not__sym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_83_order__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_84_order__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_85_order__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_86_order__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_87_order__less__subst2,axiom,
! [A2: num,B: num,F: num > rat,C: rat] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_88_order__less__subst2,axiom,
! [A2: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_89_order__less__subst2,axiom,
! [A2: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_90_order__less__subst2,axiom,
! [A2: num,B: num,F: num > int,C: int] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_91_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_92_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_93_order__less__subst1,axiom,
! [A2: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_94_order__less__subst1,axiom,
! [A2: rat,F: num > rat,B: num,C: num] :
( ( ord_less_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_95_order__less__subst1,axiom,
! [A2: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_96_order__less__subst1,axiom,
! [A2: rat,F: int > rat,B: int,C: int] :
( ( ord_less_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_97_order__less__subst1,axiom,
! [A2: num,F: rat > num,B: rat,C: rat] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_98_order__less__subst1,axiom,
! [A2: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_99_order__less__subst1,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_100_order__less__subst1,axiom,
! [A2: num,F: int > num,B: int,C: int] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_101_order__less__subst1,axiom,
! [A2: nat,F: rat > nat,B: rat,C: rat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_102_order__less__subst1,axiom,
! [A2: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_103_order__less__irrefl,axiom,
! [X2: set_rat] :
~ ( ord_less_set_rat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_104_order__less__irrefl,axiom,
! [X2: rat] :
~ ( ord_less_rat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_105_order__less__irrefl,axiom,
! [X2: num] :
~ ( ord_less_num @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_106_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_107_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_108_ord__less__eq__subst,axiom,
! [A2: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_109_ord__less__eq__subst,axiom,
! [A2: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_110_ord__less__eq__subst,axiom,
! [A2: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_111_ord__less__eq__subst,axiom,
! [A2: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_112_ord__less__eq__subst,axiom,
! [A2: num,B: num,F: num > rat,C: rat] :
( ( ord_less_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_113_ord__less__eq__subst,axiom,
! [A2: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_114_ord__less__eq__subst,axiom,
! [A2: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_115_ord__less__eq__subst,axiom,
! [A2: num,B: num,F: num > int,C: int] :
( ( ord_less_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_116_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_117_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_118_ord__eq__less__subst,axiom,
! [A2: rat,F: rat > rat,B: rat,C: rat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_119_ord__eq__less__subst,axiom,
! [A2: num,F: rat > num,B: rat,C: rat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_120_ord__eq__less__subst,axiom,
! [A2: nat,F: rat > nat,B: rat,C: rat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_121_ord__eq__less__subst,axiom,
! [A2: int,F: rat > int,B: rat,C: rat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_122_ord__eq__less__subst,axiom,
! [A2: rat,F: num > rat,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_123_ord__eq__less__subst,axiom,
! [A2: num,F: num > num,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_124_ord__eq__less__subst,axiom,
! [A2: nat,F: num > nat,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_125_ord__eq__less__subst,axiom,
! [A2: int,F: num > int,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_126_ord__eq__less__subst,axiom,
! [A2: rat,F: nat > rat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_127_ord__eq__less__subst,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_128_preal__less__def,axiom,
( ord_le5708704896291381698_preal
= ( ^ [R: dedekind_preal,S: dedekind_preal] : ( ord_less_set_rat @ ( dedekind_Rep_preal @ R ) @ ( dedekind_Rep_preal @ S ) ) ) ) ).
% preal_less_def
thf(fact_129_Rep__preal__inject,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ( dedekind_Rep_preal @ X2 )
= ( dedekind_Rep_preal @ Y ) )
= ( X2 = Y ) ) ).
% Rep_preal_inject
thf(fact_130_Rep__preal,axiom,
! [X2: dedekind_preal] : ( member_set_rat @ ( dedekind_Rep_preal @ X2 ) @ ( collect_set_rat @ dedekind_cut ) ) ).
% Rep_preal
thf(fact_131_Rep__preal__cases,axiom,
! [Y: set_rat] :
( ( member_set_rat @ Y @ ( collect_set_rat @ dedekind_cut ) )
=> ~ ! [X: dedekind_preal] :
( Y
!= ( dedekind_Rep_preal @ X ) ) ) ).
% Rep_preal_cases
thf(fact_132_Rep__preal__induct,axiom,
! [Y: set_rat,P: set_rat > $o] :
( ( member_set_rat @ Y @ ( collect_set_rat @ dedekind_cut ) )
=> ( ! [X: dedekind_preal] : ( P @ ( dedekind_Rep_preal @ X ) )
=> ( P @ Y ) ) ) ).
% Rep_preal_induct
thf(fact_133_cut__Rep__preal,axiom,
! [X2: dedekind_preal] : ( dedekind_cut @ ( dedekind_Rep_preal @ X2 ) ) ).
% cut_Rep_preal
thf(fact_134_Rep__preal__inverse,axiom,
! [X2: dedekind_preal] :
( ( dedekind_Abs_preal @ ( dedekind_Rep_preal @ X2 ) )
= X2 ) ).
% Rep_preal_inverse
thf(fact_135_lt__ex,axiom,
! [X2: rat] :
? [Y2: rat] : ( ord_less_rat @ Y2 @ X2 ) ).
% lt_ex
thf(fact_136_lt__ex,axiom,
! [X2: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X2 ) ).
% lt_ex
thf(fact_137_gt__ex,axiom,
! [X2: rat] :
? [X_1: rat] : ( ord_less_rat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_138_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_139_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_140_dense,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ? [Z: rat] :
( ( ord_less_rat @ X2 @ Z )
& ( ord_less_rat @ Z @ Y ) ) ) ).
% dense
thf(fact_141_less__imp__neq,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_142_less__imp__neq,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_143_less__imp__neq,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_144_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_145_less__imp__neq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_146_order_Oasym,axiom,
! [A2: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A2 @ B )
=> ~ ( ord_less_set_rat @ B @ A2 ) ) ).
% order.asym
thf(fact_147_order_Oasym,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ B )
=> ~ ( ord_less_rat @ B @ A2 ) ) ).
% order.asym
thf(fact_148_order_Oasym,axiom,
! [A2: num,B: num] :
( ( ord_less_num @ A2 @ B )
=> ~ ( ord_less_num @ B @ A2 ) ) ).
% order.asym
thf(fact_149_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_150_order_Oasym,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order.asym
thf(fact_151_ord__eq__less__trans,axiom,
! [A2: set_rat,B: set_rat,C: set_rat] :
( ( A2 = B )
=> ( ( ord_less_set_rat @ B @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_152_ord__eq__less__trans,axiom,
! [A2: rat,B: rat,C: rat] :
( ( A2 = B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_153_ord__eq__less__trans,axiom,
! [A2: num,B: num,C: num] :
( ( A2 = B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_154_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_155_ord__eq__less__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_156_ord__less__eq__trans,axiom,
! [A2: set_rat,B: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_157_ord__less__eq__trans,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_158_ord__less__eq__trans,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_159_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_160_ord__less__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_161_mem__Collect__eq,axiom,
! [A2: rat,P: rat > $o] :
( ( member_rat @ A2 @ ( collect_rat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_162_mem__Collect__eq,axiom,
! [A2: set_rat,P: set_rat > $o] :
( ( member_set_rat @ A2 @ ( collect_set_rat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_163_Collect__mem__eq,axiom,
! [A: set_rat] :
( ( collect_rat
@ ^ [X4: rat] : ( member_rat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_164_Collect__mem__eq,axiom,
! [A: set_set_rat] :
( ( collect_set_rat
@ ^ [X4: set_rat] : ( member_set_rat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_165_Collect__cong,axiom,
! [P: set_rat > $o,Q: set_rat > $o] :
( ! [X: set_rat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_set_rat @ P )
= ( collect_set_rat @ Q ) ) ) ).
% Collect_cong
thf(fact_166_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X )
=> ( P @ Y3 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_167_antisym__conv3,axiom,
! [Y: rat,X2: rat] :
( ~ ( ord_less_rat @ Y @ X2 )
=> ( ( ~ ( ord_less_rat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_168_antisym__conv3,axiom,
! [Y: num,X2: num] :
( ~ ( ord_less_num @ Y @ X2 )
=> ( ( ~ ( ord_less_num @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_169_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_170_antisym__conv3,axiom,
! [Y: int,X2: int] :
( ~ ( ord_less_int @ Y @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_171_linorder__cases,axiom,
! [X2: rat,Y: rat] :
( ~ ( ord_less_rat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_rat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_172_linorder__cases,axiom,
! [X2: num,Y: num] :
( ~ ( ord_less_num @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_num @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_173_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_174_linorder__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_175_dual__order_Oasym,axiom,
! [B: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B @ A2 )
=> ~ ( ord_less_set_rat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_176_dual__order_Oasym,axiom,
! [B: rat,A2: rat] :
( ( ord_less_rat @ B @ A2 )
=> ~ ( ord_less_rat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_177_dual__order_Oasym,axiom,
! [B: num,A2: num] :
( ( ord_less_num @ B @ A2 )
=> ~ ( ord_less_num @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_178_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_179_dual__order_Oasym,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ~ ( ord_less_int @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_180_dual__order_Oirrefl,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_181_dual__order_Oirrefl,axiom,
! [A2: rat] :
~ ( ord_less_rat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_182_dual__order_Oirrefl,axiom,
! [A2: num] :
~ ( ord_less_num @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_183_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_184_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_185_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [N: nat] :
( ( P4 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P4 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_186_linorder__less__wlog,axiom,
! [P: rat > rat > $o,A2: rat,B: rat] :
( ! [A3: rat,B2: rat] :
( ( ord_less_rat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: rat] : ( P @ A3 @ A3 )
=> ( ! [A3: rat,B2: rat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_187_linorder__less__wlog,axiom,
! [P: num > num > $o,A2: num,B: num] :
( ! [A3: num,B2: num] :
( ( ord_less_num @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: num] : ( P @ A3 @ A3 )
=> ( ! [A3: num,B2: num] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_188_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: 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 @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_189_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: 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 @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_190_order_Ostrict__trans,axiom,
! [A2: set_rat,B: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ B )
=> ( ( ord_less_set_rat @ B @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_191_order_Ostrict__trans,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_192_order_Ostrict__trans,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_193_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_194_order_Ostrict__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_195_not__less__iff__gr__or__eq,axiom,
! [X2: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X2 @ Y ) )
= ( ( ord_less_rat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_196_not__less__iff__gr__or__eq,axiom,
! [X2: num,Y: num] :
( ( ~ ( ord_less_num @ X2 @ Y ) )
= ( ( ord_less_num @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_197_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_198_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( ( ord_less_int @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_199_dual__order_Ostrict__trans,axiom,
! [B: set_rat,A2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ B @ A2 )
=> ( ( ord_less_set_rat @ C @ B )
=> ( ord_less_set_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_200_dual__order_Ostrict__trans,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B @ A2 )
=> ( ( ord_less_rat @ C @ B )
=> ( ord_less_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_201_dual__order_Ostrict__trans,axiom,
! [B: num,A2: num,C: num] :
( ( ord_less_num @ B @ A2 )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_202_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_203_dual__order_Ostrict__trans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_204_order_Ostrict__implies__not__eq,axiom,
! [A2: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_205_order_Ostrict__implies__not__eq,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_206_order_Ostrict__implies__not__eq,axiom,
! [A2: num,B: num] :
( ( ord_less_num @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_207_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_208_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_209_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_210_dual__order_Ostrict__implies__not__eq,axiom,
! [B: rat,A2: rat] :
( ( ord_less_rat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_211_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A2: num] :
( ( ord_less_num @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_212_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_213_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_214_linorder__neqE,axiom,
! [X2: rat,Y: rat] :
( ( X2 != Y )
=> ( ~ ( ord_less_rat @ X2 @ Y )
=> ( ord_less_rat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_215_linorder__neqE,axiom,
! [X2: num,Y: num] :
( ( X2 != Y )
=> ( ~ ( ord_less_num @ X2 @ Y )
=> ( ord_less_num @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_216_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_217_linorder__neqE,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_218_order__less__asym,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ~ ( ord_less_set_rat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_219_order__less__asym,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ~ ( ord_less_rat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_220_order__less__asym,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ~ ( ord_less_num @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_221_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_222_order__less__asym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_223_linorder__neq__iff,axiom,
! [X2: rat,Y: rat] :
( ( X2 != Y )
= ( ( ord_less_rat @ X2 @ Y )
| ( ord_less_rat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_224_linorder__neq__iff,axiom,
! [X2: num,Y: num] :
( ( X2 != Y )
= ( ( ord_less_num @ X2 @ Y )
| ( ord_less_num @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_225_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_226_linorder__neq__iff,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
= ( ( ord_less_int @ X2 @ Y )
| ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_227_order__less__asym_H,axiom,
! [A2: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A2 @ B )
=> ~ ( ord_less_set_rat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_228_order__less__asym_H,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ B )
=> ~ ( ord_less_rat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_229_order__less__asym_H,axiom,
! [A2: num,B: num] :
( ( ord_less_num @ A2 @ B )
=> ~ ( ord_less_num @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_230_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_231_order__less__asym_H,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_232_order__less__trans,axiom,
! [X2: set_rat,Y: set_rat,Z3: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ( ( ord_less_set_rat @ Y @ Z3 )
=> ( ord_less_set_rat @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_233_order__less__trans,axiom,
! [X2: rat,Y: rat,Z3: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ( ( ord_less_rat @ Y @ Z3 )
=> ( ord_less_rat @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_234_order__less__trans,axiom,
! [X2: num,Y: num,Z3: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ( ord_less_num @ Y @ Z3 )
=> ( ord_less_num @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_235_order__less__trans,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_236_order__less__trans,axiom,
! [X2: int,Y: int,Z3: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_237_type__definition__preal,axiom,
type_d4900610042970207096et_rat @ dedekind_Rep_preal @ dedekind_Abs_preal @ ( collect_set_rat @ dedekind_cut ) ).
% type_definition_preal
thf(fact_238_verit__comp__simplify1_I1_J,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_239_verit__comp__simplify1_I1_J,axiom,
! [A2: rat] :
~ ( ord_less_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_240_verit__comp__simplify1_I1_J,axiom,
! [A2: num] :
~ ( ord_less_num @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_241_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_242_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_243_linordered__field__no__lb,axiom,
! [X3: rat] :
? [Y2: rat] : ( ord_less_rat @ Y2 @ X3 ) ).
% linordered_field_no_lb
thf(fact_244_linordered__field__no__ub,axiom,
! [X3: rat] :
? [X_1: rat] : ( ord_less_rat @ X3 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_245_linorder__neqE__linordered__idom,axiom,
! [X2: rat,Y: rat] :
( ( X2 != Y )
=> ( ~ ( ord_less_rat @ X2 @ Y )
=> ( ord_less_rat @ Y @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_246_linorder__neqE__linordered__idom,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_247_preal__nonempty,axiom,
! [A: set_rat] :
( ( dedekind_cut @ A )
=> ? [X: rat] :
( ( member_rat @ X @ A )
& ( ord_less_rat @ zero_zero_rat @ X ) ) ) ).
% preal_nonempty
thf(fact_248_preal__exists__bound,axiom,
! [A: set_rat] :
( ( dedekind_cut @ A )
=> ? [X: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
& ~ ( member_rat @ X @ A ) ) ) ).
% preal_exists_bound
thf(fact_249_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_250_field__lbound__gt__zero,axiom,
! [D1: rat,D2: rat] :
( ( ord_less_rat @ zero_zero_rat @ D1 )
=> ( ( ord_less_rat @ zero_zero_rat @ D2 )
=> ? [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
& ( ord_less_rat @ E @ D1 )
& ( ord_less_rat @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_251_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_252_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_253_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_254_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_255_less__numeral__extra_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% less_numeral_extra(3)
thf(fact_256_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_257_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_258_type__definition_ORep,axiom,
! [Rep: dedekind_preal > set_rat,Abs: set_rat > dedekind_preal,A: set_set_rat,X2: dedekind_preal] :
( ( type_d4900610042970207096et_rat @ Rep @ Abs @ A )
=> ( member_set_rat @ ( Rep @ X2 ) @ A ) ) ).
% type_definition.Rep
thf(fact_259_type__definition_Ointro,axiom,
! [Rep: dedekind_preal > set_rat,A: set_set_rat,Abs: set_rat > dedekind_preal] :
( ! [X: dedekind_preal] : ( member_set_rat @ ( Rep @ X ) @ A )
=> ( ! [X: dedekind_preal] :
( ( Abs @ ( Rep @ X ) )
= X )
=> ( ! [Y2: set_rat] :
( ( member_set_rat @ Y2 @ A )
=> ( ( Rep @ ( Abs @ Y2 ) )
= Y2 ) )
=> ( type_d4900610042970207096et_rat @ Rep @ Abs @ A ) ) ) ) ).
% type_definition.intro
thf(fact_260_zero__reorient,axiom,
! [X2: rat] :
( ( zero_zero_rat = X2 )
= ( X2 = zero_zero_rat ) ) ).
% zero_reorient
thf(fact_261_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_262_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_263_type__definition__def,axiom,
( type_d4900610042970207096et_rat
= ( ^ [Rep2: dedekind_preal > set_rat,Abs2: set_rat > dedekind_preal,A4: set_set_rat] :
( ! [X4: dedekind_preal] : ( member_set_rat @ ( Rep2 @ X4 ) @ A4 )
& ! [X4: dedekind_preal] :
( ( Abs2 @ ( Rep2 @ X4 ) )
= X4 )
& ! [Y4: set_rat] :
( ( member_set_rat @ Y4 @ A4 )
=> ( ( Rep2 @ ( Abs2 @ Y4 ) )
= Y4 ) ) ) ) ) ).
% type_definition_def
thf(fact_264_type__definition_ORep__inverse,axiom,
! [Rep: dedekind_preal > set_rat,Abs: set_rat > dedekind_preal,A: set_set_rat,X2: dedekind_preal] :
( ( type_d4900610042970207096et_rat @ Rep @ Abs @ A )
=> ( ( Abs @ ( Rep @ X2 ) )
= X2 ) ) ).
% type_definition.Rep_inverse
thf(fact_265_type__definition_OAbs__inverse,axiom,
! [Rep: dedekind_preal > set_rat,Abs: set_rat > dedekind_preal,A: set_set_rat,Y: set_rat] :
( ( type_d4900610042970207096et_rat @ Rep @ Abs @ A )
=> ( ( member_set_rat @ Y @ A )
=> ( ( Rep @ ( Abs @ Y ) )
= Y ) ) ) ).
% type_definition.Abs_inverse
thf(fact_266_type__definition_ORep__inject,axiom,
! [Rep: dedekind_preal > set_rat,Abs: set_rat > dedekind_preal,A: set_set_rat,X2: dedekind_preal,Y: dedekind_preal] :
( ( type_d4900610042970207096et_rat @ Rep @ Abs @ A )
=> ( ( ( Rep @ X2 )
= ( Rep @ Y ) )
= ( X2 = Y ) ) ) ).
% type_definition.Rep_inject
thf(fact_267_type__definition_ORep__induct,axiom,
! [Rep: dedekind_preal > set_rat,Abs: set_rat > dedekind_preal,A: set_set_rat,Y: set_rat,P: set_rat > $o] :
( ( type_d4900610042970207096et_rat @ Rep @ Abs @ A )
=> ( ( member_set_rat @ Y @ A )
=> ( ! [X: dedekind_preal] : ( P @ ( Rep @ X ) )
=> ( P @ Y ) ) ) ) ).
% type_definition.Rep_induct
thf(fact_268_type__definition_OAbs__inject,axiom,
! [Rep: dedekind_preal > set_rat,Abs: set_rat > dedekind_preal,A: set_set_rat,X2: set_rat,Y: set_rat] :
( ( type_d4900610042970207096et_rat @ Rep @ Abs @ A )
=> ( ( member_set_rat @ X2 @ A )
=> ( ( member_set_rat @ Y @ A )
=> ( ( ( Abs @ X2 )
= ( Abs @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% type_definition.Abs_inject
thf(fact_269_type__definition_OAbs__induct,axiom,
! [Rep: dedekind_preal > set_rat,Abs: set_rat > dedekind_preal,A: set_set_rat,P: dedekind_preal > $o,X2: dedekind_preal] :
( ( type_d4900610042970207096et_rat @ Rep @ Abs @ A )
=> ( ! [Y2: set_rat] :
( ( member_set_rat @ Y2 @ A )
=> ( P @ ( Abs @ Y2 ) ) )
=> ( P @ X2 ) ) ) ).
% type_definition.Abs_induct
thf(fact_270_type__definition_ORep__cases,axiom,
! [Rep: dedekind_preal > set_rat,Abs: set_rat > dedekind_preal,A: set_set_rat,Y: set_rat] :
( ( type_d4900610042970207096et_rat @ Rep @ Abs @ A )
=> ( ( member_set_rat @ Y @ A )
=> ~ ! [X: dedekind_preal] :
( Y
!= ( Rep @ X ) ) ) ) ).
% type_definition.Rep_cases
thf(fact_271_type__definition_OAbs__cases,axiom,
! [Rep: dedekind_preal > set_rat,Abs: set_rat > dedekind_preal,A: set_set_rat,X2: dedekind_preal] :
( ( type_d4900610042970207096et_rat @ Rep @ Abs @ A )
=> ~ ! [Y2: set_rat] :
( ( X2
= ( Abs @ Y2 ) )
=> ~ ( member_set_rat @ Y2 @ A ) ) ) ).
% type_definition.Abs_cases
thf(fact_272_psubsetD,axiom,
! [A: set_set_rat,B3: set_set_rat,C: set_rat] :
( ( ord_less_set_set_rat @ A @ B3 )
=> ( ( member_set_rat @ C @ A )
=> ( member_set_rat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_273_psubsetD,axiom,
! [A: set_rat,B3: set_rat,C: rat] :
( ( ord_less_set_rat @ A @ B3 )
=> ( ( member_rat @ C @ A )
=> ( member_rat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_274_psubset__trans,axiom,
! [A: set_rat,B3: set_rat,C2: set_rat] :
( ( ord_less_set_rat @ A @ B3 )
=> ( ( ord_less_set_rat @ B3 @ C2 )
=> ( ord_less_set_rat @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_275_of__rat__less__0__iff,axiom,
! [R2: rat] :
( ( ord_less_rat @ ( field_2639924705303425560at_rat @ R2 ) @ zero_zero_rat )
= ( ord_less_rat @ R2 @ zero_zero_rat ) ) ).
% of_rat_less_0_iff
thf(fact_276_zero__less__of__rat__iff,axiom,
! [R2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( field_2639924705303425560at_rat @ R2 ) )
= ( ord_less_rat @ zero_zero_rat @ R2 ) ) ).
% zero_less_of_rat_iff
thf(fact_277_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% dbl_simps(2)
thf(fact_278_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_279_Dedekind__Real_Ocut__def,axiom,
( dedekind_cut
= ( ^ [A4: set_rat] :
( ( ord_less_set_rat @ bot_bot_set_rat @ A4 )
& ( ord_less_set_rat @ A4 @ ( set_or575021546402375026an_rat @ zero_zero_rat ) )
& ! [X4: rat] :
( ( member_rat @ X4 @ A4 )
=> ( ! [Z4: rat] :
( ( ( ord_less_rat @ zero_zero_rat @ Z4 )
& ( ord_less_rat @ Z4 @ X4 ) )
=> ( member_rat @ Z4 @ A4 ) )
& ? [Y4: rat] :
( ( member_rat @ Y4 @ A4 )
& ( ord_less_rat @ X4 @ Y4 ) ) ) ) ) ) ) ).
% Dedekind_Real.cut_def
thf(fact_280_add__less__same__cancel1,axiom,
! [B: rat,A2: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ B @ A2 ) @ B )
= ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% add_less_same_cancel1
thf(fact_281_add__less__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_282_add__less__same__cancel1,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_283_add__less__same__cancel2,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ B )
= ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% add_less_same_cancel2
thf(fact_284_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_285_add__less__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_286_less__add__same__cancel1,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ B ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel1
thf(fact_287_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_288_less__add__same__cancel1,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_289_less__add__same__cancel2,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ ( plus_plus_rat @ B @ A2 ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel2
thf(fact_290_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_291_less__add__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_292_add__right__cancel,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A2 )
= ( plus_plus_rat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_293_add__right__cancel,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_294_add__right__cancel,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_295_add__left__cancel,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A2 @ B )
= ( plus_plus_rat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_296_add__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_297_add__left__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_298_empty__iff,axiom,
! [C: set_rat] :
~ ( member_set_rat @ C @ bot_bot_set_set_rat ) ).
% empty_iff
thf(fact_299_empty__iff,axiom,
! [C: rat] :
~ ( member_rat @ C @ bot_bot_set_rat ) ).
% empty_iff
thf(fact_300_all__not__in__conv,axiom,
! [A: set_set_rat] :
( ( ! [X4: set_rat] :
~ ( member_set_rat @ X4 @ A ) )
= ( A = bot_bot_set_set_rat ) ) ).
% all_not_in_conv
thf(fact_301_all__not__in__conv,axiom,
! [A: set_rat] :
( ( ! [X4: rat] :
~ ( member_rat @ X4 @ A ) )
= ( A = bot_bot_set_rat ) ) ).
% all_not_in_conv
thf(fact_302_Collect__empty__eq,axiom,
! [P: set_rat > $o] :
( ( ( collect_set_rat @ P )
= bot_bot_set_set_rat )
= ( ! [X4: set_rat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_303_Collect__empty__eq,axiom,
! [P: rat > $o] :
( ( ( collect_rat @ P )
= bot_bot_set_rat )
= ( ! [X4: rat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_304_empty__Collect__eq,axiom,
! [P: set_rat > $o] :
( ( bot_bot_set_set_rat
= ( collect_set_rat @ P ) )
= ( ! [X4: set_rat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_305_empty__Collect__eq,axiom,
! [P: rat > $o] :
( ( bot_bot_set_rat
= ( collect_rat @ P ) )
= ( ! [X4: rat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_306_add_Oright__neutral,axiom,
! [A2: rat] :
( ( plus_plus_rat @ A2 @ zero_zero_rat )
= A2 ) ).
% add.right_neutral
thf(fact_307_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_308_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_309_double__zero__sym,axiom,
! [A2: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A2 @ A2 ) )
= ( A2 = zero_zero_rat ) ) ).
% double_zero_sym
thf(fact_310_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_311_add__cancel__left__left,axiom,
! [B: rat,A2: rat] :
( ( ( plus_plus_rat @ B @ A2 )
= A2 )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_left
thf(fact_312_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_313_add__cancel__left__left,axiom,
! [B: int,A2: int] :
( ( ( plus_plus_int @ B @ A2 )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_314_add__cancel__left__right,axiom,
! [A2: rat,B: rat] :
( ( ( plus_plus_rat @ A2 @ B )
= A2 )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_right
thf(fact_315_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_316_add__cancel__left__right,axiom,
! [A2: int,B: int] :
( ( ( plus_plus_int @ A2 @ B )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_317_add__cancel__right__left,axiom,
! [A2: rat,B: rat] :
( ( A2
= ( plus_plus_rat @ B @ A2 ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_left
thf(fact_318_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_319_add__cancel__right__left,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ B @ A2 ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_320_add__cancel__right__right,axiom,
! [A2: rat,B: rat] :
( ( A2
= ( plus_plus_rat @ A2 @ B ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_right
thf(fact_321_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_322_add__cancel__right__right,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ A2 @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_323_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_324_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y ) )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_325_add__0,axiom,
! [A2: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A2 )
= A2 ) ).
% add_0
thf(fact_326_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_327_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_328_add__less__cancel__right,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
= ( ord_less_rat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_329_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_330_add__less__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_331_add__less__cancel__left,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
= ( ord_less_rat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_332_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_333_add__less__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_334_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
= ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_335_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_336_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
= ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_337_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_338_of__rat__0,axiom,
( ( field_2639924705303425560at_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% of_rat_0
thf(fact_339_of__rat__eq__0__iff,axiom,
! [A2: rat] :
( ( ( field_2639924705303425560at_rat @ A2 )
= zero_zero_rat )
= ( A2 = zero_zero_rat ) ) ).
% of_rat_eq_0_iff
thf(fact_340_zero__eq__of__rat__iff,axiom,
! [A2: rat] :
( ( zero_zero_rat
= ( field_2639924705303425560at_rat @ A2 ) )
= ( zero_zero_rat = A2 ) ) ).
% zero_eq_of_rat_iff
thf(fact_341_add__right__imp__eq,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A2 )
= ( plus_plus_rat @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_342_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_343_add__right__imp__eq,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_344_add__left__imp__eq,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A2 @ B )
= ( plus_plus_rat @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_345_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_346_add__left__imp__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_347_add_Oleft__commute,axiom,
! [B: rat,A2: rat,C: rat] :
( ( plus_plus_rat @ B @ ( plus_plus_rat @ A2 @ C ) )
= ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_348_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_349_add_Oleft__commute,axiom,
! [B: int,A2: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A2 @ C ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_350_add_Ocommute,axiom,
( plus_plus_rat
= ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_351_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_352_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_353_add_Oright__cancel,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A2 )
= ( plus_plus_rat @ C @ A2 ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_354_add_Oright__cancel,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_355_add_Oleft__cancel,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A2 @ B )
= ( plus_plus_rat @ A2 @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_356_add_Oleft__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_357_add_Oassoc,axiom,
! [A2: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B ) @ C )
= ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B @ C ) ) ) ).
% add.assoc
thf(fact_358_add_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_359_add_Oassoc,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_360_dbl__def,axiom,
( neg_numeral_dbl_rat
= ( ^ [X4: rat] : ( plus_plus_rat @ X4 @ X4 ) ) ) ).
% dbl_def
thf(fact_361_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X4: int] : ( plus_plus_int @ X4 @ X4 ) ) ) ).
% dbl_def
thf(fact_362_group__cancel_Oadd2,axiom,
! [B3: rat,K: rat,B: rat,A2: rat] :
( ( B3
= ( plus_plus_rat @ K @ B ) )
=> ( ( plus_plus_rat @ A2 @ B3 )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_363_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A2: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_364_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A2: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A2 @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_365_group__cancel_Oadd1,axiom,
! [A: rat,K: rat,A2: rat,B: rat] :
( ( A
= ( plus_plus_rat @ K @ A2 ) )
=> ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_366_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_367_group__cancel_Oadd1,axiom,
! [A: int,K: int,A2: int,B: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_368_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_rat @ I @ K )
= ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_369_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_370_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_371_is__num__normalize_I1_J,axiom,
! [A2: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B ) @ C )
= ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_372_is__num__normalize_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_373_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B ) @ C )
= ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_374_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_375_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_376_emptyE,axiom,
! [A2: set_rat] :
~ ( member_set_rat @ A2 @ bot_bot_set_set_rat ) ).
% emptyE
thf(fact_377_emptyE,axiom,
! [A2: rat] :
~ ( member_rat @ A2 @ bot_bot_set_rat ) ).
% emptyE
thf(fact_378_equals0D,axiom,
! [A: set_set_rat,A2: set_rat] :
( ( A = bot_bot_set_set_rat )
=> ~ ( member_set_rat @ A2 @ A ) ) ).
% equals0D
thf(fact_379_equals0D,axiom,
! [A: set_rat,A2: rat] :
( ( A = bot_bot_set_rat )
=> ~ ( member_rat @ A2 @ A ) ) ).
% equals0D
thf(fact_380_equals0I,axiom,
! [A: set_set_rat] :
( ! [Y2: set_rat] :
~ ( member_set_rat @ Y2 @ A )
=> ( A = bot_bot_set_set_rat ) ) ).
% equals0I
thf(fact_381_equals0I,axiom,
! [A: set_rat] :
( ! [Y2: rat] :
~ ( member_rat @ Y2 @ A )
=> ( A = bot_bot_set_rat ) ) ).
% equals0I
thf(fact_382_of__rat__add,axiom,
! [A2: rat,B: rat] :
( ( field_2639924705303425560at_rat @ ( plus_plus_rat @ A2 @ B ) )
= ( plus_plus_rat @ ( field_2639924705303425560at_rat @ A2 ) @ ( field_2639924705303425560at_rat @ B ) ) ) ).
% of_rat_add
thf(fact_383_ex__in__conv,axiom,
! [A: set_set_rat] :
( ( ? [X4: set_rat] : ( member_set_rat @ X4 @ A ) )
= ( A != bot_bot_set_set_rat ) ) ).
% ex_in_conv
thf(fact_384_ex__in__conv,axiom,
! [A: set_rat] :
( ( ? [X4: rat] : ( member_rat @ X4 @ A ) )
= ( A != bot_bot_set_rat ) ) ).
% ex_in_conv
thf(fact_385_add__eq__exists,axiom,
! [A2: rat,B: rat] :
? [X: rat] :
( ( plus_plus_rat @ A2 @ X )
= B ) ).
% add_eq_exists
thf(fact_386_add__eq__exists,axiom,
! [A2: int,B: int] :
? [X: int] :
( ( plus_plus_int @ A2 @ X )
= B ) ).
% add_eq_exists
thf(fact_387_not__psubset__empty,axiom,
! [A: set_rat] :
~ ( ord_less_set_rat @ A @ bot_bot_set_rat ) ).
% not_psubset_empty
thf(fact_388_bot_Onot__eq__extremum,axiom,
! [A2: set_rat] :
( ( A2 != bot_bot_set_rat )
= ( ord_less_set_rat @ bot_bot_set_rat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_389_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_390_bot_Oextremum__strict,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ A2 @ bot_bot_set_rat ) ).
% bot.extremum_strict
thf(fact_391_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_392_verit__sum__simplify,axiom,
! [A2: rat] :
( ( plus_plus_rat @ A2 @ zero_zero_rat )
= A2 ) ).
% verit_sum_simplify
thf(fact_393_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_394_verit__sum__simplify,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% verit_sum_simplify
thf(fact_395_comm__monoid__add__class_Oadd__0,axiom,
! [A2: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_396_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_397_comm__monoid__add__class_Oadd__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_398_add_Ocomm__neutral,axiom,
! [A2: rat] :
( ( plus_plus_rat @ A2 @ zero_zero_rat )
= A2 ) ).
% add.comm_neutral
thf(fact_399_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_400_add_Ocomm__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.comm_neutral
thf(fact_401_add_Ogroup__left__neutral,axiom,
! [A2: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_402_add_Ogroup__left__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_403_add__less__imp__less__right,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
=> ( ord_less_rat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_404_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_405_add__less__imp__less__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_406_add__less__imp__less__left,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
=> ( ord_less_rat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_407_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_408_add__less__imp__less__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_409_add__strict__right__mono,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_410_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_411_add__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_412_add__strict__left__mono,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_413_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_414_add__strict__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_415_add__strict__mono,axiom,
! [A2: rat,B: rat,C: rat,D: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_rat @ C @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_416_add__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_417_add__strict__mono,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_418_add__mono__thms__linordered__field_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( K = L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_419_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_420_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_421_add__mono__thms__linordered__field_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( ord_less_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_422_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_423_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_424_add__mono__thms__linordered__field_I5_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( ord_less_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_425_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_426_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_427_of__rat__less,axiom,
! [R2: rat,S2: rat] :
( ( ord_less_rat @ ( field_2639924705303425560at_rat @ R2 ) @ ( field_2639924705303425560at_rat @ S2 ) )
= ( ord_less_rat @ R2 @ S2 ) ) ).
% of_rat_less
thf(fact_428_add__less__zeroD,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ X2 @ Y ) @ zero_zero_rat )
=> ( ( ord_less_rat @ X2 @ zero_zero_rat )
| ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).
% add_less_zeroD
thf(fact_429_add__less__zeroD,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X2 @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X2 @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_430_pos__add__strict,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_431_pos__add__strict,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_432_pos__add__strict,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_433_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A2 @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_434_add__pos__pos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_435_add__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_436_add__pos__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_437_add__neg__neg,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_neg
thf(fact_438_add__neg__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_439_add__neg__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_440_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ~ ! [S3: rat] :
( ( ord_less_rat @ zero_zero_rat @ S3 )
=> ! [T2: rat] :
( ( ord_less_rat @ zero_zero_rat @ T2 )
=> ( R2
!= ( plus_plus_rat @ S3 @ T2 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_441_greaterThan__iff,axiom,
! [I: set_rat,K: set_rat] :
( ( member_set_rat @ I @ ( set_or6174011595382531368et_rat @ K ) )
= ( ord_less_set_rat @ K @ I ) ) ).
% greaterThan_iff
thf(fact_442_greaterThan__iff,axiom,
! [I: num,K: num] :
( ( member_num @ I @ ( set_or6990855429499425204an_num @ K ) )
= ( ord_less_num @ K @ I ) ) ).
% greaterThan_iff
thf(fact_443_greaterThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_or1210151606488870762an_nat @ K ) )
= ( ord_less_nat @ K @ I ) ) ).
% greaterThan_iff
thf(fact_444_greaterThan__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_or1207661135979820486an_int @ K ) )
= ( ord_less_int @ K @ I ) ) ).
% greaterThan_iff
thf(fact_445_greaterThan__iff,axiom,
! [I: rat,K: rat] :
( ( member_rat @ I @ ( set_or575021546402375026an_rat @ K ) )
= ( ord_less_rat @ K @ I ) ) ).
% greaterThan_iff
thf(fact_446_double__eq__0__iff,axiom,
! [A2: rat] :
( ( ( plus_plus_rat @ A2 @ A2 )
= zero_zero_rat )
= ( A2 = zero_zero_rat ) ) ).
% double_eq_0_iff
thf(fact_447_double__eq__0__iff,axiom,
! [A2: int] :
( ( ( plus_plus_int @ A2 @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_448_preal__add__def,axiom,
( plus_p3173629198307831117_preal
= ( ^ [R: dedekind_preal,S: dedekind_preal] : ( dedekind_Abs_preal @ ( dedekind_add_set @ ( dedekind_Rep_preal @ R ) @ ( dedekind_Rep_preal @ S ) ) ) ) ) ).
% preal_add_def
thf(fact_449_greaterThan__eq__iff,axiom,
! [X2: rat,Y: rat] :
( ( ( set_or575021546402375026an_rat @ X2 )
= ( set_or575021546402375026an_rat @ Y ) )
= ( X2 = Y ) ) ).
% greaterThan_eq_iff
thf(fact_450_greaterThan__non__empty,axiom,
! [X2: rat] :
( ( set_or575021546402375026an_rat @ X2 )
!= bot_bot_set_rat ) ).
% greaterThan_non_empty
thf(fact_451_add__0__iff,axiom,
! [B: rat,A2: rat] :
( ( B
= ( plus_plus_rat @ B @ A2 ) )
= ( A2 = zero_zero_rat ) ) ).
% add_0_iff
thf(fact_452_add__0__iff,axiom,
! [B: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ B @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_453_add__0__iff,axiom,
! [B: int,A2: int] :
( ( B
= ( plus_plus_int @ B @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% add_0_iff
thf(fact_454_Set_Ois__empty__def,axiom,
( is_empty_rat
= ( ^ [A4: set_rat] : ( A4 = bot_bot_set_rat ) ) ) ).
% Set.is_empty_def
thf(fact_455_zero__less__two,axiom,
ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% zero_less_two
thf(fact_456_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_457_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_458_field__le__epsilon,axiom,
! [X2: rat,Y: rat] :
( ! [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
=> ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ Y @ E ) ) )
=> ( ord_less_eq_rat @ X2 @ Y ) ) ).
% field_le_epsilon
thf(fact_459_add__strict__increasing2,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_460_add__strict__increasing2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_461_add__strict__increasing2,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_462_order__refl,axiom,
! [X2: set_rat] : ( ord_less_eq_set_rat @ X2 @ X2 ) ).
% order_refl
thf(fact_463_order__refl,axiom,
! [X2: rat] : ( ord_less_eq_rat @ X2 @ X2 ) ).
% order_refl
thf(fact_464_order__refl,axiom,
! [X2: num] : ( ord_less_eq_num @ X2 @ X2 ) ).
% order_refl
thf(fact_465_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_466_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_467_dual__order_Orefl,axiom,
! [A2: set_rat] : ( ord_less_eq_set_rat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_468_dual__order_Orefl,axiom,
! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_469_dual__order_Orefl,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_470_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_471_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_472_empty__subsetI,axiom,
! [A: set_rat] : ( ord_less_eq_set_rat @ bot_bot_set_rat @ A ) ).
% empty_subsetI
thf(fact_473_subset__empty,axiom,
! [A: set_rat] :
( ( ord_less_eq_set_rat @ A @ bot_bot_set_rat )
= ( A = bot_bot_set_rat ) ) ).
% subset_empty
thf(fact_474_psubsetI,axiom,
! [A: set_rat,B3: set_rat] :
( ( ord_less_eq_set_rat @ A @ B3 )
=> ( ( A != B3 )
=> ( ord_less_set_rat @ A @ B3 ) ) ) ).
% psubsetI
thf(fact_475_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_476_add__le__cancel__left,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
= ( ord_less_eq_rat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_477_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_478_add__le__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_479_add__le__cancel__right,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
= ( ord_less_eq_rat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_480_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_481_add__le__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_482_greaterThan__subset__iff,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_set_num @ ( set_or6990855429499425204an_num @ X2 ) @ ( set_or6990855429499425204an_num @ Y ) )
= ( ord_less_eq_num @ Y @ X2 ) ) ).
% greaterThan_subset_iff
thf(fact_483_greaterThan__subset__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_or1210151606488870762an_nat @ X2 ) @ ( set_or1210151606488870762an_nat @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% greaterThan_subset_iff
thf(fact_484_greaterThan__subset__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_or1207661135979820486an_int @ X2 ) @ ( set_or1207661135979820486an_int @ Y ) )
= ( ord_less_eq_int @ Y @ X2 ) ) ).
% greaterThan_subset_iff
thf(fact_485_greaterThan__subset__iff,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_set_rat @ ( set_or575021546402375026an_rat @ X2 ) @ ( set_or575021546402375026an_rat @ Y ) )
= ( ord_less_eq_rat @ Y @ X2 ) ) ).
% greaterThan_subset_iff
thf(fact_486_add__le__same__cancel1,axiom,
! [B: rat,A2: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A2 ) @ B )
= ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% add_le_same_cancel1
thf(fact_487_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_488_add__le__same__cancel1,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_489_add__le__same__cancel2,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B ) @ B )
= ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% add_le_same_cancel2
thf(fact_490_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_491_add__le__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_492_le__add__same__cancel1,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ A2 @ B ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel1
thf(fact_493_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_494_le__add__same__cancel1,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_495_le__add__same__cancel2,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ B @ A2 ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel2
thf(fact_496_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_497_le__add__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_498_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_499_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_500_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_501_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_502_of__rat__le__1__iff,axiom,
! [R2: rat] :
( ( ord_less_eq_rat @ ( field_2639924705303425560at_rat @ R2 ) @ one_one_rat )
= ( ord_less_eq_rat @ R2 @ one_one_rat ) ) ).
% of_rat_le_1_iff
thf(fact_503_one__le__of__rat__iff,axiom,
! [R2: rat] :
( ( ord_less_eq_rat @ one_one_rat @ ( field_2639924705303425560at_rat @ R2 ) )
= ( ord_less_eq_rat @ one_one_rat @ R2 ) ) ).
% one_le_of_rat_iff
thf(fact_504_zero__le__of__rat__iff,axiom,
! [R2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( field_2639924705303425560at_rat @ R2 ) )
= ( ord_less_eq_rat @ zero_zero_rat @ R2 ) ) ).
% zero_le_of_rat_iff
thf(fact_505_of__rat__le__0__iff,axiom,
! [R2: rat] :
( ( ord_less_eq_rat @ ( field_2639924705303425560at_rat @ R2 ) @ zero_zero_rat )
= ( ord_less_eq_rat @ R2 @ zero_zero_rat ) ) ).
% of_rat_le_0_iff
thf(fact_506_one__less__of__rat__iff,axiom,
! [R2: rat] :
( ( ord_less_rat @ one_one_rat @ ( field_2639924705303425560at_rat @ R2 ) )
= ( ord_less_rat @ one_one_rat @ R2 ) ) ).
% one_less_of_rat_iff
thf(fact_507_of__rat__less__1__iff,axiom,
! [R2: rat] :
( ( ord_less_rat @ ( field_2639924705303425560at_rat @ R2 ) @ one_one_rat )
= ( ord_less_rat @ R2 @ one_one_rat ) ) ).
% of_rat_less_1_iff
thf(fact_508_verit__comp__simplify1_I2_J,axiom,
! [A2: set_rat] : ( ord_less_eq_set_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_509_verit__comp__simplify1_I2_J,axiom,
! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_510_verit__comp__simplify1_I2_J,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_511_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_512_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_513_verit__la__disequality,axiom,
! [A2: rat,B: rat] :
( ( A2 = B )
| ~ ( ord_less_eq_rat @ A2 @ B )
| ~ ( ord_less_eq_rat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_514_verit__la__disequality,axiom,
! [A2: num,B: num] :
( ( A2 = B )
| ~ ( ord_less_eq_num @ A2 @ B )
| ~ ( ord_less_eq_num @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_515_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_516_verit__la__disequality,axiom,
! [A2: int,B: int] :
( ( A2 = B )
| ~ ( ord_less_eq_int @ A2 @ B )
| ~ ( ord_less_eq_int @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_517_nle__le,axiom,
! [A2: rat,B: rat] :
( ( ~ ( ord_less_eq_rat @ A2 @ B ) )
= ( ( ord_less_eq_rat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_518_nle__le,axiom,
! [A2: num,B: num] :
( ( ~ ( ord_less_eq_num @ A2 @ B ) )
= ( ( ord_less_eq_num @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_519_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_520_nle__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B ) )
= ( ( ord_less_eq_int @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_521_le__cases3,axiom,
! [X2: rat,Y: rat,Z3: rat] :
( ( ( ord_less_eq_rat @ X2 @ Y )
=> ~ ( ord_less_eq_rat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_rat @ Y @ X2 )
=> ~ ( ord_less_eq_rat @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_rat @ X2 @ Z3 )
=> ~ ( ord_less_eq_rat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_rat @ Z3 @ Y )
=> ~ ( ord_less_eq_rat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_rat @ Y @ Z3 )
=> ~ ( ord_less_eq_rat @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_rat @ Z3 @ X2 )
=> ~ ( ord_less_eq_rat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_522_le__cases3,axiom,
! [X2: num,Y: num,Z3: num] :
( ( ( ord_less_eq_num @ X2 @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_num @ Y @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_num @ X2 @ Z3 )
=> ~ ( ord_less_eq_num @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_num @ Z3 @ Y )
=> ~ ( ord_less_eq_num @ Y @ X2 ) )
=> ( ( ( ord_less_eq_num @ Y @ Z3 )
=> ~ ( ord_less_eq_num @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_num @ Z3 @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_523_le__cases3,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_524_le__cases3,axiom,
! [X2: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_525_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_rat,Z5: set_rat] : ( Y5 = Z5 ) )
= ( ^ [X4: set_rat,Y4: set_rat] :
( ( ord_less_eq_set_rat @ X4 @ Y4 )
& ( ord_less_eq_set_rat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_526_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: rat,Z5: rat] : ( Y5 = Z5 ) )
= ( ^ [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
& ( ord_less_eq_rat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_527_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z5: num] : ( Y5 = Z5 ) )
= ( ^ [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
& ( ord_less_eq_num @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_528_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_529_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z5: int] : ( Y5 = Z5 ) )
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_530_ord__eq__le__trans,axiom,
! [A2: set_rat,B: set_rat,C: set_rat] :
( ( A2 = B )
=> ( ( ord_less_eq_set_rat @ B @ C )
=> ( ord_less_eq_set_rat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_531_ord__eq__le__trans,axiom,
! [A2: rat,B: rat,C: rat] :
( ( A2 = B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_532_ord__eq__le__trans,axiom,
! [A2: num,B: num,C: num] :
( ( A2 = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_533_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_534_ord__eq__le__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_535_ord__le__eq__trans,axiom,
! [A2: set_rat,B: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_rat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_536_ord__le__eq__trans,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_537_ord__le__eq__trans,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_538_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_539_ord__le__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_540_order__antisym,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ Y )
=> ( ( ord_less_eq_set_rat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_541_order__antisym,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
=> ( ( ord_less_eq_rat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_542_order__antisym,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ord_less_eq_num @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_543_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_544_order__antisym,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_545_order_Otrans,axiom,
! [A2: set_rat,B: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B )
=> ( ( ord_less_eq_set_rat @ B @ C )
=> ( ord_less_eq_set_rat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_546_order_Otrans,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_547_order_Otrans,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% order.trans
thf(fact_548_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_549_order_Otrans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_550_order__trans,axiom,
! [X2: set_rat,Y: set_rat,Z3: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ Y )
=> ( ( ord_less_eq_set_rat @ Y @ Z3 )
=> ( ord_less_eq_set_rat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_551_order__trans,axiom,
! [X2: rat,Y: rat,Z3: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
=> ( ( ord_less_eq_rat @ Y @ Z3 )
=> ( ord_less_eq_rat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_552_order__trans,axiom,
! [X2: num,Y: num,Z3: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ord_less_eq_num @ Y @ Z3 )
=> ( ord_less_eq_num @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_553_order__trans,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_554_order__trans,axiom,
! [X2: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_555_linorder__wlog,axiom,
! [P: rat > rat > $o,A2: rat,B: rat] :
( ! [A3: rat,B2: rat] :
( ( ord_less_eq_rat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: rat,B2: rat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_556_linorder__wlog,axiom,
! [P: num > num > $o,A2: 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 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_557_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: 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 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_558_linorder__wlog,axiom,
! [P: int > int > $o,A2: 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 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_559_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_rat,Z5: set_rat] : ( Y5 = Z5 ) )
= ( ^ [A5: set_rat,B4: set_rat] :
( ( ord_less_eq_set_rat @ B4 @ A5 )
& ( ord_less_eq_set_rat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_560_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: rat,Z5: rat] : ( Y5 = Z5 ) )
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_eq_rat @ B4 @ A5 )
& ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_561_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: num,Z5: num] : ( Y5 = Z5 ) )
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ B4 @ A5 )
& ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_562_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_563_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z5: int] : ( Y5 = Z5 ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_564_dual__order_Oantisym,axiom,
! [B: set_rat,A2: set_rat] :
( ( ord_less_eq_set_rat @ B @ A2 )
=> ( ( ord_less_eq_set_rat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_565_dual__order_Oantisym,axiom,
! [B: rat,A2: rat] :
( ( ord_less_eq_rat @ B @ A2 )
=> ( ( ord_less_eq_rat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_566_dual__order_Oantisym,axiom,
! [B: num,A2: num] :
( ( ord_less_eq_num @ B @ A2 )
=> ( ( ord_less_eq_num @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_567_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_568_dual__order_Oantisym,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_569_dual__order_Otrans,axiom,
! [B: set_rat,A2: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ B @ A2 )
=> ( ( ord_less_eq_set_rat @ C @ B )
=> ( ord_less_eq_set_rat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_570_dual__order_Otrans,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A2 )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_571_dual__order_Otrans,axiom,
! [B: num,A2: num,C: num] :
( ( ord_less_eq_num @ B @ A2 )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_572_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_573_dual__order_Otrans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_574_antisym,axiom,
! [A2: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B )
=> ( ( ord_less_eq_set_rat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_575_antisym,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_576_antisym,axiom,
! [A2: num,B: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_num @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_577_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_578_antisym,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_579_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_rat,Z5: set_rat] : ( Y5 = Z5 ) )
= ( ^ [A5: set_rat,B4: set_rat] :
( ( ord_less_eq_set_rat @ A5 @ B4 )
& ( ord_less_eq_set_rat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_580_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: rat,Z5: rat] : ( Y5 = Z5 ) )
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_eq_rat @ A5 @ B4 )
& ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_581_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z5: num] : ( Y5 = Z5 ) )
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
& ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_582_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_583_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z5: int] : ( Y5 = Z5 ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_584_order__subst1,axiom,
! [A2: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_585_order__subst1,axiom,
! [A2: rat,F: num > rat,B: num,C: num] :
( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_586_order__subst1,axiom,
! [A2: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_587_order__subst1,axiom,
! [A2: rat,F: int > rat,B: int,C: int] :
( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_588_order__subst1,axiom,
! [A2: num,F: rat > num,B: rat,C: rat] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_589_order__subst1,axiom,
! [A2: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_590_order__subst1,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_591_order__subst1,axiom,
! [A2: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_592_order__subst1,axiom,
! [A2: nat,F: rat > nat,B: rat,C: rat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_593_order__subst1,axiom,
! [A2: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_594_order__subst2,axiom,
! [A2: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_595_order__subst2,axiom,
! [A2: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_596_order__subst2,axiom,
! [A2: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_597_order__subst2,axiom,
! [A2: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_598_order__subst2,axiom,
! [A2: num,B: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_599_order__subst2,axiom,
! [A2: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_600_order__subst2,axiom,
! [A2: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_601_order__subst2,axiom,
! [A2: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_602_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_603_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_604_order__eq__refl,axiom,
! [X2: set_rat,Y: set_rat] :
( ( X2 = Y )
=> ( ord_less_eq_set_rat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_605_order__eq__refl,axiom,
! [X2: rat,Y: rat] :
( ( X2 = Y )
=> ( ord_less_eq_rat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_606_order__eq__refl,axiom,
! [X2: num,Y: num] :
( ( X2 = Y )
=> ( ord_less_eq_num @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_607_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_608_order__eq__refl,axiom,
! [X2: int,Y: int] :
( ( X2 = Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_609_linorder__linear,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
| ( ord_less_eq_rat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_610_linorder__linear,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
| ( ord_less_eq_num @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_611_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_612_linorder__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_613_ord__eq__le__subst,axiom,
! [A2: rat,F: rat > rat,B: rat,C: rat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_614_ord__eq__le__subst,axiom,
! [A2: num,F: rat > num,B: rat,C: rat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_615_ord__eq__le__subst,axiom,
! [A2: nat,F: rat > nat,B: rat,C: rat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_616_ord__eq__le__subst,axiom,
! [A2: int,F: rat > int,B: rat,C: rat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_617_ord__eq__le__subst,axiom,
! [A2: rat,F: num > rat,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_618_ord__eq__le__subst,axiom,
! [A2: num,F: num > num,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_619_ord__eq__le__subst,axiom,
! [A2: nat,F: num > nat,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_620_ord__eq__le__subst,axiom,
! [A2: int,F: num > int,B: num,C: num] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_621_ord__eq__le__subst,axiom,
! [A2: rat,F: nat > rat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_622_ord__eq__le__subst,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_623_ord__le__eq__subst,axiom,
! [A2: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_624_ord__le__eq__subst,axiom,
! [A2: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_625_ord__le__eq__subst,axiom,
! [A2: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_626_ord__le__eq__subst,axiom,
! [A2: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_627_ord__le__eq__subst,axiom,
! [A2: num,B: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_628_ord__le__eq__subst,axiom,
! [A2: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_629_ord__le__eq__subst,axiom,
! [A2: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_630_ord__le__eq__subst,axiom,
! [A2: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_631_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_632_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_633_linorder__le__cases,axiom,
! [X2: rat,Y: rat] :
( ~ ( ord_less_eq_rat @ X2 @ Y )
=> ( ord_less_eq_rat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_634_linorder__le__cases,axiom,
! [X2: num,Y: num] :
( ~ ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_num @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_635_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_636_linorder__le__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_637_order__antisym__conv,axiom,
! [Y: set_rat,X2: set_rat] :
( ( ord_less_eq_set_rat @ Y @ X2 )
=> ( ( ord_less_eq_set_rat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_638_order__antisym__conv,axiom,
! [Y: rat,X2: rat] :
( ( ord_less_eq_rat @ Y @ X2 )
=> ( ( ord_less_eq_rat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_639_order__antisym__conv,axiom,
! [Y: num,X2: num] :
( ( ord_less_eq_num @ Y @ X2 )
=> ( ( ord_less_eq_num @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_640_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_641_order__antisym__conv,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_642_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one_class.zero_le_one
thf(fact_643_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_644_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_645_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_646_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_647_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_648_not__one__le__zero,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_le_zero
thf(fact_649_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_650_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_651_le__numeral__extra_I4_J,axiom,
ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% le_numeral_extra(4)
thf(fact_652_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_653_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_654_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_655_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_656_of__rat__less__eq,axiom,
! [R2: rat,S2: rat] :
( ( ord_less_eq_rat @ ( field_2639924705303425560at_rat @ R2 ) @ ( field_2639924705303425560at_rat @ S2 ) )
= ( ord_less_eq_rat @ R2 @ S2 ) ) ).
% of_rat_less_eq
thf(fact_657_preal__le__def,axiom,
( ord_le5604041210740703414_preal
= ( ^ [R: dedekind_preal,S: dedekind_preal] : ( ord_less_eq_set_rat @ ( dedekind_Rep_preal @ R ) @ ( dedekind_Rep_preal @ S ) ) ) ) ).
% preal_le_def
thf(fact_658_bot__set__def,axiom,
( bot_bot_set_set_rat
= ( collect_set_rat @ bot_bot_set_rat_o ) ) ).
% bot_set_def
thf(fact_659_bot__set__def,axiom,
( bot_bot_set_rat
= ( collect_rat @ bot_bot_rat_o ) ) ).
% bot_set_def
thf(fact_660_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_661_le__numeral__extra_I3_J,axiom,
ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% le_numeral_extra(3)
thf(fact_662_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_663_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_664_leD,axiom,
! [Y: set_rat,X2: set_rat] :
( ( ord_less_eq_set_rat @ Y @ X2 )
=> ~ ( ord_less_set_rat @ X2 @ Y ) ) ).
% leD
thf(fact_665_leD,axiom,
! [Y: rat,X2: rat] :
( ( ord_less_eq_rat @ Y @ X2 )
=> ~ ( ord_less_rat @ X2 @ Y ) ) ).
% leD
thf(fact_666_leD,axiom,
! [Y: num,X2: num] :
( ( ord_less_eq_num @ Y @ X2 )
=> ~ ( ord_less_num @ X2 @ Y ) ) ).
% leD
thf(fact_667_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_668_leD,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_int @ X2 @ Y ) ) ).
% leD
thf(fact_669_leI,axiom,
! [X2: rat,Y: rat] :
( ~ ( ord_less_rat @ X2 @ Y )
=> ( ord_less_eq_rat @ Y @ X2 ) ) ).
% leI
thf(fact_670_leI,axiom,
! [X2: num,Y: num] :
( ~ ( ord_less_num @ X2 @ Y )
=> ( ord_less_eq_num @ Y @ X2 ) ) ).
% leI
thf(fact_671_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_672_leI,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% leI
thf(fact_673_nless__le,axiom,
! [A2: set_rat,B: set_rat] :
( ( ~ ( ord_less_set_rat @ A2 @ B ) )
= ( ~ ( ord_less_eq_set_rat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_674_nless__le,axiom,
! [A2: rat,B: rat] :
( ( ~ ( ord_less_rat @ A2 @ B ) )
= ( ~ ( ord_less_eq_rat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_675_nless__le,axiom,
! [A2: num,B: num] :
( ( ~ ( ord_less_num @ A2 @ B ) )
= ( ~ ( ord_less_eq_num @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_676_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_677_nless__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_int @ A2 @ B ) )
= ( ~ ( ord_less_eq_int @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_678_antisym__conv1,axiom,
! [X2: set_rat,Y: set_rat] :
( ~ ( ord_less_set_rat @ X2 @ Y )
=> ( ( ord_less_eq_set_rat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_679_antisym__conv1,axiom,
! [X2: rat,Y: rat] :
( ~ ( ord_less_rat @ X2 @ Y )
=> ( ( ord_less_eq_rat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_680_antisym__conv1,axiom,
! [X2: num,Y: num] :
( ~ ( ord_less_num @ X2 @ Y )
=> ( ( ord_less_eq_num @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_681_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_682_antisym__conv1,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_683_antisym__conv2,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ Y )
=> ( ( ~ ( ord_less_set_rat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_684_antisym__conv2,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
=> ( ( ~ ( ord_less_rat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_685_antisym__conv2,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ~ ( ord_less_num @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_686_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_687_antisym__conv2,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_688_dense__ge,axiom,
! [Z3: rat,Y: rat] :
( ! [X: rat] :
( ( ord_less_rat @ Z3 @ X )
=> ( ord_less_eq_rat @ Y @ X ) )
=> ( ord_less_eq_rat @ Y @ Z3 ) ) ).
% dense_ge
thf(fact_689_dense__le,axiom,
! [Y: rat,Z3: rat] :
( ! [X: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_eq_rat @ X @ Z3 ) )
=> ( ord_less_eq_rat @ Y @ Z3 ) ) ).
% dense_le
thf(fact_690_less__le__not__le,axiom,
( ord_less_set_rat
= ( ^ [X4: set_rat,Y4: set_rat] :
( ( ord_less_eq_set_rat @ X4 @ Y4 )
& ~ ( ord_less_eq_set_rat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_691_less__le__not__le,axiom,
( ord_less_rat
= ( ^ [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
& ~ ( ord_less_eq_rat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_692_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
& ~ ( ord_less_eq_num @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_693_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_694_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_695_not__le__imp__less,axiom,
! [Y: rat,X2: rat] :
( ~ ( ord_less_eq_rat @ Y @ X2 )
=> ( ord_less_rat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_696_not__le__imp__less,axiom,
! [Y: num,X2: num] :
( ~ ( ord_less_eq_num @ Y @ X2 )
=> ( ord_less_num @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_697_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_698_not__le__imp__less,axiom,
! [Y: int,X2: int] :
( ~ ( ord_less_eq_int @ Y @ X2 )
=> ( ord_less_int @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_699_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_rat
= ( ^ [A5: set_rat,B4: set_rat] :
( ( ord_less_set_rat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_700_order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_rat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_701_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_num @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_702_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_703_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_int @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_704_order_Ostrict__iff__order,axiom,
( ord_less_set_rat
= ( ^ [A5: set_rat,B4: set_rat] :
( ( ord_less_eq_set_rat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_705_order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_eq_rat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_706_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_707_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_708_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_709_order_Ostrict__trans1,axiom,
! [A2: set_rat,B: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B )
=> ( ( ord_less_set_rat @ B @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_710_order_Ostrict__trans1,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_711_order_Ostrict__trans1,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_712_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_713_order_Ostrict__trans1,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_714_order_Ostrict__trans2,axiom,
! [A2: set_rat,B: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ B )
=> ( ( ord_less_eq_set_rat @ B @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_715_order_Ostrict__trans2,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_716_order_Ostrict__trans2,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_717_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_718_order_Ostrict__trans2,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_719_order_Ostrict__iff__not,axiom,
( ord_less_set_rat
= ( ^ [A5: set_rat,B4: set_rat] :
( ( ord_less_eq_set_rat @ A5 @ B4 )
& ~ ( ord_less_eq_set_rat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_720_order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_eq_rat @ A5 @ B4 )
& ~ ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_721_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
& ~ ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_722_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_723_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_724_dense__ge__bounded,axiom,
! [Z3: rat,X2: rat,Y: rat] :
( ( ord_less_rat @ Z3 @ X2 )
=> ( ! [W: rat] :
( ( ord_less_rat @ Z3 @ W )
=> ( ( ord_less_rat @ W @ X2 )
=> ( ord_less_eq_rat @ Y @ W ) ) )
=> ( ord_less_eq_rat @ Y @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_725_dense__le__bounded,axiom,
! [X2: rat,Y: rat,Z3: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ( ! [W: rat] :
( ( ord_less_rat @ X2 @ W )
=> ( ( ord_less_rat @ W @ Y )
=> ( ord_less_eq_rat @ W @ Z3 ) ) )
=> ( ord_less_eq_rat @ Y @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_726_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_rat
= ( ^ [B4: set_rat,A5: set_rat] :
( ( ord_less_set_rat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_727_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [B4: rat,A5: rat] :
( ( ord_less_rat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_728_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_num @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_729_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_730_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_int @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_731_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_rat
= ( ^ [B4: set_rat,A5: set_rat] :
( ( ord_less_eq_set_rat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_732_dual__order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [B4: rat,A5: rat] :
( ( ord_less_eq_rat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_733_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_eq_num @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_734_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_735_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_736_dual__order_Ostrict__trans1,axiom,
! [B: set_rat,A2: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ B @ A2 )
=> ( ( ord_less_set_rat @ C @ B )
=> ( ord_less_set_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_737_dual__order_Ostrict__trans1,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A2 )
=> ( ( ord_less_rat @ C @ B )
=> ( ord_less_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_738_dual__order_Ostrict__trans1,axiom,
! [B: num,A2: num,C: num] :
( ( ord_less_eq_num @ B @ A2 )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_739_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_740_dual__order_Ostrict__trans1,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_741_dual__order_Ostrict__trans2,axiom,
! [B: set_rat,A2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ B @ A2 )
=> ( ( ord_less_eq_set_rat @ C @ B )
=> ( ord_less_set_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_742_dual__order_Ostrict__trans2,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B @ A2 )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_743_dual__order_Ostrict__trans2,axiom,
! [B: num,A2: num,C: num] :
( ( ord_less_num @ B @ A2 )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_744_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_745_dual__order_Ostrict__trans2,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_746_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_rat
= ( ^ [B4: set_rat,A5: set_rat] :
( ( ord_less_eq_set_rat @ B4 @ A5 )
& ~ ( ord_less_eq_set_rat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_747_dual__order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [B4: rat,A5: rat] :
( ( ord_less_eq_rat @ B4 @ A5 )
& ~ ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_748_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_eq_num @ B4 @ A5 )
& ~ ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_749_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_750_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_751_order_Ostrict__implies__order,axiom,
! [A2: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A2 @ B )
=> ( ord_less_eq_set_rat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_752_order_Ostrict__implies__order,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ord_less_eq_rat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_753_order_Ostrict__implies__order,axiom,
! [A2: num,B: num] :
( ( ord_less_num @ A2 @ B )
=> ( ord_less_eq_num @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_754_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_755_order_Ostrict__implies__order,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_756_dual__order_Ostrict__implies__order,axiom,
! [B: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B @ A2 )
=> ( ord_less_eq_set_rat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_757_dual__order_Ostrict__implies__order,axiom,
! [B: rat,A2: rat] :
( ( ord_less_rat @ B @ A2 )
=> ( ord_less_eq_rat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_758_dual__order_Ostrict__implies__order,axiom,
! [B: num,A2: num] :
( ( ord_less_num @ B @ A2 )
=> ( ord_less_eq_num @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_759_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_760_dual__order_Ostrict__implies__order,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_761_order__le__less,axiom,
( ord_less_eq_set_rat
= ( ^ [X4: set_rat,Y4: set_rat] :
( ( ord_less_set_rat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_762_order__le__less,axiom,
( ord_less_eq_rat
= ( ^ [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_763_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_764_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_765_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_766_order__less__le,axiom,
( ord_less_set_rat
= ( ^ [X4: set_rat,Y4: set_rat] :
( ( ord_less_eq_set_rat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_767_order__less__le,axiom,
( ord_less_rat
= ( ^ [X4: rat,Y4: rat] :
( ( ord_less_eq_rat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_768_order__less__le,axiom,
( ord_less_num
= ( ^ [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_769_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_770_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_771_linorder__not__le,axiom,
! [X2: rat,Y: rat] :
( ( ~ ( ord_less_eq_rat @ X2 @ Y ) )
= ( ord_less_rat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_772_linorder__not__le,axiom,
! [X2: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X2 @ Y ) )
= ( ord_less_num @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_773_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_774_linorder__not__le,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y ) )
= ( ord_less_int @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_775_linorder__not__less,axiom,
! [X2: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X2 @ Y ) )
= ( ord_less_eq_rat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_776_linorder__not__less,axiom,
! [X2: num,Y: num] :
( ( ~ ( ord_less_num @ X2 @ Y ) )
= ( ord_less_eq_num @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_777_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_778_linorder__not__less,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_779_order__less__imp__le,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ( ord_less_eq_set_rat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_780_order__less__imp__le,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ( ord_less_eq_rat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_781_order__less__imp__le,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_eq_num @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_782_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_783_order__less__imp__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_784_order__le__neq__trans,axiom,
! [A2: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_rat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_785_order__le__neq__trans,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_rat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_786_order__le__neq__trans,axiom,
! [A2: num,B: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_num @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_787_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_788_order__le__neq__trans,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_789_order__neq__le__trans,axiom,
! [A2: set_rat,B: set_rat] :
( ( A2 != B )
=> ( ( ord_less_eq_set_rat @ A2 @ B )
=> ( ord_less_set_rat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_790_order__neq__le__trans,axiom,
! [A2: rat,B: rat] :
( ( A2 != B )
=> ( ( ord_less_eq_rat @ A2 @ B )
=> ( ord_less_rat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_791_order__neq__le__trans,axiom,
! [A2: num,B: num] :
( ( A2 != B )
=> ( ( ord_less_eq_num @ A2 @ B )
=> ( ord_less_num @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_792_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_793_order__neq__le__trans,axiom,
! [A2: int,B: int] :
( ( A2 != B )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_794_order__le__less__trans,axiom,
! [X2: set_rat,Y: set_rat,Z3: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ Y )
=> ( ( ord_less_set_rat @ Y @ Z3 )
=> ( ord_less_set_rat @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_795_order__le__less__trans,axiom,
! [X2: rat,Y: rat,Z3: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
=> ( ( ord_less_rat @ Y @ Z3 )
=> ( ord_less_rat @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_796_order__le__less__trans,axiom,
! [X2: num,Y: num,Z3: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ord_less_num @ Y @ Z3 )
=> ( ord_less_num @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_797_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_798_order__le__less__trans,axiom,
! [X2: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_799_order__less__le__trans,axiom,
! [X2: set_rat,Y: set_rat,Z3: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ( ( ord_less_eq_set_rat @ Y @ Z3 )
=> ( ord_less_set_rat @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_800_order__less__le__trans,axiom,
! [X2: rat,Y: rat,Z3: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ( ( ord_less_eq_rat @ Y @ Z3 )
=> ( ord_less_rat @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_801_order__less__le__trans,axiom,
! [X2: num,Y: num,Z3: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ( ord_less_eq_num @ Y @ Z3 )
=> ( ord_less_num @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_802_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_803_order__less__le__trans,axiom,
! [X2: int,Y: int,Z3: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_804_order__le__less__subst1,axiom,
! [A2: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_805_order__le__less__subst1,axiom,
! [A2: rat,F: num > rat,B: num,C: num] :
( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_806_order__le__less__subst1,axiom,
! [A2: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_807_order__le__less__subst1,axiom,
! [A2: rat,F: int > rat,B: int,C: int] :
( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_808_order__le__less__subst1,axiom,
! [A2: num,F: rat > num,B: rat,C: rat] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_809_order__le__less__subst1,axiom,
! [A2: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_810_order__le__less__subst1,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_811_order__le__less__subst1,axiom,
! [A2: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_812_order__le__less__subst1,axiom,
! [A2: nat,F: rat > nat,B: rat,C: rat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_813_order__le__less__subst1,axiom,
! [A2: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_814_order__le__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_815_order__le__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_816_order__le__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_817_order__le__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_818_order__le__less__subst2,axiom,
! [A2: num,B: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_819_order__le__less__subst2,axiom,
! [A2: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_820_order__le__less__subst2,axiom,
! [A2: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_821_order__le__less__subst2,axiom,
! [A2: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_822_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_823_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_824_order__less__le__subst1,axiom,
! [A2: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_825_order__less__le__subst1,axiom,
! [A2: num,F: rat > num,B: rat,C: rat] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_826_order__less__le__subst1,axiom,
! [A2: nat,F: rat > nat,B: rat,C: rat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_827_order__less__le__subst1,axiom,
! [A2: int,F: rat > int,B: rat,C: rat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_828_order__less__le__subst1,axiom,
! [A2: rat,F: num > rat,B: num,C: num] :
( ( ord_less_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_829_order__less__le__subst1,axiom,
! [A2: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_830_order__less__le__subst1,axiom,
! [A2: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_831_order__less__le__subst1,axiom,
! [A2: int,F: num > int,B: num,C: num] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_832_order__less__le__subst1,axiom,
! [A2: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_833_order__less__le__subst1,axiom,
! [A2: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_834_order__less__le__subst2,axiom,
! [A2: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_835_order__less__le__subst2,axiom,
! [A2: num,B: num,F: num > rat,C: rat] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_836_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_837_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > rat,C: rat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_838_order__less__le__subst2,axiom,
! [A2: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_839_order__less__le__subst2,axiom,
! [A2: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_840_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_841_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > num,C: num] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_842_order__less__le__subst2,axiom,
! [A2: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_843_order__less__le__subst2,axiom,
! [A2: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_num @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_844_linorder__le__less__linear,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
| ( ord_less_rat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_845_linorder__le__less__linear,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
| ( ord_less_num @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_846_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_847_linorder__le__less__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_int @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_848_order__le__imp__less__or__eq,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ Y )
=> ( ( ord_less_set_rat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_849_order__le__imp__less__or__eq,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
=> ( ( ord_less_rat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_850_order__le__imp__less__or__eq,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ord_less_num @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_851_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_852_order__le__imp__less__or__eq,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_int @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_853_minf_I8_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z )
=> ~ ( ord_less_eq_rat @ T @ X3 ) ) ).
% minf(8)
thf(fact_854_minf_I8_J,axiom,
! [T: num] :
? [Z: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z )
=> ~ ( ord_less_eq_num @ T @ X3 ) ) ).
% minf(8)
thf(fact_855_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% minf(8)
thf(fact_856_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z )
=> ~ ( ord_less_eq_int @ T @ X3 ) ) ).
% minf(8)
thf(fact_857_minf_I6_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z )
=> ( ord_less_eq_rat @ X3 @ T ) ) ).
% minf(6)
thf(fact_858_minf_I6_J,axiom,
! [T: num] :
? [Z: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z )
=> ( ord_less_eq_num @ X3 @ T ) ) ).
% minf(6)
thf(fact_859_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( ord_less_eq_nat @ X3 @ T ) ) ).
% minf(6)
thf(fact_860_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z )
=> ( ord_less_eq_int @ X3 @ T ) ) ).
% minf(6)
thf(fact_861_pinf_I8_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ( ord_less_eq_rat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_862_pinf_I8_J,axiom,
! [T: num] :
? [Z: num] :
! [X3: num] :
( ( ord_less_num @ Z @ X3 )
=> ( ord_less_eq_num @ T @ X3 ) ) ).
% pinf(8)
thf(fact_863_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_864_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X3: int] :
( ( ord_less_int @ Z @ X3 )
=> ( ord_less_eq_int @ T @ X3 ) ) ).
% pinf(8)
thf(fact_865_pinf_I6_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ~ ( ord_less_eq_rat @ X3 @ T ) ) ).
% pinf(6)
thf(fact_866_pinf_I6_J,axiom,
! [T: num] :
? [Z: num] :
! [X3: num] :
( ( ord_less_num @ Z @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ T ) ) ).
% pinf(6)
thf(fact_867_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T ) ) ).
% pinf(6)
thf(fact_868_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X3: int] :
( ( ord_less_int @ Z @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ T ) ) ).
% pinf(6)
thf(fact_869_verit__comp__simplify1_I3_J,axiom,
! [B5: rat,A6: rat] :
( ( ~ ( ord_less_eq_rat @ B5 @ A6 ) )
= ( ord_less_rat @ A6 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_870_verit__comp__simplify1_I3_J,axiom,
! [B5: num,A6: num] :
( ( ~ ( ord_less_eq_num @ B5 @ A6 ) )
= ( ord_less_num @ A6 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_871_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A6 ) )
= ( ord_less_nat @ A6 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_872_verit__comp__simplify1_I3_J,axiom,
! [B5: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B5 @ A6 ) )
= ( ord_less_int @ A6 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_873_complete__interval,axiom,
! [A2: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A2 @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ C3 ) )
=> ( P @ X3 ) )
& ! [D3: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A2 @ X )
& ( ord_less_nat @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_874_complete__interval,axiom,
! [A2: int,B: int,P: int > $o] :
( ( ord_less_int @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C3: int] :
( ( ord_less_eq_int @ A2 @ C3 )
& ( ord_less_eq_int @ C3 @ B )
& ! [X3: int] :
( ( ( ord_less_eq_int @ A2 @ X3 )
& ( ord_less_int @ X3 @ C3 ) )
=> ( P @ X3 ) )
& ! [D3: int] :
( ! [X: int] :
( ( ( ord_less_eq_int @ A2 @ X )
& ( ord_less_int @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_int @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_875_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_876_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_877_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_878_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( ord_less_eq_rat @ K @ L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_879_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_880_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_881_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( ord_less_eq_rat @ K @ L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_882_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_883_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_884_add__mono,axiom,
! [A2: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_885_add__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_886_add__mono,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_887_add__left__mono,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_888_add__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_889_add__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_890_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A2 @ C3 ) ) ) ).
% less_eqE
thf(fact_891_add__right__mono,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_892_add__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_893_add__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_894_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C4: nat] :
( B4
= ( plus_plus_nat @ A5 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_895_add__le__imp__le__left,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
=> ( ord_less_eq_rat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_896_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_897_add__le__imp__le__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_898_add__le__imp__le__right,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
=> ( ord_less_eq_rat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_899_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_900_add__le__imp__le__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_901_bot_Oextremum,axiom,
! [A2: set_rat] : ( ord_less_eq_set_rat @ bot_bot_set_rat @ A2 ) ).
% bot.extremum
thf(fact_902_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_903_bot_Oextremum__unique,axiom,
! [A2: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ bot_bot_set_rat )
= ( A2 = bot_bot_set_rat ) ) ).
% bot.extremum_unique
thf(fact_904_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_905_bot_Oextremum__uniqueI,axiom,
! [A2: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ bot_bot_set_rat )
=> ( A2 = bot_bot_set_rat ) ) ).
% bot.extremum_uniqueI
thf(fact_906_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_907_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
% zero_neq_one
thf(fact_908_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_909_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_910_less__numeral__extra_I4_J,axiom,
~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% less_numeral_extra(4)
thf(fact_911_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_912_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_913_psubsetE,axiom,
! [A: set_rat,B3: set_rat] :
( ( ord_less_set_rat @ A @ B3 )
=> ~ ( ( ord_less_eq_set_rat @ A @ B3 )
=> ( ord_less_eq_set_rat @ B3 @ A ) ) ) ).
% psubsetE
thf(fact_914_psubset__eq,axiom,
( ord_less_set_rat
= ( ^ [A4: set_rat,B6: set_rat] :
( ( ord_less_eq_set_rat @ A4 @ B6 )
& ( A4 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_915_psubset__imp__subset,axiom,
! [A: set_rat,B3: set_rat] :
( ( ord_less_set_rat @ A @ B3 )
=> ( ord_less_eq_set_rat @ A @ B3 ) ) ).
% psubset_imp_subset
thf(fact_916_psubset__subset__trans,axiom,
! [A: set_rat,B3: set_rat,C2: set_rat] :
( ( ord_less_set_rat @ A @ B3 )
=> ( ( ord_less_eq_set_rat @ B3 @ C2 )
=> ( ord_less_set_rat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_917_subset__not__subset__eq,axiom,
( ord_less_set_rat
= ( ^ [A4: set_rat,B6: set_rat] :
( ( ord_less_eq_set_rat @ A4 @ B6 )
& ~ ( ord_less_eq_set_rat @ B6 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_918_subset__psubset__trans,axiom,
! [A: set_rat,B3: set_rat,C2: set_rat] :
( ( ord_less_eq_set_rat @ A @ B3 )
=> ( ( ord_less_set_rat @ B3 @ C2 )
=> ( ord_less_set_rat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_919_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_rat
= ( ^ [A4: set_rat,B6: set_rat] :
( ( ord_less_set_rat @ A4 @ B6 )
| ( A4 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_920_less__eq__rat__def,axiom,
( ord_less_eq_rat
= ( ^ [X4: rat,Y4: rat] :
( ( ord_less_rat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% less_eq_rat_def
thf(fact_921_add__nonpos__eq__0__iff,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ( ( plus_plus_rat @ X2 @ Y )
= zero_zero_rat )
= ( ( X2 = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_922_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_923_add__nonpos__eq__0__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X2 @ Y )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_924_add__nonneg__eq__0__iff,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ( plus_plus_rat @ X2 @ Y )
= zero_zero_rat )
= ( ( X2 = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_925_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_926_add__nonneg__eq__0__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X2 @ Y )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_927_add__nonpos__nonpos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% add_nonpos_nonpos
thf(fact_928_add__nonpos__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_929_add__nonpos__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_930_add__nonneg__nonneg,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_931_add__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_932_add__nonneg__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_933_add__increasing2,axiom,
! [C: rat,B: rat,A2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ B @ A2 )
=> ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_934_add__increasing2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_935_add__increasing2,axiom,
! [C: int,B: int,A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_936_add__decreasing2,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A2 @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_937_add__decreasing2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_938_add__decreasing2,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_939_add__increasing,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_940_add__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_941_add__increasing,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_942_add__decreasing,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_943_add__decreasing,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_944_add__decreasing,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_945_add__mono__thms__linordered__field_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( ord_less_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_946_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_947_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_948_add__mono__thms__linordered__field_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( ord_less_eq_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_949_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_950_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_951_add__le__less__mono,axiom,
! [A2: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_rat @ C @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_952_add__le__less__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_953_add__le__less__mono,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_954_add__less__le__mono,axiom,
! [A2: rat,B: rat,C: rat,D: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_955_add__less__le__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_956_add__less__le__mono,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_957_less__numeral__extra_I1_J,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% less_numeral_extra(1)
thf(fact_958_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_959_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_960_zero__less__one,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one
thf(fact_961_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_962_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_963_not__one__less__zero,axiom,
~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_less_zero
thf(fact_964_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_965_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_966_add__mono1,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% add_mono1
thf(fact_967_add__mono1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_968_add__mono1,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_969_less__add__one,axiom,
! [A2: rat] : ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ one_one_rat ) ) ).
% less_add_one
thf(fact_970_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_971_less__add__one,axiom,
! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).
% less_add_one
thf(fact_972_add__neg__nonpos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_nonpos
thf(fact_973_add__neg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_974_add__neg__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_975_add__nonneg__pos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_976_add__nonneg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_977_add__nonneg__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_978_add__nonpos__neg,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% add_nonpos_neg
thf(fact_979_add__nonpos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_980_add__nonpos__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_981_add__pos__nonneg,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_982_add__pos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_983_add__pos__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_984_add__strict__increasing,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_985_add__strict__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_986_add__strict__increasing,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_987_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
= one_one_rat ) ).
% dbl_inc_simps(2)
thf(fact_988_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_989_convex__bound__lt,axiom,
! [X2: rat,A2: rat,Y: rat,U: rat,V: rat] :
( ( ord_less_rat @ X2 @ A2 )
=> ( ( ord_less_rat @ Y @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V )
=> ( ( ( plus_plus_rat @ U @ V )
= one_one_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X2 ) @ ( times_times_rat @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_990_convex__bound__lt,axiom,
! [X2: int,A2: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X2 @ A2 )
=> ( ( ord_less_int @ Y @ A2 )
=> ( ( 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 @ X2 ) @ ( times_times_int @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_991_divide__le__eq__1__neg,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A2 ) @ one_one_rat )
= ( ord_less_eq_rat @ A2 @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_992_divide__le__eq__1__pos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A2 ) @ one_one_rat )
= ( ord_less_eq_rat @ B @ A2 ) ) ) ).
% divide_le_eq_1_pos
thf(fact_993_le__divide__eq__1__neg,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A2 ) )
= ( ord_less_eq_rat @ B @ A2 ) ) ) ).
% le_divide_eq_1_neg
thf(fact_994_le__divide__eq__1__pos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A2 ) )
= ( ord_less_eq_rat @ A2 @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_995_dbl__inc__def,axiom,
( neg_nu5219082963157363817nc_rat
= ( ^ [X4: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X4 @ X4 ) @ one_one_rat ) ) ) ).
% dbl_inc_def
thf(fact_996_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X4: int] : ( plus_plus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_997_subsetI,axiom,
! [A: set_set_rat,B3: set_set_rat] :
( ! [X: set_rat] :
( ( member_set_rat @ X @ A )
=> ( member_set_rat @ X @ B3 ) )
=> ( ord_le513522071413781156et_rat @ A @ B3 ) ) ).
% subsetI
thf(fact_998_subsetI,axiom,
! [A: set_rat,B3: set_rat] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( member_rat @ X @ B3 ) )
=> ( ord_less_eq_set_rat @ A @ B3 ) ) ).
% subsetI
thf(fact_999_subset__antisym,axiom,
! [A: set_rat,B3: set_rat] :
( ( ord_less_eq_set_rat @ A @ B3 )
=> ( ( ord_less_eq_set_rat @ B3 @ A )
=> ( A = B3 ) ) ) ).
% subset_antisym
thf(fact_1000_mult__zero__left,axiom,
! [A2: rat] :
( ( times_times_rat @ zero_zero_rat @ A2 )
= zero_zero_rat ) ).
% mult_zero_left
thf(fact_1001_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1002_mult__zero__left,axiom,
! [A2: int] :
( ( times_times_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1003_mult__zero__right,axiom,
! [A2: rat] :
( ( times_times_rat @ A2 @ zero_zero_rat )
= zero_zero_rat ) ).
% mult_zero_right
thf(fact_1004_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1005_mult__zero__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1006_mult__eq__0__iff,axiom,
! [A2: rat,B: rat] :
( ( ( times_times_rat @ A2 @ B )
= zero_zero_rat )
= ( ( A2 = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% mult_eq_0_iff
thf(fact_1007_mult__eq__0__iff,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1008_mult__eq__0__iff,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_1009_mult__cancel__left,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ( times_times_rat @ C @ A2 )
= ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_1010_mult__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_1011_mult__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_1012_mult__cancel__right,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ( times_times_rat @ A2 @ C )
= ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_1013_mult__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_1014_mult__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_1015_div__0,axiom,
! [A2: rat] :
( ( divide_divide_rat @ zero_zero_rat @ A2 )
= zero_zero_rat ) ).
% div_0
thf(fact_1016_div__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% div_0
thf(fact_1017_div__0,axiom,
! [A2: int] :
( ( divide_divide_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% div_0
thf(fact_1018_divide__eq__0__iff,axiom,
! [A2: rat,B: rat] :
( ( ( divide_divide_rat @ A2 @ B )
= zero_zero_rat )
= ( ( A2 = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divide_eq_0_iff
thf(fact_1019_div__by__0,axiom,
! [A2: rat] :
( ( divide_divide_rat @ A2 @ zero_zero_rat )
= zero_zero_rat ) ).
% div_by_0
thf(fact_1020_div__by__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_1021_div__by__0,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_1022_divide__cancel__left,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ( divide_divide_rat @ C @ A2 )
= ( divide_divide_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( A2 = B ) ) ) ).
% divide_cancel_left
thf(fact_1023_divide__cancel__right,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ( divide_divide_rat @ A2 @ C )
= ( divide_divide_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( A2 = B ) ) ) ).
% divide_cancel_right
thf(fact_1024_division__ring__divide__zero,axiom,
! [A2: rat] :
( ( divide_divide_rat @ A2 @ zero_zero_rat )
= zero_zero_rat ) ).
% division_ring_divide_zero
thf(fact_1025_mult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% mult_1
thf(fact_1026_mult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% mult_1
thf(fact_1027_mult_Oright__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_1028_mult_Oright__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.right_neutral
thf(fact_1029_div__by__1,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ one_one_nat )
= A2 ) ).
% div_by_1
thf(fact_1030_div__by__1,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ one_one_int )
= A2 ) ).
% div_by_1
thf(fact_1031_mult__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( C
= ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_left1
thf(fact_1032_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_1033_mult__cancel__left2,axiom,
! [C: rat,A2: rat] :
( ( ( times_times_rat @ C @ A2 )
= C )
= ( ( C = zero_zero_rat )
| ( A2 = one_one_rat ) ) ) ).
% mult_cancel_left2
thf(fact_1034_mult__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ( times_times_int @ C @ A2 )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_1035_mult__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( C
= ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_right1
thf(fact_1036_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_1037_mult__cancel__right2,axiom,
! [A2: rat,C: rat] :
( ( ( times_times_rat @ A2 @ C )
= C )
= ( ( C = zero_zero_rat )
| ( A2 = one_one_rat ) ) ) ).
% mult_cancel_right2
thf(fact_1038_mult__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ( times_times_int @ A2 @ C )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1039_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: rat,A2: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1040_nonzero__mult__div__cancel__right,axiom,
! [B: rat,A2: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A2 @ B ) @ B )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1041_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1042_nonzero__mult__div__cancel__right,axiom,
! [B: int,A2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ B ) @ B )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1043_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: rat,A2: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
= ( divide_divide_rat @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1044_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: rat,A2: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ B @ C ) )
= ( divide_divide_rat @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1045_nonzero__mult__div__cancel__left,axiom,
! [A2: rat,B: rat] :
( ( A2 != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A2 @ B ) @ A2 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1046_nonzero__mult__div__cancel__left,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B ) @ A2 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1047_nonzero__mult__div__cancel__left,axiom,
! [A2: int,B: int] :
( ( A2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ B ) @ A2 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1048_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: rat,A2: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1049_mult__divide__mult__cancel__left__if,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ( C = zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
= zero_zero_rat ) )
& ( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A2 @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1050_divide__eq__1__iff,axiom,
! [A2: rat,B: rat] :
( ( ( divide_divide_rat @ A2 @ B )
= one_one_rat )
= ( ( B != zero_zero_rat )
& ( A2 = B ) ) ) ).
% divide_eq_1_iff
thf(fact_1051_div__self,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( divide_divide_rat @ A2 @ A2 )
= one_one_rat ) ) ).
% div_self
thf(fact_1052_div__self,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( divide_divide_nat @ A2 @ A2 )
= one_one_nat ) ) ).
% div_self
thf(fact_1053_div__self,axiom,
! [A2: int] :
( ( A2 != zero_zero_int )
=> ( ( divide_divide_int @ A2 @ A2 )
= one_one_int ) ) ).
% div_self
thf(fact_1054_one__eq__divide__iff,axiom,
! [A2: rat,B: rat] :
( ( one_one_rat
= ( divide_divide_rat @ A2 @ B ) )
= ( ( B != zero_zero_rat )
& ( A2 = B ) ) ) ).
% one_eq_divide_iff
thf(fact_1055_divide__self,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( divide_divide_rat @ A2 @ A2 )
= one_one_rat ) ) ).
% divide_self
thf(fact_1056_divide__self__if,axiom,
! [A2: rat] :
( ( ( A2 = zero_zero_rat )
=> ( ( divide_divide_rat @ A2 @ A2 )
= zero_zero_rat ) )
& ( ( A2 != zero_zero_rat )
=> ( ( divide_divide_rat @ A2 @ A2 )
= one_one_rat ) ) ) ).
% divide_self_if
thf(fact_1057_divide__eq__eq__1,axiom,
! [B: rat,A2: rat] :
( ( ( divide_divide_rat @ B @ A2 )
= one_one_rat )
= ( ( A2 != zero_zero_rat )
& ( A2 = B ) ) ) ).
% divide_eq_eq_1
thf(fact_1058_eq__divide__eq__1,axiom,
! [B: rat,A2: rat] :
( ( one_one_rat
= ( divide_divide_rat @ B @ A2 ) )
= ( ( A2 != zero_zero_rat )
& ( A2 = B ) ) ) ).
% eq_divide_eq_1
thf(fact_1059_one__divide__eq__0__iff,axiom,
! [A2: rat] :
( ( ( divide_divide_rat @ one_one_rat @ A2 )
= zero_zero_rat )
= ( A2 = zero_zero_rat ) ) ).
% one_divide_eq_0_iff
thf(fact_1060_zero__eq__1__divide__iff,axiom,
! [A2: rat] :
( ( zero_zero_rat
= ( divide_divide_rat @ one_one_rat @ A2 ) )
= ( A2 = zero_zero_rat ) ) ).
% zero_eq_1_divide_iff
thf(fact_1061_divide__le__0__1__iff,axiom,
! [A2: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% divide_le_0_1_iff
thf(fact_1062_zero__le__divide__1__iff,axiom,
! [A2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% zero_le_divide_1_iff
thf(fact_1063_zero__less__divide__1__iff,axiom,
! [A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) )
= ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% zero_less_divide_1_iff
thf(fact_1064_less__divide__eq__1__pos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A2 ) )
= ( ord_less_rat @ A2 @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1065_less__divide__eq__1__neg,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A2 ) )
= ( ord_less_rat @ B @ A2 ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1066_divide__less__eq__1__pos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ A2 ) @ one_one_rat )
= ( ord_less_rat @ B @ A2 ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1067_divide__less__eq__1__neg,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ A2 ) @ one_one_rat )
= ( ord_less_rat @ A2 @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1068_divide__less__0__1__iff,axiom,
! [A2: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ zero_zero_rat )
= ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% divide_less_0_1_iff
thf(fact_1069_nonzero__divide__mult__cancel__left,axiom,
! [A2: rat,B: rat] :
( ( A2 != zero_zero_rat )
=> ( ( divide_divide_rat @ A2 @ ( times_times_rat @ A2 @ B ) )
= ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1070_nonzero__divide__mult__cancel__right,axiom,
! [B: rat,A2: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ B @ ( times_times_rat @ A2 @ B ) )
= ( divide_divide_rat @ one_one_rat @ A2 ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1071_in__mono,axiom,
! [A: set_set_rat,B3: set_set_rat,X2: set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B3 )
=> ( ( member_set_rat @ X2 @ A )
=> ( member_set_rat @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_1072_in__mono,axiom,
! [A: set_rat,B3: set_rat,X2: rat] :
( ( ord_less_eq_set_rat @ A @ B3 )
=> ( ( member_rat @ X2 @ A )
=> ( member_rat @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_1073_subsetD,axiom,
! [A: set_set_rat,B3: set_set_rat,C: set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B3 )
=> ( ( member_set_rat @ C @ A )
=> ( member_set_rat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_1074_subsetD,axiom,
! [A: set_rat,B3: set_rat,C: rat] :
( ( ord_less_eq_set_rat @ A @ B3 )
=> ( ( member_rat @ C @ A )
=> ( member_rat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_1075_equalityE,axiom,
! [A: set_rat,B3: set_rat] :
( ( A = B3 )
=> ~ ( ( ord_less_eq_set_rat @ A @ B3 )
=> ~ ( ord_less_eq_set_rat @ B3 @ A ) ) ) ).
% equalityE
thf(fact_1076_subset__eq,axiom,
( ord_le513522071413781156et_rat
= ( ^ [A4: set_set_rat,B6: set_set_rat] :
! [X4: set_rat] :
( ( member_set_rat @ X4 @ A4 )
=> ( member_set_rat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_1077_subset__eq,axiom,
( ord_less_eq_set_rat
= ( ^ [A4: set_rat,B6: set_rat] :
! [X4: rat] :
( ( member_rat @ X4 @ A4 )
=> ( member_rat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_1078_equalityD1,axiom,
! [A: set_rat,B3: set_rat] :
( ( A = B3 )
=> ( ord_less_eq_set_rat @ A @ B3 ) ) ).
% equalityD1
thf(fact_1079_equalityD2,axiom,
! [A: set_rat,B3: set_rat] :
( ( A = B3 )
=> ( ord_less_eq_set_rat @ B3 @ A ) ) ).
% equalityD2
thf(fact_1080_subset__iff,axiom,
( ord_le513522071413781156et_rat
= ( ^ [A4: set_set_rat,B6: set_set_rat] :
! [T3: set_rat] :
( ( member_set_rat @ T3 @ A4 )
=> ( member_set_rat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_1081_subset__iff,axiom,
( ord_less_eq_set_rat
= ( ^ [A4: set_rat,B6: set_rat] :
! [T3: rat] :
( ( member_rat @ T3 @ A4 )
=> ( member_rat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_1082_subset__refl,axiom,
! [A: set_rat] : ( ord_less_eq_set_rat @ A @ A ) ).
% subset_refl
thf(fact_1083_Collect__mono,axiom,
! [P: set_rat > $o,Q: set_rat > $o] :
( ! [X: set_rat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le513522071413781156et_rat @ ( collect_set_rat @ P ) @ ( collect_set_rat @ Q ) ) ) ).
% Collect_mono
thf(fact_1084_Collect__mono,axiom,
! [P: rat > $o,Q: rat > $o] :
( ! [X: rat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_rat @ ( collect_rat @ P ) @ ( collect_rat @ Q ) ) ) ).
% Collect_mono
thf(fact_1085_subset__trans,axiom,
! [A: set_rat,B3: set_rat,C2: set_rat] :
( ( ord_less_eq_set_rat @ A @ B3 )
=> ( ( ord_less_eq_set_rat @ B3 @ C2 )
=> ( ord_less_eq_set_rat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_1086_set__eq__subset,axiom,
( ( ^ [Y5: set_rat,Z5: set_rat] : ( Y5 = Z5 ) )
= ( ^ [A4: set_rat,B6: set_rat] :
( ( ord_less_eq_set_rat @ A4 @ B6 )
& ( ord_less_eq_set_rat @ B6 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_1087_Collect__mono__iff,axiom,
! [P: set_rat > $o,Q: set_rat > $o] :
( ( ord_le513522071413781156et_rat @ ( collect_set_rat @ P ) @ ( collect_set_rat @ Q ) )
= ( ! [X4: set_rat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1088_Collect__mono__iff,axiom,
! [P: rat > $o,Q: rat > $o] :
( ( ord_less_eq_set_rat @ ( collect_rat @ P ) @ ( collect_rat @ Q ) )
= ( ! [X4: rat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1089_preal__mult__def,axiom,
( times_3000655703912201937_preal
= ( ^ [R: dedekind_preal,S: dedekind_preal] : ( dedekind_Abs_preal @ ( dedekind_mult_set @ ( dedekind_Rep_preal @ R ) @ ( dedekind_Rep_preal @ S ) ) ) ) ) ).
% preal_mult_def
thf(fact_1090_add__inc,axiom,
! [X2: num,Y: num] :
( ( plus_plus_num @ X2 @ ( inc @ Y ) )
= ( inc @ ( plus_plus_num @ X2 @ Y ) ) ) ).
% add_inc
thf(fact_1091_mult__inc,axiom,
! [X2: num,Y: num] :
( ( times_times_num @ X2 @ ( inc @ Y ) )
= ( plus_plus_num @ ( times_times_num @ X2 @ Y ) @ X2 ) ) ).
% mult_inc
thf(fact_1092_semiring__norm_I75_J,axiom,
! [M2: num] :
~ ( ord_less_num @ M2 @ one ) ).
% semiring_norm(75)
thf(fact_1093_num__induct,axiom,
! [P: num > $o,X2: num] :
( ( P @ one )
=> ( ! [X: num] :
( ( P @ X )
=> ( P @ ( inc @ X ) ) )
=> ( P @ X2 ) ) ) ).
% num_induct
thf(fact_1094_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1095_le__num__One__iff,axiom,
! [X2: num] :
( ( ord_less_eq_num @ X2 @ one )
= ( X2 = one ) ) ).
% le_num_One_iff
thf(fact_1096_add__One__commute,axiom,
! [N2: num] :
( ( plus_plus_num @ one @ N2 )
= ( plus_plus_num @ N2 @ one ) ) ).
% add_One_commute
thf(fact_1097_add__One,axiom,
! [X2: num] :
( ( plus_plus_num @ X2 @ one )
= ( inc @ X2 ) ) ).
% add_One
thf(fact_1098_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1099_nat__mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1100_nat__1__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1101_div__eq__dividend__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N2 )
= M2 )
= ( N2 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1102_div__less__dividend,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1103_mult__eq__self__implies__10,axiom,
! [M2: nat,N2: nat] :
( ( M2
= ( times_times_nat @ M2 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1104_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1105_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1106_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_1107_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1108_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1109_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1110_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1111_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1112_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1113_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1114_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1115_mult__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1116_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1117_mult__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1118_mult__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1119_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1120_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1121_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1122_nat__0__less__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1123_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1124_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1125_div__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1126_div__mult__self__is__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N2 ) @ N2 )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1127_div__mult__self1__is__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M2 ) @ N2 )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1128_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N2 @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q3 ) @ N2 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1129_dividend__less__times__div,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M2 @ N2 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1130_dividend__less__div__times,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% dividend_less_div_times
thf(fact_1131_split__div,axiom,
! [P: nat > $o,M2: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N2 != zero_zero_nat )
=> ! [I2: nat,J2: nat] :
( ( ( ord_less_nat @ J2 @ N2 )
& ( M2
= ( plus_plus_nat @ ( times_times_nat @ N2 @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_div
thf(fact_1132_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1133_add__mult__distrib,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1134_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1135_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1136_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_1137_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_1138_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_1139_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1140_less__mult__imp__div__less,axiom,
! [M2: nat,I: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I @ N2 ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1141_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_1142_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_1143_div__less__iff__less__mult,axiom,
! [Q3: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q3 ) @ N2 )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N2 @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1144_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
? [K2: nat] :
( N
= ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1145_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1146_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1147_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_1148_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_1149_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_1150_add__leD2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1151_add__leD1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_1152_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_1153_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_1154_add__leE,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1155_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1156_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1157_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1158_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1159_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1160_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1161_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1162_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_1163_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_1164_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1165_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1166_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_1167_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1168_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1169_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1170_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_1171_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_1172_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_1173_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_1174_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_1175_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_1176_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_1177_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_1178_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1179_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_1180_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1181_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_1182_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1183_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1184_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_1185_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N2 )
| ( N2 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1186_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_1187_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1188_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: 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 @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1189_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1190_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1191_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1192_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_1193_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1194_div__le__mono2,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1195_div__greater__zero__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ N2 @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_1196_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1197_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1198_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1199_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1200_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_1201_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1202_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( P @ A3 @ B2 )
= ( P @ B2 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B2: nat] :
( ( P @ A3 @ B2 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B2 ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1203_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1204_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_1205_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_1206_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_1207_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_1208_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_1209_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N2 @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K3 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1210_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_1211_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_1212_semiring__norm_I6_J,axiom,
! [M2: num,N2: num] :
( ( plus_plus_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) )
= ( bit0 @ ( plus_plus_num @ M2 @ N2 ) ) ) ).
% semiring_norm(6)
thf(fact_1213_semiring__norm_I78_J,axiom,
! [M2: num,N2: num] :
( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) )
= ( ord_less_num @ M2 @ N2 ) ) ).
% semiring_norm(78)
thf(fact_1214_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_1215_num__double,axiom,
! [N2: num] :
( ( times_times_num @ ( bit0 @ one ) @ N2 )
= ( bit0 @ N2 ) ) ).
% num_double
thf(fact_1216_semiring__norm_I76_J,axiom,
! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).
% semiring_norm(76)
thf(fact_1217_add__self__div__2,axiom,
! [M2: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M2 ) ).
% add_self_div_2
thf(fact_1218_pos__zdiv__mult__2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( 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 ) ) @ A2 ) )
= ( divide_divide_int @ B @ A2 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1219_neg__zdiv__mult__2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ 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 ) ) @ A2 ) )
= ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A2 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1220_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_1221_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_1222_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K3: int] :
( ( P @ K3 )
=> ( ( K3 != zero_zero_int )
=> ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K3: int] :
( ( P @ K3 )
=> ( ( K3
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_1223_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_1224_nat__induct2,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct2
thf(fact_1225_zle__add1__eq__le,axiom,
! [W2: int,Z3: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_1226_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_1227_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_1228_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_1229_int__div__less__self,axiom,
! [X2: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X2 )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X2 @ K ) @ X2 ) ) ) ).
% int_div_less_self
thf(fact_1230_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_1231_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_1232_zless__add1__eq,axiom,
! [W2: int,Z3: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z3 )
| ( W2 = Z3 ) ) ) ).
% zless_add1_eq
thf(fact_1233_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1234_pos__zmult__eq__1__iff,axiom,
! [M2: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N2 )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N2 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1235_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_1236_int__distrib_I2_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1237_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W2 )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(1)
thf(fact_1238_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1239_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1240_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1241_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1242_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1243_div__neg__pos__less0,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1244_verit__la__generic,axiom,
! [A2: int,X2: int] :
( ( ord_less_eq_int @ A2 @ X2 )
| ( A2 = X2 )
| ( ord_less_eq_int @ X2 @ A2 ) ) ).
% verit_la_generic
thf(fact_1245_verit__less__mono__div__int2,axiom,
! [A: int,B3: int,N2: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B3 @ N2 ) @ ( divide_divide_int @ A @ N2 ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1246_nonneg1__imp__zdiv__pos__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B ) )
= ( ( ord_less_eq_int @ B @ A2 )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1247_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1248_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1249_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_1250_div__nonpos__pos__le0,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1251_div__nonneg__neg__le0,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1252_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_1253_zdiv__mono2__neg,axiom,
! [A2: int,B5: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B5 ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1254_zdiv__mono1__neg,axiom,
! [A2: int,A6: int,B: int] :
( ( ord_less_eq_int @ A2 @ A6 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1255_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_1256_zdiv__mono2,axiom,
! [A2: int,B5: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ ( divide_divide_int @ A2 @ B5 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1257_zdiv__mono1,axiom,
! [A2: int,A6: int,B: int] :
( ( ord_less_eq_int @ A2 @ A6 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ ( divide_divide_int @ A6 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1258_conj__le__cong,axiom,
! [X2: int,X6: int,P: $o,P2: $o] :
( ( X2 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_1259_imp__le__cong,axiom,
! [X2: int,X6: int,P: $o,P2: $o] :
( ( X2 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_1260_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 )
=> ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1261_split__zdiv,axiom,
! [P: int > $o,N2: int,K: int] :
( ( P @ ( divide_divide_int @ N2 @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I2: int,J2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
& ( ord_less_int @ J2 @ K )
& ( N2
= ( 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 )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_zdiv
thf(fact_1262_int__div__neg__eq,axiom,
! [A2: int,B: int,Q3: int,R2: int] :
( ( A2
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( divide_divide_int @ A2 @ B )
= Q3 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1263_int__div__pos__eq,axiom,
! [A2: int,B: int,Q3: int,R2: int] :
( ( A2
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( divide_divide_int @ A2 @ B )
= Q3 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1264_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_1265_zless__imp__add1__zle,axiom,
! [W2: int,Z3: int] :
( ( ord_less_int @ W2 @ Z3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z3 ) ) ).
% zless_imp_add1_zle
% Conjectures (3)
thf(conj_0,hypothesis,
dedekind_cut @ a ).
thf(conj_1,hypothesis,
member_rat @ y @ a ).
thf(conj_2,conjecture,
? [X3: rat] :
( ( member_rat @ X3 @ a )
& ( ord_less_rat @ y @ X3 ) ) ).
%------------------------------------------------------------------------------