TPTP Problem File: SLH0179^1.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 : Youngs_Inequality/0000_Youngs/prob_00366_014416__13029172_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1369 ( 476 unt; 97 typ; 0 def)
% Number of atoms : 4424 (1041 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 12398 ( 360 ~; 123 |; 258 &;9537 @)
% ( 0 <=>;2120 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 788 ( 788 >; 0 *; 0 +; 0 <<)
% Number of symbols : 93 ( 90 usr; 12 con; 0-4 aty)
% Number of variables : 3836 ( 134 ^;3552 !; 150 ?;3836 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:30:15.831
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
set_set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $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__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (90)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Fun_Omonotone__on_001t__Int__Oint_001t__Int__Oint,type,
monotone_on_int_int: set_int > ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Int__Oint_001t__Nat__Onat,type,
monotone_on_int_nat: set_int > ( int > int > $o ) > ( nat > nat > $o ) > ( int > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Int__Oint_001t__Real__Oreal,type,
monotone_on_int_real: set_int > ( int > int > $o ) > ( real > real > $o ) > ( int > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Int__Oint,type,
monotone_on_nat_int: set_nat > ( nat > nat > $o ) > ( int > int > $o ) > ( nat > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
monotone_on_nat_nat: set_nat > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Real__Oreal,type,
monotone_on_nat_real: set_nat > ( nat > nat > $o ) > ( real > real > $o ) > ( nat > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Int__Oint,type,
monotone_on_real_int: set_real > ( real > real > $o ) > ( int > int > $o ) > ( real > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Nat__Onat,type,
monotone_on_real_nat: set_real > ( real > real > $o ) > ( nat > nat > $o ) > ( real > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Real__Oreal,type,
monoto4017252874604999745l_real: set_real > ( real > real > $o ) > ( real > real > $o ) > ( real > real ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
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__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
uminus612125837232591019t_real: set_real > set_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Inner__Product_Oreal__inner__class_Oinner_001t__Real__Oreal,type,
inner_5569486634795291694r_real: real > real > real ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Integral__Test_Oantimono__fun__sum__integral__diff,type,
integr4865894440751020556l_diff: ( real > real ) > $o ).
thf(sy_c_Integral__Test_Oantimono__fun__sum__integral__diff_Osum__integral__diff__series,type,
integr3166334062659923703series: ( real > real ) > nat > real ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
bot_bot_set_set_real: set_set_real ).
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__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $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__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Int__Oint,type,
order_Greatest_int: ( int > $o ) > int ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal,type,
order_Greatest_real: ( real > $o ) > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Int__Oint,type,
image_real_int: ( real > int ) > set_real > set_int ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Nat__Onat,type,
image_real_nat: ( real > nat ) > set_real > set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Int__Oint,type,
set_ord_atLeast_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
set_ord_atLeast_real: real > set_real ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Int__Oint_001t__Int__Oint,type,
topolo2178910747331673048nt_int: set_int > ( int > int ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Int__Oint_001t__Nat__Onat,type,
topolo2181401217840723324nt_nat: set_int > ( int > nat ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Nat__Onat_001t__Int__Oint,type,
topolo1179557035430618492at_int: set_nat > ( nat > int ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Nat__Onat_001t__Nat__Onat,type,
topolo1182047505939668768at_nat: set_nat > ( nat > nat ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Nat__Onat_001t__Real__Oreal,type,
topolo6943266826644216316t_real: set_nat > ( nat > real ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Int__Oint,type,
topolo2284712892409288920al_int: set_real > ( real > int ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Nat__Onat,type,
topolo2287203362918339196al_nat: set_real > ( real > nat ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
thf(sy_c_Youngs_Osegments,type,
segments: nat > set_set_real ).
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__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_a,type,
a: real ).
thf(sy_v_b,type,
b: real ).
thf(sy_v_f,type,
f: real > real ).
thf(sy_v_g,type,
g: real > real ).
% Relevant facts (1264)
thf(fact_0_False,axiom,
a != zero_zero_real ).
% False
thf(fact_1_f_I1_J,axiom,
( ( f @ zero_zero_real )
= zero_zero_real ) ).
% f(1)
thf(fact_2_a,axiom,
ord_less_eq_real @ zero_zero_real @ a ).
% a
thf(fact_3_f_I2_J,axiom,
( ( f @ a )
= b ) ).
% f(2)
thf(fact_4_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_5_g,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ a )
=> ( ( g @ ( f @ X ) )
= X ) ) ) ).
% g
thf(fact_6_bgauge__existence__lemma,axiom,
! [S: set_real,Q: real > real > $o] :
( ( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ? [D: real] :
( ( ord_less_real @ zero_zero_real @ D )
& ( Q @ D @ X2 ) ) ) )
= ( ! [X2: real] :
? [D: real] :
( ( ord_less_real @ zero_zero_real @ D )
& ( ( member_real @ X2 @ S )
=> ( Q @ D @ X2 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_7_bgauge__existence__lemma,axiom,
! [S: set_nat,Q: real > nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ? [D: real] :
( ( ord_less_real @ zero_zero_real @ D )
& ( Q @ D @ X2 ) ) ) )
= ( ! [X2: nat] :
? [D: real] :
( ( ord_less_real @ zero_zero_real @ D )
& ( ( member_nat @ X2 @ S )
=> ( Q @ D @ X2 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_8_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_9_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_10_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_11_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_12_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_13_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_14_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_15_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_16_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_17_complete__real,axiom,
! [S2: set_real] :
( ? [X3: real] : ( member_real @ X3 @ S2 )
=> ( ? [Z: real] :
! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Z ) )
=> ? [Y: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Y ) )
& ! [Z: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ) ) ).
% complete_real
thf(fact_18_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_19_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_20_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_21_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_22_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% less_eq_real_def
thf(fact_23_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_24_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_25_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_26_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_27_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_28_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_29_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_30_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_31_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_32_order__le__imp__less__or__eq,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ( ord_less_real @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_33_order__le__imp__less__or__eq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_nat @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_34_order__le__imp__less__or__eq,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ( ord_less_int @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_35_linorder__le__less__linear,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
| ( ord_less_real @ Y3 @ X ) ) ).
% linorder_le_less_linear
thf(fact_36_linorder__le__less__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ).
% linorder_le_less_linear
thf(fact_37_linorder__le__less__linear,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
| ( ord_less_int @ Y3 @ X ) ) ).
% linorder_le_less_linear
thf(fact_38_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_39_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_40_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_41_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_42_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_43_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_44_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_45_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_46_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_47_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_48_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_49_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_50_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_51_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_52_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_53_order__less__le__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_54_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_55_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_56_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_57_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_58_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_59_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_60_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_61_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_62_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_63_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_64_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_65_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_66_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_67_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_68_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_69_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_70_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_71_order__le__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_72_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_73_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_74_order__less__le__trans,axiom,
! [X: real,Y3: real,Z2: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_75_order__less__le__trans,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_76_order__less__le__trans,axiom,
! [X: int,Y3: int,Z2: int] :
( ( ord_less_int @ X @ Y3 )
=> ( ( ord_less_eq_int @ Y3 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_77_order__le__less__trans,axiom,
! [X: real,Y3: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ( ord_less_real @ Y3 @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_78_order__le__less__trans,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_79_order__le__less__trans,axiom,
! [X: int,Y3: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ( ord_less_int @ Y3 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_80_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_81_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_82_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_83_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_84_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_85_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_86_order__antisym__conv,axiom,
! [Y3: real,X: real] :
( ( ord_less_eq_real @ Y3 @ X )
=> ( ( ord_less_eq_real @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_87_order__antisym__conv,axiom,
! [Y3: nat,X: nat] :
( ( ord_less_eq_nat @ Y3 @ X )
=> ( ( ord_less_eq_nat @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_88_order__antisym__conv,axiom,
! [Y3: int,X: int] :
( ( ord_less_eq_int @ Y3 @ X )
=> ( ( ord_less_eq_int @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_89_linorder__le__cases,axiom,
! [X: real,Y3: real] :
( ~ ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X ) ) ).
% linorder_le_cases
thf(fact_90_linorder__le__cases,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_le_cases
thf(fact_91_linorder__le__cases,axiom,
! [X: int,Y3: int] :
( ~ ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X ) ) ).
% linorder_le_cases
thf(fact_92_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_93_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_94_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_95_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_96_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_97_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_98_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_99_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_100_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_101_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_102_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_103_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_104_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_105_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_106_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_107_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_108_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_109_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_110_linorder__linear,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
| ( ord_less_eq_real @ Y3 @ X ) ) ).
% linorder_linear
thf(fact_111_linorder__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_linear
thf(fact_112_linorder__linear,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
| ( ord_less_eq_int @ Y3 @ X ) ) ).
% linorder_linear
thf(fact_113_order__eq__refl,axiom,
! [X: real,Y3: real] :
( ( X = Y3 )
=> ( ord_less_eq_real @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_114_order__eq__refl,axiom,
! [X: nat,Y3: nat] :
( ( X = Y3 )
=> ( ord_less_eq_nat @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_115_order__eq__refl,axiom,
! [X: int,Y3: int] :
( ( X = Y3 )
=> ( ord_less_eq_int @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_116_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_117_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_118_order__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_119_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_120_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_121_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_122_order__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_123_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_124_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_125_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_126_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_127_order__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_128_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_129_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_130_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_131_order__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_132_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_133_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_134_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_135_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_136_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_137_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_138_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_139_Collect__mem__eq,axiom,
! [A3: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_140_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_141_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_142_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_143_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_144_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_145_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_146_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_147_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_148_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_149_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_150_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_151_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_152_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_153_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_154_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_155_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_156_order__trans,axiom,
! [X: real,Y3: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ Z2 )
=> ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_157_order__trans,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_158_order__trans,axiom,
! [X: int,Y3: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ( ord_less_eq_int @ Y3 @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_159_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_160_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_161_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_162_order__antisym,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_163_order__antisym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_164_order__antisym,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ( ord_less_eq_int @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_165_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_166_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_167_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_168_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_169_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_170_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_171_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
& ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_172_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_173_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
& ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_174_le__cases3,axiom,
! [X: real,Y3: real,Z2: real] :
( ( ( ord_less_eq_real @ X @ Y3 )
=> ~ ( ord_less_eq_real @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y3 @ X )
=> ~ ( ord_less_eq_real @ X @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y3 )
=> ~ ( ord_less_eq_real @ Y3 @ X ) )
=> ( ( ( ord_less_eq_real @ Y3 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X )
=> ~ ( ord_less_eq_real @ X @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_175_le__cases3,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_176_le__cases3,axiom,
! [X: int,Y3: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y3 )
=> ~ ( ord_less_eq_int @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y3 @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y3 )
=> ~ ( ord_less_eq_int @ Y3 @ X ) )
=> ( ( ( ord_less_eq_int @ Y3 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_177_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_178_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_179_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_180_lt__ex,axiom,
! [X: real] :
? [Y: real] : ( ord_less_real @ Y @ X ) ).
% lt_ex
thf(fact_181_lt__ex,axiom,
! [X: int] :
? [Y: int] : ( ord_less_int @ Y @ X ) ).
% lt_ex
thf(fact_182_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_183_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_184_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_185_dense,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ? [Z4: real] :
( ( ord_less_real @ X @ Z4 )
& ( ord_less_real @ Z4 @ Y3 ) ) ) ).
% dense
thf(fact_186_less__imp__neq,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( X != Y3 ) ) ).
% less_imp_neq
thf(fact_187_less__imp__neq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( X != Y3 ) ) ).
% less_imp_neq
thf(fact_188_less__imp__neq,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( X != Y3 ) ) ).
% less_imp_neq
thf(fact_189_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_190_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_191_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_192_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_193_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_194_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_195_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_196_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_197_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_198_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_199_antisym__conv3,axiom,
! [Y3: real,X: real] :
( ~ ( ord_less_real @ Y3 @ X )
=> ( ( ~ ( ord_less_real @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv3
thf(fact_200_antisym__conv3,axiom,
! [Y3: nat,X: nat] :
( ~ ( ord_less_nat @ Y3 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv3
thf(fact_201_antisym__conv3,axiom,
! [Y3: int,X: int] :
( ~ ( ord_less_int @ Y3 @ X )
=> ( ( ~ ( ord_less_int @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv3
thf(fact_202_linorder__cases,axiom,
! [X: real,Y3: real] :
( ~ ( ord_less_real @ X @ Y3 )
=> ( ( X != Y3 )
=> ( ord_less_real @ Y3 @ X ) ) ) ).
% linorder_cases
thf(fact_203_linorder__cases,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_nat @ X @ Y3 )
=> ( ( X != Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_cases
thf(fact_204_linorder__cases,axiom,
! [X: int,Y3: int] :
( ~ ( ord_less_int @ X @ Y3 )
=> ( ( X != Y3 )
=> ( ord_less_int @ Y3 @ X ) ) ) ).
% linorder_cases
thf(fact_205_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_206_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_207_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_208_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_209_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_210_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_211_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_212_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_213_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_214_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_215_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_216_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_217_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_218_not__less__iff__gr__or__eq,axiom,
! [X: real,Y3: real] :
( ( ~ ( ord_less_real @ X @ Y3 ) )
= ( ( ord_less_real @ Y3 @ X )
| ( X = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_219_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X )
| ( X = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_220_not__less__iff__gr__or__eq,axiom,
! [X: int,Y3: int] :
( ( ~ ( ord_less_int @ X @ Y3 ) )
= ( ( ord_less_int @ Y3 @ X )
| ( X = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_221_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_222_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_223_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_224_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_225_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_226_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_227_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_228_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_229_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_230_linorder__neqE,axiom,
! [X: real,Y3: real] :
( ( X != Y3 )
=> ( ~ ( ord_less_real @ X @ Y3 )
=> ( ord_less_real @ Y3 @ X ) ) ) ).
% linorder_neqE
thf(fact_231_linorder__neqE,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE
thf(fact_232_linorder__neqE,axiom,
! [X: int,Y3: int] :
( ( X != Y3 )
=> ( ~ ( ord_less_int @ X @ Y3 )
=> ( ord_less_int @ Y3 @ X ) ) ) ).
% linorder_neqE
thf(fact_233_order__less__asym,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ~ ( ord_less_real @ Y3 @ X ) ) ).
% order_less_asym
thf(fact_234_order__less__asym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_asym
thf(fact_235_order__less__asym,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ~ ( ord_less_int @ Y3 @ X ) ) ).
% order_less_asym
thf(fact_236_linorder__neq__iff,axiom,
! [X: real,Y3: real] :
( ( X != Y3 )
= ( ( ord_less_real @ X @ Y3 )
| ( ord_less_real @ Y3 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_237_linorder__neq__iff,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
= ( ( ord_less_nat @ X @ Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_238_linorder__neq__iff,axiom,
! [X: int,Y3: int] :
( ( X != Y3 )
= ( ( ord_less_int @ X @ Y3 )
| ( ord_less_int @ Y3 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_239_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_240_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_241_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_242_order__less__trans,axiom,
! [X: real,Y3: real,Z2: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ( ord_less_real @ Y3 @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_243_order__less__trans,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_244_order__less__trans,axiom,
! [X: int,Y3: int,Z2: int] :
( ( ord_less_int @ X @ Y3 )
=> ( ( ord_less_int @ Y3 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_245_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_246_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_247_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_248_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_249_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_250_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_251_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_252_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_253_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_254_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_255_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_256_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_257_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_258_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_259_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_260_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_261_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_262_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_263_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_264_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_265_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_266_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_267_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_268_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_269_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_270_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_271_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_272_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_273_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_274_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_275_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_276_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_277_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: real,Y: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_278_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_279_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_280_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_281_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_282_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_283_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_284_order__less__not__sym,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ~ ( ord_less_real @ Y3 @ X ) ) ).
% order_less_not_sym
thf(fact_285_order__less__not__sym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_not_sym
thf(fact_286_order__less__not__sym,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ~ ( ord_less_int @ Y3 @ X ) ) ).
% order_less_not_sym
thf(fact_287_order__less__imp__triv,axiom,
! [X: real,Y3: real,P: $o] :
( ( ord_less_real @ X @ Y3 )
=> ( ( ord_less_real @ Y3 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_288_order__less__imp__triv,axiom,
! [X: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_289_order__less__imp__triv,axiom,
! [X: int,Y3: int,P: $o] :
( ( ord_less_int @ X @ Y3 )
=> ( ( ord_less_int @ Y3 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_290_linorder__less__linear,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
| ( X = Y3 )
| ( ord_less_real @ Y3 @ X ) ) ).
% linorder_less_linear
thf(fact_291_linorder__less__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
| ( X = Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ).
% linorder_less_linear
thf(fact_292_linorder__less__linear,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
| ( X = Y3 )
| ( ord_less_int @ Y3 @ X ) ) ).
% linorder_less_linear
thf(fact_293_order__less__imp__not__eq,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( X != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_294_order__less__imp__not__eq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( X != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_295_order__less__imp__not__eq,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( X != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_296_order__less__imp__not__eq2,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( Y3 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_297_order__less__imp__not__eq2,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( Y3 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_298_order__less__imp__not__eq2,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( Y3 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_299_order__less__imp__not__less,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ~ ( ord_less_real @ Y3 @ X ) ) ).
% order_less_imp_not_less
thf(fact_300_order__less__imp__not__less,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_imp_not_less
thf(fact_301_order__less__imp__not__less,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ~ ( ord_less_int @ Y3 @ X ) ) ).
% order_less_imp_not_less
thf(fact_302_leD,axiom,
! [Y3: real,X: real] :
( ( ord_less_eq_real @ Y3 @ X )
=> ~ ( ord_less_real @ X @ Y3 ) ) ).
% leD
thf(fact_303_leD,axiom,
! [Y3: nat,X: nat] :
( ( ord_less_eq_nat @ Y3 @ X )
=> ~ ( ord_less_nat @ X @ Y3 ) ) ).
% leD
thf(fact_304_leD,axiom,
! [Y3: int,X: int] :
( ( ord_less_eq_int @ Y3 @ X )
=> ~ ( ord_less_int @ X @ Y3 ) ) ).
% leD
thf(fact_305_leI,axiom,
! [X: real,Y3: real] :
( ~ ( ord_less_real @ X @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X ) ) ).
% leI
thf(fact_306_leI,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) ) ).
% leI
thf(fact_307_leI,axiom,
! [X: int,Y3: int] :
( ~ ( ord_less_int @ X @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X ) ) ).
% leI
thf(fact_308_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_309_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_310_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_311_antisym__conv1,axiom,
! [X: real,Y3: real] :
( ~ ( ord_less_real @ X @ Y3 )
=> ( ( ord_less_eq_real @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_312_antisym__conv1,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_313_antisym__conv1,axiom,
! [X: int,Y3: int] :
( ~ ( ord_less_int @ X @ Y3 )
=> ( ( ord_less_eq_int @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_314_antisym__conv2,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ( ~ ( ord_less_real @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_315_antisym__conv2,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_316_antisym__conv2,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ( ~ ( ord_less_int @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_317_dense__ge,axiom,
! [Z2: real,Y3: real] :
( ! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_eq_real @ Y3 @ X4 ) )
=> ( ord_less_eq_real @ Y3 @ Z2 ) ) ).
% dense_ge
thf(fact_318_dense__le,axiom,
! [Y3: real,Z2: real] :
( ! [X4: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ X4 @ Z2 ) )
=> ( ord_less_eq_real @ Y3 @ Z2 ) ) ).
% dense_le
thf(fact_319_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
& ~ ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_320_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_321_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
& ~ ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_322_not__le__imp__less,axiom,
! [Y3: real,X: real] :
( ~ ( ord_less_eq_real @ Y3 @ X )
=> ( ord_less_real @ X @ Y3 ) ) ).
% not_le_imp_less
thf(fact_323_not__le__imp__less,axiom,
! [Y3: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X )
=> ( ord_less_nat @ X @ Y3 ) ) ).
% not_le_imp_less
thf(fact_324_not__le__imp__less,axiom,
! [Y3: int,X: int] :
( ~ ( ord_less_eq_int @ Y3 @ X )
=> ( ord_less_int @ X @ Y3 ) ) ).
% not_le_imp_less
thf(fact_325_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_326_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_327_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_328_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_329_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_330_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_331_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_332_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_333_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_334_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_335_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_336_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_337_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_338_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_339_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_340_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y3: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y3 @ W ) ) )
=> ( ord_less_eq_real @ Y3 @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_341_dense__le__bounded,axiom,
! [X: real,Y3: real,Z2: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y3 )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y3 @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_342_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_343_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_344_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_345_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_346_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_347_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_348_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_349_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_350_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_351_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_352_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_353_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_354_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ~ ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_355_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_356_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_357_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_358_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_359_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_360_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_361_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_362_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_363_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_364_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_365_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_366_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_367_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_368_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_369_linorder__not__le,axiom,
! [X: real,Y3: real] :
( ( ~ ( ord_less_eq_real @ X @ Y3 ) )
= ( ord_less_real @ Y3 @ X ) ) ).
% linorder_not_le
thf(fact_370_linorder__not__le,axiom,
! [X: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y3 ) )
= ( ord_less_nat @ Y3 @ X ) ) ).
% linorder_not_le
thf(fact_371_linorder__not__le,axiom,
! [X: int,Y3: int] :
( ( ~ ( ord_less_eq_int @ X @ Y3 ) )
= ( ord_less_int @ Y3 @ X ) ) ).
% linorder_not_le
thf(fact_372_linorder__not__less,axiom,
! [X: real,Y3: real] :
( ( ~ ( ord_less_real @ X @ Y3 ) )
= ( ord_less_eq_real @ Y3 @ X ) ) ).
% linorder_not_less
thf(fact_373_linorder__not__less,axiom,
! [X: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_not_less
thf(fact_374_linorder__not__less,axiom,
! [X: int,Y3: int] :
( ( ~ ( ord_less_int @ X @ Y3 ) )
= ( ord_less_eq_int @ Y3 @ X ) ) ).
% linorder_not_less
thf(fact_375_order__less__imp__le,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_less_eq_real @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_376_order__less__imp__le,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_377_order__less__imp__le,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( ord_less_eq_int @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_378_seq__mono__lemma,axiom,
! [M: nat,D3: nat > real,E2: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_real @ ( D3 @ N3 ) @ ( E2 @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_real @ ( E2 @ N3 ) @ ( E2 @ M ) ) )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ord_less_real @ ( D3 @ N4 ) @ ( E2 @ M ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_379_minf_I8_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ~ ( ord_less_eq_real @ T @ X3 ) ) ).
% minf(8)
thf(fact_380_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% minf(8)
thf(fact_381_minf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ~ ( ord_less_eq_int @ T @ X3 ) ) ).
% minf(8)
thf(fact_382_minf_I6_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ord_less_eq_real @ X3 @ T ) ) ).
% minf(6)
thf(fact_383_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ord_less_eq_nat @ X3 @ T ) ) ).
% minf(6)
thf(fact_384_minf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ord_less_eq_int @ X3 @ T ) ) ).
% minf(6)
thf(fact_385_pinf_I8_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ord_less_eq_real @ T @ X3 ) ) ).
% pinf(8)
thf(fact_386_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ord_less_eq_nat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_387_pinf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ord_less_eq_int @ T @ X3 ) ) ).
% pinf(8)
thf(fact_388_pinf_I6_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ T ) ) ).
% pinf(6)
thf(fact_389_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T ) ) ).
% pinf(6)
thf(fact_390_pinf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ T ) ) ).
% pinf(6)
thf(fact_391_verit__comp__simplify1_I3_J,axiom,
! [B4: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B4 @ A5 ) )
= ( ord_less_real @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_392_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_393_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
= ( ord_less_int @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_394_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
& ~ ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_395_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B )
& ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ C2 ) )
=> ( P @ X3 ) )
& ! [D4: real] :
( ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_real @ X4 @ D4 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_real @ D4 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_396_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ C2 ) )
=> ( P @ X3 ) )
& ! [D4: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D4 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D4 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_397_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ C2 ) )
=> ( P @ X3 ) )
& ! [D4: int] :
( ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ D4 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_int @ D4 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_398_sm,axiom,
monoto4017252874604999745l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ ord_less_real @ ord_less_real @ f ).
% sm
thf(fact_399_cont,axiom,
topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ f ).
% cont
thf(fact_400_Greatest__equality,axiom,
! [P: real > $o,X: real] :
( ( P @ X )
=> ( ! [Y: real] :
( ( P @ Y )
=> ( ord_less_eq_real @ Y @ X ) )
=> ( ( order_Greatest_real @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_401_Greatest__equality,axiom,
! [P: int > $o,X: int] :
( ( P @ X )
=> ( ! [Y: int] :
( ( P @ Y )
=> ( ord_less_eq_int @ Y @ X ) )
=> ( ( order_Greatest_int @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_402_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_403_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_404_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_405_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_406_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_407_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_408_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_409_ex__gt__or__lt,axiom,
! [A: real] :
? [B3: real] :
( ( ord_less_real @ A @ B3 )
| ( ord_less_real @ B3 @ A ) ) ).
% ex_gt_or_lt
thf(fact_410_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_411_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_412_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_413_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z: real] :
! [X4: real] :
( ( ord_less_real @ Z @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: real] :
! [X4: real] :
( ( ord_less_real @ Z @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( ( P @ X3 )
& ( Q2 @ X3 ) )
= ( ( P4 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_414_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( ( P @ X3 )
& ( Q2 @ X3 ) )
= ( ( P4 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_415_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ Z @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ Z @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( ( P @ X3 )
& ( Q2 @ X3 ) )
= ( ( P4 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_416_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z: real] :
! [X4: real] :
( ( ord_less_real @ Z @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: real] :
! [X4: real] :
( ( ord_less_real @ Z @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( ( P @ X3 )
| ( Q2 @ X3 ) )
= ( ( P4 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_417_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( ( P @ X3 )
| ( Q2 @ X3 ) )
= ( ( P4 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_418_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ Z @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ Z @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( ( P @ X3 )
| ( Q2 @ X3 ) )
= ( ( P4 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_419_pinf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_420_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_421_pinf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_422_pinf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_423_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_424_pinf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_425_pinf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ~ ( ord_less_real @ X3 @ T ) ) ).
% pinf(5)
thf(fact_426_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ~ ( ord_less_nat @ X3 @ T ) ) ).
% pinf(5)
thf(fact_427_pinf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ~ ( ord_less_int @ X3 @ T ) ) ).
% pinf(5)
thf(fact_428_pinf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ord_less_real @ T @ X3 ) ) ).
% pinf(7)
thf(fact_429_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ord_less_nat @ T @ X3 ) ) ).
% pinf(7)
thf(fact_430_pinf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ord_less_int @ T @ X3 ) ) ).
% pinf(7)
thf(fact_431_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( ( P @ X3 )
& ( Q2 @ X3 ) )
= ( ( P4 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_432_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( ( P @ X3 )
& ( Q2 @ X3 ) )
= ( ( P4 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_433_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( ( P @ X3 )
& ( Q2 @ X3 ) )
= ( ( P4 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_434_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( ( P @ X3 )
| ( Q2 @ X3 ) )
= ( ( P4 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_435_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( ( P @ X3 )
| ( Q2 @ X3 ) )
= ( ( P4 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_436_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( ( P @ X3 )
| ( Q2 @ X3 ) )
= ( ( P4 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_437_minf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_438_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_439_minf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_440_minf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_441_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_442_minf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_443_minf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ord_less_real @ X3 @ T ) ) ).
% minf(5)
thf(fact_444_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ord_less_nat @ X3 @ T ) ) ).
% minf(5)
thf(fact_445_minf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ord_less_int @ X3 @ T ) ) ).
% minf(5)
thf(fact_446_minf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ~ ( ord_less_real @ T @ X3 ) ) ).
% minf(7)
thf(fact_447_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ~ ( ord_less_nat @ T @ X3 ) ) ).
% minf(7)
thf(fact_448_minf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ~ ( ord_less_int @ T @ X3 ) ) ).
% minf(7)
thf(fact_449_GreatestI2__order,axiom,
! [P: real > $o,X: real,Q2: real > $o] :
( ( P @ X )
=> ( ! [Y: real] :
( ( P @ Y )
=> ( ord_less_eq_real @ Y @ X ) )
=> ( ! [X4: real] :
( ( P @ X4 )
=> ( ! [Y5: real] :
( ( P @ Y5 )
=> ( ord_less_eq_real @ Y5 @ X4 ) )
=> ( Q2 @ X4 ) ) )
=> ( Q2 @ ( order_Greatest_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_450_GreatestI2__order,axiom,
! [P: int > $o,X: int,Q2: int > $o] :
( ( P @ X )
=> ( ! [Y: int] :
( ( P @ Y )
=> ( ord_less_eq_int @ Y @ X ) )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( ! [Y5: int] :
( ( P @ Y5 )
=> ( ord_less_eq_int @ Y5 @ X4 ) )
=> ( Q2 @ X4 ) ) )
=> ( Q2 @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_451_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q2: nat > $o] :
( ( P @ X )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X ) )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) )
=> ( Q2 @ X4 ) ) )
=> ( Q2 @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_452_atLeast__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atLeast_nat @ K ) )
= ( ord_less_eq_nat @ K @ I ) ) ).
% atLeast_iff
thf(fact_453_atLeast__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_atLeast_int @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ).
% atLeast_iff
thf(fact_454_atLeast__iff,axiom,
! [I: real,K: real] :
( ( member_real @ I @ ( set_ord_atLeast_real @ K ) )
= ( ord_less_eq_real @ K @ I ) ) ).
% atLeast_iff
thf(fact_455_atLeast__subset__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ X ) @ ( set_ord_atLeast_nat @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X ) ) ).
% atLeast_subset_iff
thf(fact_456_atLeast__subset__iff,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq_set_int @ ( set_ord_atLeast_int @ X ) @ ( set_ord_atLeast_int @ Y3 ) )
= ( ord_less_eq_int @ Y3 @ X ) ) ).
% atLeast_subset_iff
thf(fact_457_atLeast__subset__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_set_real @ ( set_ord_atLeast_real @ X ) @ ( set_ord_atLeast_real @ Y3 ) )
= ( ord_less_eq_real @ Y3 @ X ) ) ).
% atLeast_subset_iff
thf(fact_458_strict__mono__on__imp__mono__on,axiom,
! [A3: set_real,F: real > real] :
( ( monoto4017252874604999745l_real @ A3 @ ord_less_real @ ord_less_real @ F )
=> ( monoto4017252874604999745l_real @ A3 @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_459_strict__mono__on__imp__mono__on,axiom,
! [A3: set_real,F: real > nat] :
( ( monotone_on_real_nat @ A3 @ ord_less_real @ ord_less_nat @ F )
=> ( monotone_on_real_nat @ A3 @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_460_strict__mono__on__imp__mono__on,axiom,
! [A3: set_real,F: real > int] :
( ( monotone_on_real_int @ A3 @ ord_less_real @ ord_less_int @ F )
=> ( monotone_on_real_int @ A3 @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_461_strict__mono__on__imp__mono__on,axiom,
! [A3: set_nat,F: nat > real] :
( ( monotone_on_nat_real @ A3 @ ord_less_nat @ ord_less_real @ F )
=> ( monotone_on_nat_real @ A3 @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_462_strict__mono__on__imp__mono__on,axiom,
! [A3: set_nat,F: nat > nat] :
( ( monotone_on_nat_nat @ A3 @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_463_strict__mono__on__imp__mono__on,axiom,
! [A3: set_nat,F: nat > int] :
( ( monotone_on_nat_int @ A3 @ ord_less_nat @ ord_less_int @ F )
=> ( monotone_on_nat_int @ A3 @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_464_strict__mono__on__imp__mono__on,axiom,
! [A3: set_int,F: int > real] :
( ( monotone_on_int_real @ A3 @ ord_less_int @ ord_less_real @ F )
=> ( monotone_on_int_real @ A3 @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_465_strict__mono__on__imp__mono__on,axiom,
! [A3: set_int,F: int > nat] :
( ( monotone_on_int_nat @ A3 @ ord_less_int @ ord_less_nat @ F )
=> ( monotone_on_int_nat @ A3 @ ord_less_eq_int @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_466_strict__mono__on__imp__mono__on,axiom,
! [A3: set_int,F: int > int] :
( ( monotone_on_int_int @ A3 @ ord_less_int @ ord_less_int @ F )
=> ( monotone_on_int_int @ A3 @ ord_less_eq_int @ ord_less_eq_int @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_467_strict__mono__on__leD,axiom,
! [A3: set_real,F: real > real,X: real,Y3: real] :
( ( monoto4017252874604999745l_real @ A3 @ ord_less_real @ ord_less_real @ F )
=> ( ( member_real @ X @ A3 )
=> ( ( member_real @ Y3 @ A3 )
=> ( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_468_strict__mono__on__leD,axiom,
! [A3: set_real,F: real > nat,X: real,Y3: real] :
( ( monotone_on_real_nat @ A3 @ ord_less_real @ ord_less_nat @ F )
=> ( ( member_real @ X @ A3 )
=> ( ( member_real @ Y3 @ A3 )
=> ( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_469_strict__mono__on__leD,axiom,
! [A3: set_real,F: real > int,X: real,Y3: real] :
( ( monotone_on_real_int @ A3 @ ord_less_real @ ord_less_int @ F )
=> ( ( member_real @ X @ A3 )
=> ( ( member_real @ Y3 @ A3 )
=> ( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_470_strict__mono__on__leD,axiom,
! [A3: set_nat,F: nat > real,X: nat,Y3: nat] :
( ( monotone_on_nat_real @ A3 @ ord_less_nat @ ord_less_real @ F )
=> ( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y3 @ A3 )
=> ( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_471_strict__mono__on__leD,axiom,
! [A3: set_nat,F: nat > nat,X: nat,Y3: nat] :
( ( monotone_on_nat_nat @ A3 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y3 @ A3 )
=> ( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_472_strict__mono__on__leD,axiom,
! [A3: set_nat,F: nat > int,X: nat,Y3: nat] :
( ( monotone_on_nat_int @ A3 @ ord_less_nat @ ord_less_int @ F )
=> ( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y3 @ A3 )
=> ( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_473_strict__mono__on__leD,axiom,
! [A3: set_int,F: int > real,X: int,Y3: int] :
( ( monotone_on_int_real @ A3 @ ord_less_int @ ord_less_real @ F )
=> ( ( member_int @ X @ A3 )
=> ( ( member_int @ Y3 @ A3 )
=> ( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_474_strict__mono__on__leD,axiom,
! [A3: set_int,F: int > nat,X: int,Y3: int] :
( ( monotone_on_int_nat @ A3 @ ord_less_int @ ord_less_nat @ F )
=> ( ( member_int @ X @ A3 )
=> ( ( member_int @ Y3 @ A3 )
=> ( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_475_strict__mono__on__leD,axiom,
! [A3: set_int,F: int > int,X: int,Y3: int] :
( ( monotone_on_int_int @ A3 @ ord_less_int @ ord_less_int @ F )
=> ( ( member_int @ X @ A3 )
=> ( ( member_int @ Y3 @ A3 )
=> ( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_476_mono__on__greaterD,axiom,
! [A3: set_real,G: real > real,X: real,Y3: real] :
( ( monoto4017252874604999745l_real @ A3 @ ord_less_eq_real @ ord_less_eq_real @ G )
=> ( ( member_real @ X @ A3 )
=> ( ( member_real @ Y3 @ A3 )
=> ( ( ord_less_real @ ( G @ Y3 ) @ ( G @ X ) )
=> ( ord_less_real @ Y3 @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_477_mono__on__greaterD,axiom,
! [A3: set_real,G: real > nat,X: real,Y3: real] :
( ( monotone_on_real_nat @ A3 @ ord_less_eq_real @ ord_less_eq_nat @ G )
=> ( ( member_real @ X @ A3 )
=> ( ( member_real @ Y3 @ A3 )
=> ( ( ord_less_nat @ ( G @ Y3 ) @ ( G @ X ) )
=> ( ord_less_real @ Y3 @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_478_mono__on__greaterD,axiom,
! [A3: set_real,G: real > int,X: real,Y3: real] :
( ( monotone_on_real_int @ A3 @ ord_less_eq_real @ ord_less_eq_int @ G )
=> ( ( member_real @ X @ A3 )
=> ( ( member_real @ Y3 @ A3 )
=> ( ( ord_less_int @ ( G @ Y3 ) @ ( G @ X ) )
=> ( ord_less_real @ Y3 @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_479_mono__on__greaterD,axiom,
! [A3: set_nat,G: nat > real,X: nat,Y3: nat] :
( ( monotone_on_nat_real @ A3 @ ord_less_eq_nat @ ord_less_eq_real @ G )
=> ( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y3 @ A3 )
=> ( ( ord_less_real @ ( G @ Y3 ) @ ( G @ X ) )
=> ( ord_less_nat @ Y3 @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_480_mono__on__greaterD,axiom,
! [A3: set_nat,G: nat > nat,X: nat,Y3: nat] :
( ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ G )
=> ( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y3 @ A3 )
=> ( ( ord_less_nat @ ( G @ Y3 ) @ ( G @ X ) )
=> ( ord_less_nat @ Y3 @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_481_mono__on__greaterD,axiom,
! [A3: set_nat,G: nat > int,X: nat,Y3: nat] :
( ( monotone_on_nat_int @ A3 @ ord_less_eq_nat @ ord_less_eq_int @ G )
=> ( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y3 @ A3 )
=> ( ( ord_less_int @ ( G @ Y3 ) @ ( G @ X ) )
=> ( ord_less_nat @ Y3 @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_482_mono__on__greaterD,axiom,
! [A3: set_int,G: int > real,X: int,Y3: int] :
( ( monotone_on_int_real @ A3 @ ord_less_eq_int @ ord_less_eq_real @ G )
=> ( ( member_int @ X @ A3 )
=> ( ( member_int @ Y3 @ A3 )
=> ( ( ord_less_real @ ( G @ Y3 ) @ ( G @ X ) )
=> ( ord_less_int @ Y3 @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_483_mono__on__greaterD,axiom,
! [A3: set_int,G: int > nat,X: int,Y3: int] :
( ( monotone_on_int_nat @ A3 @ ord_less_eq_int @ ord_less_eq_nat @ G )
=> ( ( member_int @ X @ A3 )
=> ( ( member_int @ Y3 @ A3 )
=> ( ( ord_less_nat @ ( G @ Y3 ) @ ( G @ X ) )
=> ( ord_less_int @ Y3 @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_484_mono__on__greaterD,axiom,
! [A3: set_int,G: int > int,X: int,Y3: int] :
( ( monotone_on_int_int @ A3 @ ord_less_eq_int @ ord_less_eq_int @ G )
=> ( ( member_int @ X @ A3 )
=> ( ( member_int @ Y3 @ A3 )
=> ( ( ord_less_int @ ( G @ Y3 ) @ ( G @ X ) )
=> ( ord_less_int @ Y3 @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_485_atLeast__eq__iff,axiom,
! [X: real,Y3: real] :
( ( ( set_ord_atLeast_real @ X )
= ( set_ord_atLeast_real @ Y3 ) )
= ( X = Y3 ) ) ).
% atLeast_eq_iff
thf(fact_486_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_487_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_488_strict__mono__on__eqD,axiom,
! [A3: set_real,F: real > real,X: real,Y3: real] :
( ( monoto4017252874604999745l_real @ A3 @ ord_less_real @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( ( member_real @ X @ A3 )
=> ( ( member_real @ Y3 @ A3 )
=> ( Y3 = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_489_strict__mono__on__eqD,axiom,
! [A3: set_real,F: real > nat,X: real,Y3: real] :
( ( monotone_on_real_nat @ A3 @ ord_less_real @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( ( member_real @ X @ A3 )
=> ( ( member_real @ Y3 @ A3 )
=> ( Y3 = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_490_strict__mono__on__eqD,axiom,
! [A3: set_real,F: real > int,X: real,Y3: real] :
( ( monotone_on_real_int @ A3 @ ord_less_real @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( ( member_real @ X @ A3 )
=> ( ( member_real @ Y3 @ A3 )
=> ( Y3 = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_491_strict__mono__on__eqD,axiom,
! [A3: set_nat,F: nat > real,X: nat,Y3: nat] :
( ( monotone_on_nat_real @ A3 @ ord_less_nat @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y3 @ A3 )
=> ( Y3 = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_492_strict__mono__on__eqD,axiom,
! [A3: set_nat,F: nat > nat,X: nat,Y3: nat] :
( ( monotone_on_nat_nat @ A3 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y3 @ A3 )
=> ( Y3 = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_493_strict__mono__on__eqD,axiom,
! [A3: set_nat,F: nat > int,X: nat,Y3: nat] :
( ( monotone_on_nat_int @ A3 @ ord_less_nat @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y3 @ A3 )
=> ( Y3 = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_494_strict__mono__on__eqD,axiom,
! [A3: set_int,F: int > real,X: int,Y3: int] :
( ( monotone_on_int_real @ A3 @ ord_less_int @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( ( member_int @ X @ A3 )
=> ( ( member_int @ Y3 @ A3 )
=> ( Y3 = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_495_strict__mono__on__eqD,axiom,
! [A3: set_int,F: int > nat,X: int,Y3: int] :
( ( monotone_on_int_nat @ A3 @ ord_less_int @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( ( member_int @ X @ A3 )
=> ( ( member_int @ Y3 @ A3 )
=> ( Y3 = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_496_strict__mono__on__eqD,axiom,
! [A3: set_int,F: int > int,X: int,Y3: int] :
( ( monotone_on_int_int @ A3 @ ord_less_int @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( ( member_int @ X @ A3 )
=> ( ( member_int @ Y3 @ A3 )
=> ( Y3 = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_497_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_498_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_499_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_500_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_501_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_502_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_503_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_504_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_505_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_506_infinite__descent0,axiom,
! [P: nat > $o,N: 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 @ N ) ) ) ).
% infinite_descent0
thf(fact_507_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M4: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_508_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_509_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_510_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_511_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_512_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_513_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_514_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_515_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_516_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_517_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_518_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_519_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_520_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_521_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_522_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_523_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_524_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_525_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_526_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_527_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_528_monotone__on__def,axiom,
( monoto4017252874604999745l_real
= ( ^ [A6: set_real,Orda: real > real > $o,Ordb: real > real > $o,F2: real > real] :
! [X2: real] :
( ( member_real @ X2 @ A6 )
=> ! [Y2: real] :
( ( member_real @ Y2 @ A6 )
=> ( ( Orda @ X2 @ Y2 )
=> ( Ordb @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_529_monotone__onI,axiom,
! [A3: set_real,Orda2: real > real > $o,Ordb2: real > real > $o,F: real > real] :
( ! [X4: real,Y: real] :
( ( member_real @ X4 @ A3 )
=> ( ( member_real @ Y @ A3 )
=> ( ( Orda2 @ X4 @ Y )
=> ( Ordb2 @ ( F @ X4 ) @ ( F @ Y ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A3 @ Orda2 @ Ordb2 @ F ) ) ).
% monotone_onI
thf(fact_530_monotone__onD,axiom,
! [A3: set_real,Orda2: real > real > $o,Ordb2: real > real > $o,F: real > real,X: real,Y3: real] :
( ( monoto4017252874604999745l_real @ A3 @ Orda2 @ Ordb2 @ F )
=> ( ( member_real @ X @ A3 )
=> ( ( member_real @ Y3 @ A3 )
=> ( ( Orda2 @ X @ Y3 )
=> ( Ordb2 @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% monotone_onD
thf(fact_531_monotone__on__subset,axiom,
! [A3: set_real,Orda2: real > real > $o,Ordb2: real > real > $o,F: real > real,B5: set_real] :
( ( monoto4017252874604999745l_real @ A3 @ Orda2 @ Ordb2 @ F )
=> ( ( ord_less_eq_set_real @ B5 @ A3 )
=> ( monoto4017252874604999745l_real @ B5 @ Orda2 @ Ordb2 @ F ) ) ) ).
% monotone_on_subset
thf(fact_532_mono__on__subset,axiom,
! [A3: set_real,F: real > real,B5: set_real] :
( ( monoto4017252874604999745l_real @ A3 @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_real @ B5 @ A3 )
=> ( monoto4017252874604999745l_real @ B5 @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ) ).
% mono_on_subset
thf(fact_533_mono__on__subset,axiom,
! [A3: set_real,F: real > nat,B5: set_real] :
( ( monotone_on_real_nat @ A3 @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_real @ B5 @ A3 )
=> ( monotone_on_real_nat @ B5 @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_534_mono__on__subset,axiom,
! [A3: set_real,F: real > int,B5: set_real] :
( ( monotone_on_real_int @ A3 @ ord_less_eq_real @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_set_real @ B5 @ A3 )
=> ( monotone_on_real_int @ B5 @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ) ).
% mono_on_subset
thf(fact_535_mono__on__subset,axiom,
! [A3: set_nat,F: nat > real,B5: set_nat] :
( ( monotone_on_nat_real @ A3 @ ord_less_eq_nat @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_nat @ B5 @ A3 )
=> ( monotone_on_nat_real @ B5 @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ) ).
% mono_on_subset
thf(fact_536_mono__on__subset,axiom,
! [A3: set_nat,F: nat > nat,B5: set_nat] :
( ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B5 @ A3 )
=> ( monotone_on_nat_nat @ B5 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_537_mono__on__subset,axiom,
! [A3: set_nat,F: nat > int,B5: set_nat] :
( ( monotone_on_nat_int @ A3 @ ord_less_eq_nat @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_set_nat @ B5 @ A3 )
=> ( monotone_on_nat_int @ B5 @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ) ).
% mono_on_subset
thf(fact_538_mono__on__subset,axiom,
! [A3: set_int,F: int > real,B5: set_int] :
( ( monotone_on_int_real @ A3 @ ord_less_eq_int @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_int @ B5 @ A3 )
=> ( monotone_on_int_real @ B5 @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ) ).
% mono_on_subset
thf(fact_539_mono__on__subset,axiom,
! [A3: set_int,F: int > nat,B5: set_int] :
( ( monotone_on_int_nat @ A3 @ ord_less_eq_int @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_int @ B5 @ A3 )
=> ( monotone_on_int_nat @ B5 @ ord_less_eq_int @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_540_mono__on__subset,axiom,
! [A3: set_int,F: int > int,B5: set_int] :
( ( monotone_on_int_int @ A3 @ ord_less_eq_int @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_set_int @ B5 @ A3 )
=> ( monotone_on_int_int @ B5 @ ord_less_eq_int @ ord_less_eq_int @ F ) ) ) ).
% mono_on_subset
thf(fact_541_mono__onI,axiom,
! [A3: set_real,F: real > real] :
( ! [R: real,S3: real] :
( ( member_real @ R @ A3 )
=> ( ( member_real @ S3 @ A3 )
=> ( ( ord_less_eq_real @ R @ S3 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A3 @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_542_mono__onI,axiom,
! [A3: set_real,F: real > nat] :
( ! [R: real,S3: real] :
( ( member_real @ R @ A3 )
=> ( ( member_real @ S3 @ A3 )
=> ( ( ord_less_eq_real @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A3 @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_543_mono__onI,axiom,
! [A3: set_real,F: real > int] :
( ! [R: real,S3: real] :
( ( member_real @ R @ A3 )
=> ( ( member_real @ S3 @ A3 )
=> ( ( ord_less_eq_real @ R @ S3 )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_int @ A3 @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_544_mono__onI,axiom,
! [A3: set_nat,F: nat > real] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A3 )
=> ( ( member_nat @ S3 @ A3 )
=> ( ( ord_less_eq_nat @ R @ S3 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_real @ A3 @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_545_mono__onI,axiom,
! [A3: set_nat,F: nat > nat] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A3 )
=> ( ( member_nat @ S3 @ A3 )
=> ( ( ord_less_eq_nat @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_546_mono__onI,axiom,
! [A3: set_nat,F: nat > int] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A3 )
=> ( ( member_nat @ S3 @ A3 )
=> ( ( ord_less_eq_nat @ R @ S3 )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_int @ A3 @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_547_mono__onI,axiom,
! [A3: set_int,F: int > real] :
( ! [R: int,S3: int] :
( ( member_int @ R @ A3 )
=> ( ( member_int @ S3 @ A3 )
=> ( ( ord_less_eq_int @ R @ S3 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_real @ A3 @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_548_mono__onI,axiom,
! [A3: set_int,F: int > nat] :
( ! [R: int,S3: int] :
( ( member_int @ R @ A3 )
=> ( ( member_int @ S3 @ A3 )
=> ( ( ord_less_eq_int @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_nat @ A3 @ ord_less_eq_int @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_549_mono__onI,axiom,
! [A3: set_int,F: int > int] :
( ! [R: int,S3: int] :
( ( member_int @ R @ A3 )
=> ( ( member_int @ S3 @ A3 )
=> ( ( ord_less_eq_int @ R @ S3 )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_int @ A3 @ ord_less_eq_int @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_550_mono__onD,axiom,
! [A3: set_real,F: real > real,R2: real,S: real] :
( ( monoto4017252874604999745l_real @ A3 @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( member_real @ R2 @ A3 )
=> ( ( member_real @ S @ A3 )
=> ( ( ord_less_eq_real @ R2 @ S )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_551_mono__onD,axiom,
! [A3: set_real,F: real > nat,R2: real,S: real] :
( ( monotone_on_real_nat @ A3 @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( member_real @ R2 @ A3 )
=> ( ( member_real @ S @ A3 )
=> ( ( ord_less_eq_real @ R2 @ S )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_552_mono__onD,axiom,
! [A3: set_real,F: real > int,R2: real,S: real] :
( ( monotone_on_real_int @ A3 @ ord_less_eq_real @ ord_less_eq_int @ F )
=> ( ( member_real @ R2 @ A3 )
=> ( ( member_real @ S @ A3 )
=> ( ( ord_less_eq_real @ R2 @ S )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_553_mono__onD,axiom,
! [A3: set_nat,F: nat > real,R2: nat,S: nat] :
( ( monotone_on_nat_real @ A3 @ ord_less_eq_nat @ ord_less_eq_real @ F )
=> ( ( member_nat @ R2 @ A3 )
=> ( ( member_nat @ S @ A3 )
=> ( ( ord_less_eq_nat @ R2 @ S )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_554_mono__onD,axiom,
! [A3: set_nat,F: nat > nat,R2: nat,S: nat] :
( ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R2 @ A3 )
=> ( ( member_nat @ S @ A3 )
=> ( ( ord_less_eq_nat @ R2 @ S )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_555_mono__onD,axiom,
! [A3: set_nat,F: nat > int,R2: nat,S: nat] :
( ( monotone_on_nat_int @ A3 @ ord_less_eq_nat @ ord_less_eq_int @ F )
=> ( ( member_nat @ R2 @ A3 )
=> ( ( member_nat @ S @ A3 )
=> ( ( ord_less_eq_nat @ R2 @ S )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_556_mono__onD,axiom,
! [A3: set_int,F: int > real,R2: int,S: int] :
( ( monotone_on_int_real @ A3 @ ord_less_eq_int @ ord_less_eq_real @ F )
=> ( ( member_int @ R2 @ A3 )
=> ( ( member_int @ S @ A3 )
=> ( ( ord_less_eq_int @ R2 @ S )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_557_mono__onD,axiom,
! [A3: set_int,F: int > nat,R2: int,S: int] :
( ( monotone_on_int_nat @ A3 @ ord_less_eq_int @ ord_less_eq_nat @ F )
=> ( ( member_int @ R2 @ A3 )
=> ( ( member_int @ S @ A3 )
=> ( ( ord_less_eq_int @ R2 @ S )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_558_mono__onD,axiom,
! [A3: set_int,F: int > int,R2: int,S: int] :
( ( monotone_on_int_int @ A3 @ ord_less_eq_int @ ord_less_eq_int @ F )
=> ( ( member_int @ R2 @ A3 )
=> ( ( member_int @ S @ A3 )
=> ( ( ord_less_eq_int @ R2 @ S )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_559_ord_Omono__on__subset,axiom,
! [A3: set_real,Less_eq: real > real > $o,F: real > real,B5: set_real] :
( ( monoto4017252874604999745l_real @ A3 @ Less_eq @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_real @ B5 @ A3 )
=> ( monoto4017252874604999745l_real @ B5 @ Less_eq @ ord_less_eq_real @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_560_ord_Omono__on__def,axiom,
! [A3: set_nat,Less_eq: nat > nat > $o,F: nat > real] :
( ( monotone_on_nat_real @ A3 @ Less_eq @ ord_less_eq_real @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A3 )
& ( member_nat @ S4 @ A3 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_561_ord_Omono__on__def,axiom,
! [A3: set_real,Less_eq: real > real > $o,F: real > real] :
( ( monoto4017252874604999745l_real @ A3 @ Less_eq @ ord_less_eq_real @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A3 )
& ( member_real @ S4 @ A3 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_562_ord_Omono__on__def,axiom,
! [A3: set_real,Less_eq: real > real > $o,F: real > nat] :
( ( monotone_on_real_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A3 )
& ( member_real @ S4 @ A3 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_563_ord_Omono__on__def,axiom,
! [A3: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A3 )
& ( member_nat @ S4 @ A3 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_564_ord_Omono__on__def,axiom,
! [A3: set_real,Less_eq: real > real > $o,F: real > int] :
( ( monotone_on_real_int @ A3 @ Less_eq @ ord_less_eq_int @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A3 )
& ( member_real @ S4 @ A3 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_int @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_565_ord_Omono__on__def,axiom,
! [A3: set_nat,Less_eq: nat > nat > $o,F: nat > int] :
( ( monotone_on_nat_int @ A3 @ Less_eq @ ord_less_eq_int @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A3 )
& ( member_nat @ S4 @ A3 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_int @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_566_ord_Omono__onI,axiom,
! [A3: set_nat,Less_eq: nat > nat > $o,F: nat > real] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A3 )
=> ( ( member_nat @ S3 @ A3 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_real @ A3 @ Less_eq @ ord_less_eq_real @ F ) ) ).
% ord.mono_onI
thf(fact_567_ord_Omono__onI,axiom,
! [A3: set_real,Less_eq: real > real > $o,F: real > real] :
( ! [R: real,S3: real] :
( ( member_real @ R @ A3 )
=> ( ( member_real @ S3 @ A3 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A3 @ Less_eq @ ord_less_eq_real @ F ) ) ).
% ord.mono_onI
thf(fact_568_ord_Omono__onI,axiom,
! [A3: set_real,Less_eq: real > real > $o,F: real > nat] :
( ! [R: real,S3: real] :
( ( member_real @ R @ A3 )
=> ( ( member_real @ S3 @ A3 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_569_ord_Omono__onI,axiom,
! [A3: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A3 )
=> ( ( member_nat @ S3 @ A3 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_570_ord_Omono__onI,axiom,
! [A3: set_real,Less_eq: real > real > $o,F: real > int] :
( ! [R: real,S3: real] :
( ( member_real @ R @ A3 )
=> ( ( member_real @ S3 @ A3 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_int @ A3 @ Less_eq @ ord_less_eq_int @ F ) ) ).
% ord.mono_onI
thf(fact_571_ord_Omono__onI,axiom,
! [A3: set_nat,Less_eq: nat > nat > $o,F: nat > int] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A3 )
=> ( ( member_nat @ S3 @ A3 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_int @ A3 @ Less_eq @ ord_less_eq_int @ F ) ) ).
% ord.mono_onI
thf(fact_572_ord_Omono__onD,axiom,
! [A3: set_nat,Less_eq: nat > nat > $o,F: nat > real,R2: nat,S: nat] :
( ( monotone_on_nat_real @ A3 @ Less_eq @ ord_less_eq_real @ F )
=> ( ( member_nat @ R2 @ A3 )
=> ( ( member_nat @ S @ A3 )
=> ( ( Less_eq @ R2 @ S )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_573_ord_Omono__onD,axiom,
! [A3: set_real,Less_eq: real > real > $o,F: real > real,R2: real,S: real] :
( ( monoto4017252874604999745l_real @ A3 @ Less_eq @ ord_less_eq_real @ F )
=> ( ( member_real @ R2 @ A3 )
=> ( ( member_real @ S @ A3 )
=> ( ( Less_eq @ R2 @ S )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_574_ord_Omono__onD,axiom,
! [A3: set_real,Less_eq: real > real > $o,F: real > nat,R2: real,S: real] :
( ( monotone_on_real_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_real @ R2 @ A3 )
=> ( ( member_real @ S @ A3 )
=> ( ( Less_eq @ R2 @ S )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_575_ord_Omono__onD,axiom,
! [A3: set_nat,Less_eq: nat > nat > $o,F: nat > nat,R2: nat,S: nat] :
( ( monotone_on_nat_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R2 @ A3 )
=> ( ( member_nat @ S @ A3 )
=> ( ( Less_eq @ R2 @ S )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_576_ord_Omono__onD,axiom,
! [A3: set_real,Less_eq: real > real > $o,F: real > int,R2: real,S: real] :
( ( monotone_on_real_int @ A3 @ Less_eq @ ord_less_eq_int @ F )
=> ( ( member_real @ R2 @ A3 )
=> ( ( member_real @ S @ A3 )
=> ( ( Less_eq @ R2 @ S )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_577_ord_Omono__onD,axiom,
! [A3: set_nat,Less_eq: nat > nat > $o,F: nat > int,R2: nat,S: nat] :
( ( monotone_on_nat_int @ A3 @ Less_eq @ ord_less_eq_int @ F )
=> ( ( member_nat @ R2 @ A3 )
=> ( ( member_nat @ S @ A3 )
=> ( ( Less_eq @ R2 @ S )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_578_ord_Ostrict__mono__onD,axiom,
! [A3: set_nat,Less: nat > nat > $o,F: nat > real,R2: nat,S: nat] :
( ( monotone_on_nat_real @ A3 @ Less @ ord_less_real @ F )
=> ( ( member_nat @ R2 @ A3 )
=> ( ( member_nat @ S @ A3 )
=> ( ( Less @ R2 @ S )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_579_ord_Ostrict__mono__onD,axiom,
! [A3: set_real,Less: real > real > $o,F: real > real,R2: real,S: real] :
( ( monoto4017252874604999745l_real @ A3 @ Less @ ord_less_real @ F )
=> ( ( member_real @ R2 @ A3 )
=> ( ( member_real @ S @ A3 )
=> ( ( Less @ R2 @ S )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_580_ord_Ostrict__mono__onD,axiom,
! [A3: set_real,Less: real > real > $o,F: real > nat,R2: real,S: real] :
( ( monotone_on_real_nat @ A3 @ Less @ ord_less_nat @ F )
=> ( ( member_real @ R2 @ A3 )
=> ( ( member_real @ S @ A3 )
=> ( ( Less @ R2 @ S )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_581_ord_Ostrict__mono__onD,axiom,
! [A3: set_nat,Less: nat > nat > $o,F: nat > nat,R2: nat,S: nat] :
( ( monotone_on_nat_nat @ A3 @ Less @ ord_less_nat @ F )
=> ( ( member_nat @ R2 @ A3 )
=> ( ( member_nat @ S @ A3 )
=> ( ( Less @ R2 @ S )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_582_ord_Ostrict__mono__onD,axiom,
! [A3: set_real,Less: real > real > $o,F: real > int,R2: real,S: real] :
( ( monotone_on_real_int @ A3 @ Less @ ord_less_int @ F )
=> ( ( member_real @ R2 @ A3 )
=> ( ( member_real @ S @ A3 )
=> ( ( Less @ R2 @ S )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_583_ord_Ostrict__mono__onD,axiom,
! [A3: set_nat,Less: nat > nat > $o,F: nat > int,R2: nat,S: nat] :
( ( monotone_on_nat_int @ A3 @ Less @ ord_less_int @ F )
=> ( ( member_nat @ R2 @ A3 )
=> ( ( member_nat @ S @ A3 )
=> ( ( Less @ R2 @ S )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_584_ord_Ostrict__mono__onI,axiom,
! [A3: set_nat,Less: nat > nat > $o,F: nat > real] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A3 )
=> ( ( member_nat @ S3 @ A3 )
=> ( ( Less @ R @ S3 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_real @ A3 @ Less @ ord_less_real @ F ) ) ).
% ord.strict_mono_onI
thf(fact_585_ord_Ostrict__mono__onI,axiom,
! [A3: set_real,Less: real > real > $o,F: real > real] :
( ! [R: real,S3: real] :
( ( member_real @ R @ A3 )
=> ( ( member_real @ S3 @ A3 )
=> ( ( Less @ R @ S3 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A3 @ Less @ ord_less_real @ F ) ) ).
% ord.strict_mono_onI
thf(fact_586_ord_Ostrict__mono__onI,axiom,
! [A3: set_real,Less: real > real > $o,F: real > nat] :
( ! [R: real,S3: real] :
( ( member_real @ R @ A3 )
=> ( ( member_real @ S3 @ A3 )
=> ( ( Less @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A3 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_587_ord_Ostrict__mono__onI,axiom,
! [A3: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A3 )
=> ( ( member_nat @ S3 @ A3 )
=> ( ( Less @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A3 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_588_ord_Ostrict__mono__onI,axiom,
! [A3: set_real,Less: real > real > $o,F: real > int] :
( ! [R: real,S3: real] :
( ( member_real @ R @ A3 )
=> ( ( member_real @ S3 @ A3 )
=> ( ( Less @ R @ S3 )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_int @ A3 @ Less @ ord_less_int @ F ) ) ).
% ord.strict_mono_onI
thf(fact_589_ord_Ostrict__mono__onI,axiom,
! [A3: set_nat,Less: nat > nat > $o,F: nat > int] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A3 )
=> ( ( member_nat @ S3 @ A3 )
=> ( ( Less @ R @ S3 )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_int @ A3 @ Less @ ord_less_int @ F ) ) ).
% ord.strict_mono_onI
thf(fact_590_ord_Ostrict__mono__on__def,axiom,
! [A3: set_nat,Less: nat > nat > $o,F: nat > real] :
( ( monotone_on_nat_real @ A3 @ Less @ ord_less_real @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A3 )
& ( member_nat @ S4 @ A3 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_591_ord_Ostrict__mono__on__def,axiom,
! [A3: set_real,Less: real > real > $o,F: real > real] :
( ( monoto4017252874604999745l_real @ A3 @ Less @ ord_less_real @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A3 )
& ( member_real @ S4 @ A3 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_592_ord_Ostrict__mono__on__def,axiom,
! [A3: set_real,Less: real > real > $o,F: real > nat] :
( ( monotone_on_real_nat @ A3 @ Less @ ord_less_nat @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A3 )
& ( member_real @ S4 @ A3 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_593_ord_Ostrict__mono__on__def,axiom,
! [A3: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A3 @ Less @ ord_less_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A3 )
& ( member_nat @ S4 @ A3 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_594_ord_Ostrict__mono__on__def,axiom,
! [A3: set_real,Less: real > real > $o,F: real > int] :
( ( monotone_on_real_int @ A3 @ Less @ ord_less_int @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A3 )
& ( member_real @ S4 @ A3 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_int @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_595_ord_Ostrict__mono__on__def,axiom,
! [A3: set_nat,Less: nat > nat > $o,F: nat > int] :
( ( monotone_on_nat_int @ A3 @ Less @ ord_less_int @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A3 )
& ( member_nat @ S4 @ A3 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_int @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_596_strict__mono__onD,axiom,
! [A3: set_real,F: real > real,R2: real,S: real] :
( ( monoto4017252874604999745l_real @ A3 @ ord_less_real @ ord_less_real @ F )
=> ( ( member_real @ R2 @ A3 )
=> ( ( member_real @ S @ A3 )
=> ( ( ord_less_real @ R2 @ S )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_597_strict__mono__onD,axiom,
! [A3: set_real,F: real > nat,R2: real,S: real] :
( ( monotone_on_real_nat @ A3 @ ord_less_real @ ord_less_nat @ F )
=> ( ( member_real @ R2 @ A3 )
=> ( ( member_real @ S @ A3 )
=> ( ( ord_less_real @ R2 @ S )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_598_strict__mono__onD,axiom,
! [A3: set_real,F: real > int,R2: real,S: real] :
( ( monotone_on_real_int @ A3 @ ord_less_real @ ord_less_int @ F )
=> ( ( member_real @ R2 @ A3 )
=> ( ( member_real @ S @ A3 )
=> ( ( ord_less_real @ R2 @ S )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_599_strict__mono__onD,axiom,
! [A3: set_nat,F: nat > real,R2: nat,S: nat] :
( ( monotone_on_nat_real @ A3 @ ord_less_nat @ ord_less_real @ F )
=> ( ( member_nat @ R2 @ A3 )
=> ( ( member_nat @ S @ A3 )
=> ( ( ord_less_nat @ R2 @ S )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_600_strict__mono__onD,axiom,
! [A3: set_nat,F: nat > nat,R2: nat,S: nat] :
( ( monotone_on_nat_nat @ A3 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ R2 @ A3 )
=> ( ( member_nat @ S @ A3 )
=> ( ( ord_less_nat @ R2 @ S )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_601_strict__mono__onD,axiom,
! [A3: set_nat,F: nat > int,R2: nat,S: nat] :
( ( monotone_on_nat_int @ A3 @ ord_less_nat @ ord_less_int @ F )
=> ( ( member_nat @ R2 @ A3 )
=> ( ( member_nat @ S @ A3 )
=> ( ( ord_less_nat @ R2 @ S )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_602_strict__mono__onD,axiom,
! [A3: set_int,F: int > real,R2: int,S: int] :
( ( monotone_on_int_real @ A3 @ ord_less_int @ ord_less_real @ F )
=> ( ( member_int @ R2 @ A3 )
=> ( ( member_int @ S @ A3 )
=> ( ( ord_less_int @ R2 @ S )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_603_strict__mono__onD,axiom,
! [A3: set_int,F: int > nat,R2: int,S: int] :
( ( monotone_on_int_nat @ A3 @ ord_less_int @ ord_less_nat @ F )
=> ( ( member_int @ R2 @ A3 )
=> ( ( member_int @ S @ A3 )
=> ( ( ord_less_int @ R2 @ S )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_604_strict__mono__onD,axiom,
! [A3: set_int,F: int > int,R2: int,S: int] :
( ( monotone_on_int_int @ A3 @ ord_less_int @ ord_less_int @ F )
=> ( ( member_int @ R2 @ A3 )
=> ( ( member_int @ S @ A3 )
=> ( ( ord_less_int @ R2 @ S )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_605_strict__mono__onI,axiom,
! [A3: set_real,F: real > real] :
( ! [R: real,S3: real] :
( ( member_real @ R @ A3 )
=> ( ( member_real @ S3 @ A3 )
=> ( ( ord_less_real @ R @ S3 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A3 @ ord_less_real @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_606_strict__mono__onI,axiom,
! [A3: set_real,F: real > nat] :
( ! [R: real,S3: real] :
( ( member_real @ R @ A3 )
=> ( ( member_real @ S3 @ A3 )
=> ( ( ord_less_real @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A3 @ ord_less_real @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_607_strict__mono__onI,axiom,
! [A3: set_real,F: real > int] :
( ! [R: real,S3: real] :
( ( member_real @ R @ A3 )
=> ( ( member_real @ S3 @ A3 )
=> ( ( ord_less_real @ R @ S3 )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_int @ A3 @ ord_less_real @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_608_strict__mono__onI,axiom,
! [A3: set_nat,F: nat > real] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A3 )
=> ( ( member_nat @ S3 @ A3 )
=> ( ( ord_less_nat @ R @ S3 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_real @ A3 @ ord_less_nat @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_609_strict__mono__onI,axiom,
! [A3: set_nat,F: nat > nat] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A3 )
=> ( ( member_nat @ S3 @ A3 )
=> ( ( ord_less_nat @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A3 @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_610_strict__mono__onI,axiom,
! [A3: set_nat,F: nat > int] :
( ! [R: nat,S3: nat] :
( ( member_nat @ R @ A3 )
=> ( ( member_nat @ S3 @ A3 )
=> ( ( ord_less_nat @ R @ S3 )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_int @ A3 @ ord_less_nat @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_611_strict__mono__onI,axiom,
! [A3: set_int,F: int > real] :
( ! [R: int,S3: int] :
( ( member_int @ R @ A3 )
=> ( ( member_int @ S3 @ A3 )
=> ( ( ord_less_int @ R @ S3 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_real @ A3 @ ord_less_int @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_612_strict__mono__onI,axiom,
! [A3: set_int,F: int > nat] :
( ! [R: int,S3: int] :
( ( member_int @ R @ A3 )
=> ( ( member_int @ S3 @ A3 )
=> ( ( ord_less_int @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_nat @ A3 @ ord_less_int @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_613_strict__mono__onI,axiom,
! [A3: set_int,F: int > int] :
( ! [R: int,S3: int] :
( ( member_int @ R @ A3 )
=> ( ( member_int @ S3 @ A3 )
=> ( ( ord_less_int @ R @ S3 )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_int @ A3 @ ord_less_int @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_614_subsetI,axiom,
! [A3: set_real,B5: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A3 )
=> ( member_real @ X4 @ B5 ) )
=> ( ord_less_eq_set_real @ A3 @ B5 ) ) ).
% subsetI
thf(fact_615_subsetI,axiom,
! [A3: set_nat,B5: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_nat @ X4 @ B5 ) )
=> ( ord_less_eq_set_nat @ A3 @ B5 ) ) ).
% subsetI
thf(fact_616_continuous__on__subset,axiom,
! [S: set_real,F: real > real,T: set_real] :
( ( topolo5044208981011980120l_real @ S @ F )
=> ( ( ord_less_eq_set_real @ T @ S )
=> ( topolo5044208981011980120l_real @ T @ F ) ) ) ).
% continuous_on_subset
thf(fact_617_strict__mono__continuous__invD,axiom,
! [A: real,F: real > real,G: real > real] :
( ( monoto4017252874604999745l_real @ ( set_ord_atLeast_real @ A ) @ ord_less_real @ ord_less_real @ F )
=> ( ( topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ A ) @ F )
=> ( ( ( image_real_real @ F @ ( set_ord_atLeast_real @ A ) )
= ( set_ord_atLeast_real @ ( F @ A ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( G @ ( F @ X4 ) )
= X4 ) )
=> ( topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ ( F @ A ) ) @ G ) ) ) ) ) ).
% strict_mono_continuous_invD
thf(fact_618_antimono__fun__sum__integral__diff__def,axiom,
( integr4865894440751020556l_diff
= ( ^ [F2: real > real] :
( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) )
& ! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X2 ) ) )
& ( topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ F2 ) ) ) ) ).
% antimono_fun_sum_integral_diff_def
thf(fact_619_antimono__fun__sum__integral__diff_Ointro,axiom,
! [F: real > real] :
( ! [X4: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ X4 @ Y )
=> ( ord_less_eq_real @ ( F @ Y ) @ ( F @ X4 ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ F )
=> ( integr4865894440751020556l_diff @ F ) ) ) ) ).
% antimono_fun_sum_integral_diff.intro
thf(fact_620_inner__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( inner_5569486634795291694r_real @ A @ B ) ) ) ) ).
% inner_nonneg_nonneg
thf(fact_621_image__eqI,axiom,
! [B: real,F: real > real,X: real,A3: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A3 )
=> ( member_real @ B @ ( image_real_real @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_622_image__eqI,axiom,
! [B: nat,F: real > nat,X: real,A3: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A3 )
=> ( member_nat @ B @ ( image_real_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_623_image__eqI,axiom,
! [B: int,F: nat > int,X: nat,A3: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A3 )
=> ( member_int @ B @ ( image_nat_int @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_624_image__eqI,axiom,
! [B: real,F: nat > real,X: nat,A3: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A3 )
=> ( member_real @ B @ ( image_nat_real @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_625_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A3: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A3 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_626_image__mono,axiom,
! [A3: set_real,B5: set_real,F: real > real] :
( ( ord_less_eq_set_real @ A3 @ B5 )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A3 ) @ ( image_real_real @ F @ B5 ) ) ) ).
% image_mono
thf(fact_627_image__mono,axiom,
! [A3: set_nat,B5: set_nat,F: nat > int] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A3 ) @ ( image_nat_int @ F @ B5 ) ) ) ).
% image_mono
thf(fact_628_image__mono,axiom,
! [A3: set_nat,B5: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ ( image_nat_nat @ F @ B5 ) ) ) ).
% image_mono
thf(fact_629_image__subsetI,axiom,
! [A3: set_real,F: real > real,B5: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A3 )
=> ( member_real @ ( F @ X4 ) @ B5 ) )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A3 ) @ B5 ) ) ).
% image_subsetI
thf(fact_630_image__subsetI,axiom,
! [A3: set_real,F: real > nat,B5: set_nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A3 )
=> ( member_nat @ ( F @ X4 ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_real_nat @ F @ A3 ) @ B5 ) ) ).
% image_subsetI
thf(fact_631_image__subsetI,axiom,
! [A3: set_nat,F: nat > int,B5: set_int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_int @ ( F @ X4 ) @ B5 ) )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A3 ) @ B5 ) ) ).
% image_subsetI
thf(fact_632_image__subsetI,axiom,
! [A3: set_nat,F: nat > real,B5: set_real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_real @ ( F @ X4 ) @ B5 ) )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ A3 ) @ B5 ) ) ).
% image_subsetI
thf(fact_633_image__subsetI,axiom,
! [A3: set_nat,F: nat > nat,B5: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_nat @ ( F @ X4 ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B5 ) ) ).
% image_subsetI
thf(fact_634_subset__imageE,axiom,
! [B5: set_real,F: real > real,A3: set_real] :
( ( ord_less_eq_set_real @ B5 @ ( image_real_real @ F @ A3 ) )
=> ~ ! [C3: set_real] :
( ( ord_less_eq_set_real @ C3 @ A3 )
=> ( B5
!= ( image_real_real @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_635_subset__imageE,axiom,
! [B5: set_int,F: nat > int,A3: set_nat] :
( ( ord_less_eq_set_int @ B5 @ ( image_nat_int @ F @ A3 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A3 )
=> ( B5
!= ( image_nat_int @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_636_subset__imageE,axiom,
! [B5: set_nat,F: nat > nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A3 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A3 )
=> ( B5
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_637_image__subset__iff,axiom,
! [F: nat > int,A3: set_nat,B5: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A3 ) @ B5 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_int @ ( F @ X2 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_638_image__subset__iff,axiom,
! [F: real > real,A3: set_real,B5: set_real] :
( ( ord_less_eq_set_real @ ( image_real_real @ F @ A3 ) @ B5 )
= ( ! [X2: real] :
( ( member_real @ X2 @ A3 )
=> ( member_real @ ( F @ X2 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_639_image__subset__iff,axiom,
! [F: nat > nat,A3: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B5 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat @ ( F @ X2 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_640_subset__image__iff,axiom,
! [B5: set_real,F: real > real,A3: set_real] :
( ( ord_less_eq_set_real @ B5 @ ( image_real_real @ F @ A3 ) )
= ( ? [AA: set_real] :
( ( ord_less_eq_set_real @ AA @ A3 )
& ( B5
= ( image_real_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_641_subset__image__iff,axiom,
! [B5: set_int,F: nat > int,A3: set_nat] :
( ( ord_less_eq_set_int @ B5 @ ( image_nat_int @ F @ A3 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A3 )
& ( B5
= ( image_nat_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_642_subset__image__iff,axiom,
! [B5: set_nat,F: nat > nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A3 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A3 )
& ( B5
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_643_psubsetD,axiom,
! [A3: set_real,B5: set_real,C: real] :
( ( ord_less_set_real @ A3 @ B5 )
=> ( ( member_real @ C @ A3 )
=> ( member_real @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_644_psubsetD,axiom,
! [A3: set_nat,B5: set_nat,C: nat] :
( ( ord_less_set_nat @ A3 @ B5 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_645_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_646_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_647_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_648_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_649_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_650_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_651_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_652_linorder__neqE__nat,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_653_imageI,axiom,
! [X: real,A3: set_real,F: real > real] :
( ( member_real @ X @ A3 )
=> ( member_real @ ( F @ X ) @ ( image_real_real @ F @ A3 ) ) ) ).
% imageI
thf(fact_654_imageI,axiom,
! [X: real,A3: set_real,F: real > nat] :
( ( member_real @ X @ A3 )
=> ( member_nat @ ( F @ X ) @ ( image_real_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_655_imageI,axiom,
! [X: nat,A3: set_nat,F: nat > int] :
( ( member_nat @ X @ A3 )
=> ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ A3 ) ) ) ).
% imageI
thf(fact_656_imageI,axiom,
! [X: nat,A3: set_nat,F: nat > real] :
( ( member_nat @ X @ A3 )
=> ( member_real @ ( F @ X ) @ ( image_nat_real @ F @ A3 ) ) ) ).
% imageI
thf(fact_657_imageI,axiom,
! [X: nat,A3: set_nat,F: nat > nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_658_image__iff,axiom,
! [Z2: int,F: nat > int,A3: set_nat] :
( ( member_int @ Z2 @ ( image_nat_int @ F @ A3 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_659_image__iff,axiom,
! [Z2: real,F: real > real,A3: set_real] :
( ( member_real @ Z2 @ ( image_real_real @ F @ A3 ) )
= ( ? [X2: real] :
( ( member_real @ X2 @ A3 )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_660_image__iff,axiom,
! [Z2: nat,F: nat > nat,A3: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A3 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_661_bex__imageD,axiom,
! [F: real > real,A3: set_real,P: real > $o] :
( ? [X3: real] :
( ( member_real @ X3 @ ( image_real_real @ F @ A3 ) )
& ( P @ X3 ) )
=> ? [X4: real] :
( ( member_real @ X4 @ A3 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_662_bex__imageD,axiom,
! [F: nat > int,A3: set_nat,P: int > $o] :
( ? [X3: int] :
( ( member_int @ X3 @ ( image_nat_int @ F @ A3 ) )
& ( P @ X3 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A3 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_663_bex__imageD,axiom,
! [F: nat > nat,A3: set_nat,P: nat > $o] :
( ? [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A3 ) )
& ( P @ X3 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A3 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_664_image__cong,axiom,
! [M4: set_real,N5: set_real,F: real > real,G: real > real] :
( ( M4 = N5 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ N5 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_real_real @ F @ M4 )
= ( image_real_real @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_665_image__cong,axiom,
! [M4: set_nat,N5: set_nat,F: nat > int,G: nat > int] :
( ( M4 = N5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_int @ F @ M4 )
= ( image_nat_int @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_666_image__cong,axiom,
! [M4: set_nat,N5: set_nat,F: nat > nat,G: nat > nat] :
( ( M4 = N5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_nat @ F @ M4 )
= ( image_nat_nat @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_667_ball__imageD,axiom,
! [F: real > real,A3: set_real,P: real > $o] :
( ! [X4: real] :
( ( member_real @ X4 @ ( image_real_real @ F @ A3 ) )
=> ( P @ X4 ) )
=> ! [X3: real] :
( ( member_real @ X3 @ A3 )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_668_ball__imageD,axiom,
! [F: nat > int,A3: set_nat,P: int > $o] :
( ! [X4: int] :
( ( member_int @ X4 @ ( image_nat_int @ F @ A3 ) )
=> ( P @ X4 ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_669_ball__imageD,axiom,
! [F: nat > nat,A3: set_nat,P: nat > $o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A3 ) )
=> ( P @ X4 ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_670_rev__image__eqI,axiom,
! [X: real,A3: set_real,B: real,F: real > real] :
( ( member_real @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_real_real @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_671_rev__image__eqI,axiom,
! [X: real,A3: set_real,B: nat,F: real > nat] :
( ( member_real @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_real_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_672_rev__image__eqI,axiom,
! [X: nat,A3: set_nat,B: int,F: nat > int] :
( ( member_nat @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_nat_int @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_673_rev__image__eqI,axiom,
! [X: nat,A3: set_nat,B: real,F: nat > real] :
( ( member_nat @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_nat_real @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_674_rev__image__eqI,axiom,
! [X: nat,A3: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_675_antimono__fun__sum__integral__diff_Ononneg,axiom,
! [F: real > real,X: real] :
( ( integr4865894440751020556l_diff @ F )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ).
% antimono_fun_sum_integral_diff.nonneg
thf(fact_676_antimono__fun__sum__integral__diff_Odec,axiom,
! [F: real > real,X: real,Y3: real] :
( ( integr4865894440751020556l_diff @ F )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X ) ) ) ) ) ).
% antimono_fun_sum_integral_diff.dec
thf(fact_677_invertible__fixpoint__property,axiom,
! [T2: set_int,I: int > nat,S2: set_nat,R2: nat > int,G: int > int] :
( ( topolo2181401217840723324nt_nat @ T2 @ I )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ I @ T2 ) @ S2 )
=> ( ( topolo1179557035430618492at_int @ S2 @ R2 )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ R2 @ S2 ) @ T2 )
=> ( ! [Y: int] :
( ( member_int @ Y @ T2 )
=> ( ( R2 @ ( I @ Y ) )
= Y ) )
=> ( ! [F3: nat > nat] :
( ( topolo1182047505939668768at_nat @ S2 @ F3 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ S2 ) @ S2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S2 )
& ( ( F3 @ X3 )
= X3 ) ) ) )
=> ( ( topolo2178910747331673048nt_int @ T2 @ G )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ G @ T2 ) @ T2 )
=> ~ ! [Y: int] :
( ( member_int @ Y @ T2 )
=> ( ( G @ Y )
!= Y ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_678_invertible__fixpoint__property,axiom,
! [T2: set_nat,I: nat > int,S2: set_int,R2: int > nat,G: nat > nat] :
( ( topolo1179557035430618492at_int @ T2 @ I )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ I @ T2 ) @ S2 )
=> ( ( topolo2181401217840723324nt_nat @ S2 @ R2 )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ R2 @ S2 ) @ T2 )
=> ( ! [Y: nat] :
( ( member_nat @ Y @ T2 )
=> ( ( R2 @ ( I @ Y ) )
= Y ) )
=> ( ! [F3: int > int] :
( ( topolo2178910747331673048nt_int @ S2 @ F3 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F3 @ S2 ) @ S2 )
=> ? [X3: int] :
( ( member_int @ X3 @ S2 )
& ( ( F3 @ X3 )
= X3 ) ) ) )
=> ( ( topolo1182047505939668768at_nat @ T2 @ G )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ T2 ) @ T2 )
=> ~ ! [Y: nat] :
( ( member_nat @ Y @ T2 )
=> ( ( G @ Y )
!= Y ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_679_invertible__fixpoint__property,axiom,
! [T2: set_nat,I: nat > nat,S2: set_nat,R2: nat > nat,G: nat > nat] :
( ( topolo1182047505939668768at_nat @ T2 @ I )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ I @ T2 ) @ S2 )
=> ( ( topolo1182047505939668768at_nat @ S2 @ R2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ S2 ) @ T2 )
=> ( ! [Y: nat] :
( ( member_nat @ Y @ T2 )
=> ( ( R2 @ ( I @ Y ) )
= Y ) )
=> ( ! [F3: nat > nat] :
( ( topolo1182047505939668768at_nat @ S2 @ F3 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ S2 ) @ S2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S2 )
& ( ( F3 @ X3 )
= X3 ) ) ) )
=> ( ( topolo1182047505939668768at_nat @ T2 @ G )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ T2 ) @ T2 )
=> ~ ! [Y: nat] :
( ( member_nat @ Y @ T2 )
=> ( ( G @ Y )
!= Y ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_680_invertible__fixpoint__property,axiom,
! [T2: set_real,I: real > nat,S2: set_nat,R2: nat > real,G: real > real] :
( ( topolo2287203362918339196al_nat @ T2 @ I )
=> ( ( ord_less_eq_set_nat @ ( image_real_nat @ I @ T2 ) @ S2 )
=> ( ( topolo6943266826644216316t_real @ S2 @ R2 )
=> ( ( ord_less_eq_set_real @ ( image_nat_real @ R2 @ S2 ) @ T2 )
=> ( ! [Y: real] :
( ( member_real @ Y @ T2 )
=> ( ( R2 @ ( I @ Y ) )
= Y ) )
=> ( ! [F3: nat > nat] :
( ( topolo1182047505939668768at_nat @ S2 @ F3 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ S2 ) @ S2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S2 )
& ( ( F3 @ X3 )
= X3 ) ) ) )
=> ( ( topolo5044208981011980120l_real @ T2 @ G )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ G @ T2 ) @ T2 )
=> ~ ! [Y: real] :
( ( member_real @ Y @ T2 )
=> ( ( G @ Y )
!= Y ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_681_invertible__fixpoint__property,axiom,
! [T2: set_nat,I: nat > real,S2: set_real,R2: real > nat,G: nat > nat] :
( ( topolo6943266826644216316t_real @ T2 @ I )
=> ( ( ord_less_eq_set_real @ ( image_nat_real @ I @ T2 ) @ S2 )
=> ( ( topolo2287203362918339196al_nat @ S2 @ R2 )
=> ( ( ord_less_eq_set_nat @ ( image_real_nat @ R2 @ S2 ) @ T2 )
=> ( ! [Y: nat] :
( ( member_nat @ Y @ T2 )
=> ( ( R2 @ ( I @ Y ) )
= Y ) )
=> ( ! [F3: real > real] :
( ( topolo5044208981011980120l_real @ S2 @ F3 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ F3 @ S2 ) @ S2 )
=> ? [X3: real] :
( ( member_real @ X3 @ S2 )
& ( ( F3 @ X3 )
= X3 ) ) ) )
=> ( ( topolo1182047505939668768at_nat @ T2 @ G )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ T2 ) @ T2 )
=> ~ ! [Y: nat] :
( ( member_nat @ Y @ T2 )
=> ( ( G @ Y )
!= Y ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_682_invertible__fixpoint__property,axiom,
! [T2: set_real,I: real > real,S2: set_real,R2: real > real,G: real > real] :
( ( topolo5044208981011980120l_real @ T2 @ I )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ I @ T2 ) @ S2 )
=> ( ( topolo5044208981011980120l_real @ S2 @ R2 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ R2 @ S2 ) @ T2 )
=> ( ! [Y: real] :
( ( member_real @ Y @ T2 )
=> ( ( R2 @ ( I @ Y ) )
= Y ) )
=> ( ! [F3: real > real] :
( ( topolo5044208981011980120l_real @ S2 @ F3 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ F3 @ S2 ) @ S2 )
=> ? [X3: real] :
( ( member_real @ X3 @ S2 )
& ( ( F3 @ X3 )
= X3 ) ) ) )
=> ( ( topolo5044208981011980120l_real @ T2 @ G )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ G @ T2 ) @ T2 )
=> ~ ! [Y: real] :
( ( member_real @ Y @ T2 )
=> ( ( G @ Y )
!= Y ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_683_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [T3: real] :
( ( member_real @ T3 @ A6 )
=> ( member_real @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_684_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A6 )
=> ( member_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_685_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [X2: real] :
( ( member_real @ X2 @ A6 )
=> ( member_real @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_686_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( member_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_687_subsetD,axiom,
! [A3: set_real,B5: set_real,C: real] :
( ( ord_less_eq_set_real @ A3 @ B5 )
=> ( ( member_real @ C @ A3 )
=> ( member_real @ C @ B5 ) ) ) ).
% subsetD
thf(fact_688_subsetD,axiom,
! [A3: set_nat,B5: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_689_in__mono,axiom,
! [A3: set_real,B5: set_real,X: real] :
( ( ord_less_eq_set_real @ A3 @ B5 )
=> ( ( member_real @ X @ A3 )
=> ( member_real @ X @ B5 ) ) ) ).
% in_mono
thf(fact_690_in__mono,axiom,
! [A3: set_nat,B5: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
=> ( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_691_antimono__fun__sum__integral__diff_Ocont,axiom,
! [F: real > real] :
( ( integr4865894440751020556l_diff @ F )
=> ( topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ F ) ) ).
% antimono_fun_sum_integral_diff.cont
thf(fact_692_continuous__on__cong,axiom,
! [S: set_real,T: set_real,F: real > real,G: real > real] :
( ( S = T )
=> ( ! [X4: real] :
( ( member_real @ X4 @ T )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( topolo5044208981011980120l_real @ S @ F )
= ( topolo5044208981011980120l_real @ T @ G ) ) ) ) ).
% continuous_on_cong
thf(fact_693_inner__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( inner_5569486634795291694r_real @ X @ X ) )
= ( X != zero_zero_real ) ) ).
% inner_gt_zero_iff
thf(fact_694_inner__eq__zero__iff,axiom,
! [X: real] :
( ( ( inner_5569486634795291694r_real @ X @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% inner_eq_zero_iff
thf(fact_695_inner__zero__right,axiom,
! [X: real] :
( ( inner_5569486634795291694r_real @ X @ zero_zero_real )
= zero_zero_real ) ).
% inner_zero_right
thf(fact_696_inner__zero__left,axiom,
! [X: real] :
( ( inner_5569486634795291694r_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% inner_zero_left
thf(fact_697_all__zero__iff,axiom,
! [X: real] :
( ( ! [U: real] :
( ( inner_5569486634795291694r_real @ X @ U )
= zero_zero_real ) )
= ( X = zero_zero_real ) ) ).
% all_zero_iff
thf(fact_698_inner__ge__zero,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( inner_5569486634795291694r_real @ X @ X ) ) ).
% inner_ge_zero
thf(fact_699_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_700_inner__commute,axiom,
( inner_5569486634795291694r_real
= ( ^ [X2: real,Y2: real] : ( inner_5569486634795291694r_real @ Y2 @ X2 ) ) ) ).
% inner_commute
thf(fact_701_hyperplane__eq__Ex,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ~ ! [X4: real] :
( ( inner_5569486634795291694r_real @ A @ X4 )
!= B ) ) ).
% hyperplane_eq_Ex
thf(fact_702_all__subset__image,axiom,
! [F: real > real,A3: set_real,P: set_real > $o] :
( ( ! [B6: set_real] :
( ( ord_less_eq_set_real @ B6 @ ( image_real_real @ F @ A3 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_real] :
( ( ord_less_eq_set_real @ B6 @ A3 )
=> ( P @ ( image_real_real @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_703_all__subset__image,axiom,
! [F: nat > int,A3: set_nat,P: set_int > $o] :
( ( ! [B6: set_int] :
( ( ord_less_eq_set_int @ B6 @ ( image_nat_int @ F @ A3 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A3 )
=> ( P @ ( image_nat_int @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_704_all__subset__image,axiom,
! [F: nat > nat,A3: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A3 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A3 )
=> ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_705_antimono__fun__sum__integral__diff_Osum__integral__diff__series__antimono,axiom,
! [F: real > real,M: nat,N: nat] :
( ( integr4865894440751020556l_diff @ F )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_real @ ( integr3166334062659923703series @ F @ N ) @ ( integr3166334062659923703series @ F @ M ) ) ) ) ).
% antimono_fun_sum_integral_diff.sum_integral_diff_series_antimono
thf(fact_706_antimono__fun__sum__integral__diff_Osum__integral__diff__series__nonneg,axiom,
! [F: real > real,N: nat] :
( ( integr4865894440751020556l_diff @ F )
=> ( ord_less_eq_real @ zero_zero_real @ ( integr3166334062659923703series @ F @ N ) ) ) ).
% antimono_fun_sum_integral_diff.sum_integral_diff_series_nonneg
thf(fact_707_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_708_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_709_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_710_strict__mono__image__endpoints,axiom,
! [A: real,B: real,F: real > real] :
( ( monoto4017252874604999745l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ord_less_real @ ord_less_real @ F )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% strict_mono_image_endpoints
thf(fact_711_strict__mono__image__endpoints,axiom,
! [A: real,B: real,F: real > nat] :
( ( monotone_on_real_nat @ ( set_or1222579329274155063t_real @ A @ B ) @ ord_less_real @ ord_less_nat @ F )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( image_real_nat @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1269000886237332187st_nat @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% strict_mono_image_endpoints
thf(fact_712_strict__mono__image__endpoints,axiom,
! [A: real,B: real,F: real > int] :
( ( monotone_on_real_int @ ( set_or1222579329274155063t_real @ A @ B ) @ ord_less_real @ ord_less_int @ F )
=> ( ( topolo2284712892409288920al_int @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( image_real_int @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1266510415728281911st_int @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% strict_mono_image_endpoints
thf(fact_713_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_714_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_715_atLeastAtMost__iff,axiom,
! [I: real,L: real,U2: real] :
( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U2 ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_eq_real @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_716_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U2: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U2 ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_717_atLeastAtMost__iff,axiom,
! [I: int,L: int,U2: int] :
( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U2 ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_eq_int @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_718_Icc__eq__Icc,axiom,
! [L: real,H: real,L2: real,H2: real] :
( ( ( set_or1222579329274155063t_real @ L @ H )
= ( set_or1222579329274155063t_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_real @ L @ H )
& ~ ( ord_less_eq_real @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_719_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_720_Icc__eq__Icc,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or1266510415728281911st_int @ L @ H )
= ( set_or1266510415728281911st_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_int @ L @ H )
& ~ ( ord_less_eq_int @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_721_atLeastatMost__subset__iff,axiom,
! [A: real,B: real,C: real,D3: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D3 ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D3 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_722_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D3 ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D3 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_723_atLeastatMost__subset__iff,axiom,
! [A: int,B: int,C: int,D3: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D3 ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D3 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_724_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_725_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_726_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_727_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_728_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_729_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_730_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_731_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_732_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_733_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_734_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_735_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_736_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_737_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_738_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_739_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_740_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_741_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_742_Icc__subset__Ici__iff,axiom,
! [L: real,H: real,L2: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H ) @ ( set_ord_atLeast_real @ L2 ) )
= ( ~ ( ord_less_eq_real @ L @ H )
| ( ord_less_eq_real @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_743_Icc__subset__Ici__iff,axiom,
! [L: nat,H: nat,L2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atLeast_nat @ L2 ) )
= ( ~ ( ord_less_eq_nat @ L @ H )
| ( ord_less_eq_nat @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_744_Icc__subset__Ici__iff,axiom,
! [L: int,H: int,L2: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H ) @ ( set_ord_atLeast_int @ L2 ) )
= ( ~ ( ord_less_eq_int @ L @ H )
| ( ord_less_eq_int @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_745_antimono__fun__sum__integral__diff_Osum__integral__diff__series_Ocong,axiom,
integr3166334062659923703series = integr3166334062659923703series ).
% antimono_fun_sum_integral_diff.sum_integral_diff_series.cong
thf(fact_746_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_747_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_748_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_749_not__Ici__eq__Icc,axiom,
! [L2: real,L: real,H: real] :
( ( set_ord_atLeast_real @ L2 )
!= ( set_or1222579329274155063t_real @ L @ H ) ) ).
% not_Ici_eq_Icc
thf(fact_750_not__Ici__eq__Icc,axiom,
! [L2: nat,L: nat,H: nat] :
( ( set_ord_atLeast_nat @ L2 )
!= ( set_or1269000886237332187st_nat @ L @ H ) ) ).
% not_Ici_eq_Icc
thf(fact_751_not__Ici__eq__Icc,axiom,
! [L2: int,L: int,H: int] :
( ( set_ord_atLeast_int @ L2 )
!= ( set_or1266510415728281911st_int @ L @ H ) ) ).
% not_Ici_eq_Icc
thf(fact_752_IVT_H,axiom,
! [F: real > nat,A: real,Y3: nat,B: real] :
( ( ord_less_eq_nat @ ( F @ A ) @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ( F @ X4 )
= Y3 ) ) ) ) ) ) ).
% IVT'
thf(fact_753_IVT_H,axiom,
! [F: real > int,A: real,Y3: int,B: real] :
( ( ord_less_eq_int @ ( F @ A ) @ Y3 )
=> ( ( ord_less_eq_int @ Y3 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2284712892409288920al_int @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ( F @ X4 )
= Y3 ) ) ) ) ) ) ).
% IVT'
thf(fact_754_IVT_H,axiom,
! [F: real > real,A: real,Y3: real,B: real] :
( ( ord_less_eq_real @ ( F @ A ) @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ( F @ X4 )
= Y3 ) ) ) ) ) ) ).
% IVT'
thf(fact_755_IVT2_H,axiom,
! [F: real > nat,B: real,Y3: nat,A: real] :
( ( ord_less_eq_nat @ ( F @ B ) @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ ( F @ A ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ( F @ X4 )
= Y3 ) ) ) ) ) ) ).
% IVT2'
thf(fact_756_IVT2_H,axiom,
! [F: real > int,B: real,Y3: int,A: real] :
( ( ord_less_eq_int @ ( F @ B ) @ Y3 )
=> ( ( ord_less_eq_int @ Y3 @ ( F @ A ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2284712892409288920al_int @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ( F @ X4 )
= Y3 ) ) ) ) ) ) ).
% IVT2'
thf(fact_757_IVT2_H,axiom,
! [F: real > real,B: real,Y3: real,A: real] :
( ( ord_less_eq_real @ ( F @ B ) @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ ( F @ A ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ( F @ X4 )
= Y3 ) ) ) ) ) ) ).
% IVT2'
thf(fact_758_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_759_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_760_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_761_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_762_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_763_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_764_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_765_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_766_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_767_not__Ici__le__Icc,axiom,
! [L: real,L2: real,H2: real] :
~ ( ord_less_eq_set_real @ ( set_ord_atLeast_real @ L ) @ ( set_or1222579329274155063t_real @ L2 @ H2 ) ) ).
% not_Ici_le_Icc
thf(fact_768_not__Ici__le__Icc,axiom,
! [L: nat,L2: nat,H2: nat] :
~ ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ L ) @ ( set_or1269000886237332187st_nat @ L2 @ H2 ) ) ).
% not_Ici_le_Icc
thf(fact_769_not__Ici__le__Icc,axiom,
! [L: int,L2: int,H2: int] :
~ ( ord_less_eq_set_int @ ( set_ord_atLeast_int @ L ) @ ( set_or1266510415728281911st_int @ L2 @ H2 ) ) ).
% not_Ici_le_Icc
thf(fact_770_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_771_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_772_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_773_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_774_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_775_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_776_atLeastatMost__psubset__iff,axiom,
! [A: real,B: real,C: real,D3: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D3 ) )
= ( ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D3 )
& ( ( ord_less_real @ C @ A )
| ( ord_less_real @ B @ D3 ) ) ) )
& ( ord_less_eq_real @ C @ D3 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_777_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D3 ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D3 )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B @ D3 ) ) ) )
& ( ord_less_eq_nat @ C @ D3 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_778_atLeastatMost__psubset__iff,axiom,
! [A: int,B: int,C: int,D3: int] :
( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D3 ) )
= ( ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D3 )
& ( ( ord_less_int @ C @ A )
| ( ord_less_int @ B @ D3 ) ) ) )
& ( ord_less_eq_int @ C @ D3 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_779_interval__inner__leI_I2_J,axiom,
! [X: real,A: real,B: real,I: real] :
( ( member_real @ X @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ I )
=> ( ord_less_eq_real @ ( inner_5569486634795291694r_real @ X @ I ) @ ( inner_5569486634795291694r_real @ B @ I ) ) ) ) ).
% interval_inner_leI(2)
thf(fact_780_interval__inner__leI_I1_J,axiom,
! [X: real,A: real,B: real,I: real] :
( ( member_real @ X @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ I )
=> ( ord_less_eq_real @ ( inner_5569486634795291694r_real @ A @ I ) @ ( inner_5569486634795291694r_real @ X @ I ) ) ) ) ).
% interval_inner_leI(1)
thf(fact_781_continuous__image__closed__interval,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [C2: real,D5: real] :
( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ C2 @ D5 ) )
& ( ord_less_eq_real @ C2 @ D5 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_782_ivt__decreasing__component__on__1,axiom,
! [A: real,B: real,F: real > real,K: real,Y3: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ( ord_less_eq_real @ ( inner_5569486634795291694r_real @ ( F @ B ) @ K ) @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ ( inner_5569486634795291694r_real @ ( F @ A ) @ K ) )
=> ? [X4: real] :
( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
& ( ( inner_5569486634795291694r_real @ ( F @ X4 ) @ K )
= Y3 ) ) ) ) ) ) ).
% ivt_decreasing_component_on_1
thf(fact_783_ivt__increasing__component__on__1,axiom,
! [A: real,B: real,F: real > real,K: real,Y3: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ( ord_less_eq_real @ ( inner_5569486634795291694r_real @ ( F @ A ) @ K ) @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ ( inner_5569486634795291694r_real @ ( F @ B ) @ K ) )
=> ? [X4: real] :
( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
& ( ( inner_5569486634795291694r_real @ ( F @ X4 ) @ K )
= Y3 ) ) ) ) ) ) ).
% ivt_increasing_component_on_1
thf(fact_784_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_785_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_786_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less
thf(fact_787_image__int__atLeastAtMost,axiom,
! [A: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastAtMost
thf(fact_788_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_789_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A2 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_790_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_791_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_792_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less
thf(fact_793_reals__Archimedean2,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% reals_Archimedean2
thf(fact_794_real__arch__simple,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_795_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_796_image__mult__atLeastAtMost,axiom,
! [D3: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
=> ( ( image_real_real @ ( times_times_real @ D3 ) @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ ( times_times_real @ D3 @ A ) @ ( times_times_real @ D3 @ B ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_797_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_798_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_799_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_800_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_801_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_802_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_803_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_804_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_805_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_806_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_807_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_808_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_809_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_810_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_811_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_812_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_813_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_814_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_815_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_816_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_817_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_818_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_819_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_820_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_821_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_822_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_823_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_824_inner__minus__left,axiom,
! [X: real,Y3: real] :
( ( inner_5569486634795291694r_real @ ( uminus_uminus_real @ X ) @ Y3 )
= ( uminus_uminus_real @ ( inner_5569486634795291694r_real @ X @ Y3 ) ) ) ).
% inner_minus_left
thf(fact_825_inner__minus__right,axiom,
! [X: real,Y3: real] :
( ( inner_5569486634795291694r_real @ X @ ( uminus_uminus_real @ Y3 ) )
= ( uminus_uminus_real @ ( inner_5569486634795291694r_real @ X @ Y3 ) ) ) ).
% inner_minus_right
thf(fact_826_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_827_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_828_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_829_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_830_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_831_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_832_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_833_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_834_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_835_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_836_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_837_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_838_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_839_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_840_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_841_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_842_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_843_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_844_image__uminus__atLeastAtMost,axiom,
! [X: real,Y3: real] :
( ( image_real_real @ uminus_uminus_real @ ( set_or1222579329274155063t_real @ X @ Y3 ) )
= ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ Y3 ) @ ( uminus_uminus_real @ X ) ) ) ).
% image_uminus_atLeastAtMost
thf(fact_845_image__uminus__atLeastAtMost,axiom,
! [X: int,Y3: int] :
( ( image_int_int @ uminus_uminus_int @ ( set_or1266510415728281911st_int @ X @ Y3 ) )
= ( set_or1266510415728281911st_int @ ( uminus_uminus_int @ Y3 ) @ ( uminus_uminus_int @ X ) ) ) ).
% image_uminus_atLeastAtMost
thf(fact_846_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_847_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_848_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_849_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_850_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_851_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_852_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_853_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A2: int,B2: int] : ( times_times_int @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_854_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A2: real,B2: real] : ( times_times_real @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_855_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_856_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_857_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_858_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_859_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_860_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_861_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_862_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_863_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_864_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_865_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_866_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_867_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_868_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_869_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_870_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_871_le__imp__neg__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_872_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_873_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_874_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_875_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_876_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_877_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_878_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_879_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_880_mult__of__nat__commute,axiom,
! [X: nat,Y3: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y3 )
= ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_881_mult__of__nat__commute,axiom,
! [X: nat,Y3: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y3 )
= ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_882_mult__of__nat__commute,axiom,
! [X: nat,Y3: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y3 )
= ( times_times_real @ Y3 @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_883_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_884_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_885_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_886_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_887_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_888_continuous__on__mult__const,axiom,
! [S: set_real,C: real] : ( topolo5044208981011980120l_real @ S @ ( times_times_real @ C ) ) ).
% continuous_on_mult_const
thf(fact_889_ex__less__of__nat__mult,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_890_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y5: real] :
? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_891_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_892_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_893_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_894_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_895_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_896_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_897_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_898_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_899_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_900_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_901_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_902_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_903_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_904_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_905_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_906_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_907_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_908_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_909_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_910_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_911_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_912_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_913_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_914_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_915_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_916_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_917_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_918_inner__real__def,axiom,
inner_5569486634795291694r_real = times_times_real ).
% inner_real_def
thf(fact_919_real__minus__mult__self__le,axiom,
! [U2: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U2 @ U2 ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_920_linorder__neqE__linordered__idom,axiom,
! [X: real,Y3: real] :
( ( X != Y3 )
=> ( ~ ( ord_less_real @ X @ Y3 )
=> ( ord_less_real @ Y3 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_921_linorder__neqE__linordered__idom,axiom,
! [X: int,Y3: int] :
( ( X != Y3 )
=> ( ~ ( ord_less_int @ X @ Y3 )
=> ( ord_less_int @ Y3 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_922_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_923_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_924_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_925_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_926_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_927_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_928_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_929_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_930_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_931_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_932_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_933_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_934_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_935_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_936_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_937_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_938_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_939_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_940_mult__mono,axiom,
! [A: real,B: real,C: real,D3: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_941_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_942_mult__mono,axiom,
! [A: int,B: int,C: int,D3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_943_mult__mono_H,axiom,
! [A: real,B: real,C: real,D3: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_944_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_945_mult__mono_H,axiom,
! [A: int,B: int,C: int,D3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_946_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_947_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_948_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_949_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_950_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_951_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_952_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_953_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_954_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_955_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_956_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_957_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_958_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_959_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_960_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_961_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_962_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_963_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_964_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_965_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_966_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_967_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_968_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_969_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_970_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_971_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_972_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_973_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_974_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_975_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_976_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_977_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_978_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_979_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_980_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_981_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_982_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_983_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_984_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_985_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_986_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_987_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_988_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_989_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_990_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_991_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_992_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_993_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_994_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_995_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_996_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_997_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_998_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_999_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_1000_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_1001_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_1002_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1003_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1004_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1005_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1006_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1007_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1008_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1009_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1010_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1011_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1012_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1013_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1014_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1015_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1016_mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1017_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1018_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1019_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1020_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1021_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1022_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1023_mult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1024_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1025_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1026_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1027_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1028_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1029_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1030_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1031_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1032_mult__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1033_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1034_mult__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1035_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1036_mult__left__less__imp__less,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1037_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1038_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1039_mult__strict__mono,axiom,
! [A: real,B: real,C: real,D3: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D3 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D3 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1040_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D3 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1041_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D3: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D3 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D3 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1042_mult__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1043_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1044_mult__right__less__imp__less,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1045_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1046_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1047_mult__strict__mono_H,axiom,
! [A: real,B: real,C: real,D3: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D3 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1048_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D3 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1049_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D3: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D3 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1050_mult__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1051_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1052_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1053_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1054_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1055_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1056_mult__left__le__imp__le,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1057_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1058_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1059_mult__right__le__imp__le,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1060_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1061_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1062_mult__le__less__imp__less,axiom,
! [A: real,B: real,C: real,D3: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D3 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D3 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1063_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D3 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D3 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1064_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D3 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1065_mult__less__le__imp__less,axiom,
! [A: real,B: real,C: real,D3: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D3 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1066_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D3 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1067_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D3: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D3 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1068_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X: real] :
( ( ( times_times_real @ A @ X )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_1069_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: real] :
( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_1070_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_1071_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X: real,Y3: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y3 ) )
= ( ( X = Y3 )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_1072_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X: real,B: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ B @ X ) )
= ( ( A = B )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_1073_Compl__iff,axiom,
! [C: real,A3: set_real] :
( ( member_real @ C @ ( uminus612125837232591019t_real @ A3 ) )
= ( ~ ( member_real @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_1074_Compl__iff,axiom,
! [C: nat,A3: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) )
= ( ~ ( member_nat @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_1075_ComplI,axiom,
! [C: real,A3: set_real] :
( ~ ( member_real @ C @ A3 )
=> ( member_real @ C @ ( uminus612125837232591019t_real @ A3 ) ) ) ).
% ComplI
thf(fact_1076_ComplI,axiom,
! [C: nat,A3: set_nat] :
( ~ ( member_nat @ C @ A3 )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) ) ) ).
% ComplI
thf(fact_1077_ComplD,axiom,
! [C: real,A3: set_real] :
( ( member_real @ C @ ( uminus612125837232591019t_real @ A3 ) )
=> ~ ( member_real @ C @ A3 ) ) ).
% ComplD
thf(fact_1078_ComplD,axiom,
! [C: nat,A3: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) )
=> ~ ( member_nat @ C @ A3 ) ) ).
% ComplD
thf(fact_1079_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: real,X: real,Y3: real] :
( ( A != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y3 ) )
=> ( X = Y3 ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_1080_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: real,A: real,B: real] :
( ( X != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ B @ X ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_1081_mult__le__cancel__iff2,axiom,
! [Z2: real,X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ ( times_times_real @ Z2 @ Y3 ) )
= ( ord_less_eq_real @ X @ Y3 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1082_mult__le__cancel__iff2,axiom,
! [Z2: int,X: int,Y3: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X ) @ ( times_times_int @ Z2 @ Y3 ) )
= ( ord_less_eq_int @ X @ Y3 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1083_mult__le__cancel__iff1,axiom,
! [Z2: real,X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y3 @ Z2 ) )
= ( ord_less_eq_real @ X @ Y3 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1084_mult__le__cancel__iff1,axiom,
! [Z2: int,X: int,Y3: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y3 @ Z2 ) )
= ( ord_less_eq_int @ X @ Y3 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1085_real__eq__0__iff__le__ge__0,axiom,
! [X: real] :
( ( X = zero_zero_real )
= ( ( ord_less_eq_real @ zero_zero_real @ X )
& ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_1086_mult__less__iff1,axiom,
! [Z2: real,X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y3 @ Z2 ) )
= ( ord_less_real @ X @ Y3 ) ) ) ).
% mult_less_iff1
thf(fact_1087_mult__less__iff1,axiom,
! [Z2: int,X: int,Y3: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y3 @ Z2 ) )
= ( ord_less_int @ X @ Y3 ) ) ) ).
% mult_less_iff1
thf(fact_1088_mult__delta__right,axiom,
! [B: $o,X: int,Y3: int] :
( ( B
=> ( ( times_times_int @ X @ ( if_int @ B @ Y3 @ zero_zero_int ) )
= ( times_times_int @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_int @ X @ ( if_int @ B @ Y3 @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_1089_mult__delta__right,axiom,
! [B: $o,X: real,Y3: real] :
( ( B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y3 @ zero_zero_real ) )
= ( times_times_real @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y3 @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_1090_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_1091_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1092_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1093_zless__nat__conj,axiom,
! [W2: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W2 @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_1094_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_1095_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_1096_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1097_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1098_nat__mono,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ).
% nat_mono
thf(fact_1099_nat__mono__iff,axiom,
! [Z2: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_1100_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_1101_zless__nat__eq__int__zless,axiom,
! [M: nat,Z2: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1102_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( ( nat2 @ W2 )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1103_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1104_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N2: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N2 ) )
=> ( P @ N2 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1105_nat__le__eq__zle,axiom,
! [W2: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_1106_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1107_nat__less__eq__zless,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_1108_segments__0,axiom,
( ( segments @ zero_zero_nat )
= bot_bot_set_set_real ) ).
% segments_0
thf(fact_1109_nat__ceiling__le__eq,axiom,
! [X: real,A: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% nat_ceiling_le_eq
thf(fact_1110_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_1111_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D5: real,E: real] :
( ( ord_less_real @ D5 @ E )
=> ( ( P @ D5 )
=> ( P @ E ) ) )
=> ( ! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_1112_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
= ( ? [N2: nat] :
( ( N2 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_1113_real__arch__invD,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [N3: nat] :
( ( N3 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E2 ) ) ) ).
% real_arch_invD
thf(fact_1114_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1115_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1116_real__inner__1__right,axiom,
! [X: real] :
( ( inner_5569486634795291694r_real @ X @ one_one_real )
= X ) ).
% real_inner_1_right
thf(fact_1117_real__inner__1__left,axiom,
! [X: real] :
( ( inner_5569486634795291694r_real @ one_one_real @ X )
= X ) ).
% real_inner_1_left
thf(fact_1118_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1119_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1120_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1121_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1122_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1123_real__of__nat__ge__one__iff,axiom,
! [N: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ one_one_nat @ N ) ) ).
% real_of_nat_ge_one_iff
thf(fact_1124_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1125_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1126_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1127_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q2: nat > $o] :
( ! [X4: nat > real] :
( ( P @ X4 )
=> ( P @ ( F @ X4 ) ) )
=> ( ! [X4: nat > real] :
( ( P @ X4 )
=> ! [I2: nat] :
( ( Q2 @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I2 ) )
& ( ord_less_eq_real @ ( X4 @ I2 ) @ one_one_real ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X3: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L3 @ X3 @ I3 ) @ one_one_nat )
& ! [X3: nat > real,I3: nat] :
( ( ( P @ X3 )
& ( Q2 @ I3 )
& ( ( X3 @ I3 )
= zero_zero_real ) )
=> ( ( L3 @ X3 @ I3 )
= zero_zero_nat ) )
& ! [X3: nat > real,I3: nat] :
( ( ( P @ X3 )
& ( Q2 @ I3 )
& ( ( X3 @ I3 )
= one_one_real ) )
=> ( ( L3 @ X3 @ I3 )
= one_one_nat ) )
& ! [X3: nat > real,I3: nat] :
( ( ( P @ X3 )
& ( Q2 @ I3 )
& ( ( L3 @ X3 @ I3 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X3 @ I3 ) @ ( F @ X3 @ I3 ) ) )
& ! [X3: nat > real,I3: nat] :
( ( ( P @ X3 )
& ( Q2 @ I3 )
& ( ( L3 @ X3 @ I3 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X3 @ I3 ) @ ( X3 @ I3 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1128_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_1129_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1130_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1131_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1132_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1133_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1134_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1135_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
= ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_1136_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1137_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1138_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1139_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1140_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1141_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1142_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_1143_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1144_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1145_Suc__inject,axiom,
! [X: nat,Y3: nat] :
( ( ( suc @ X )
= ( suc @ Y3 ) )
=> ( X = Y3 ) ) ).
% Suc_inject
thf(fact_1146_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1147_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1148_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1149_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1150_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1151_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1152_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1153_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1154_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1155_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1156_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1157_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1158_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1159_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1160_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1161_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1162_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1163_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1164_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1165_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1166_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M5: nat] :
( M7
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1167_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1168_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1169_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1170_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1171_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1172_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R4: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R4 @ X4 @ X4 )
=> ( ! [X4: nat,Y: nat,Z4: nat] :
( ( R4 @ X4 @ Y )
=> ( ( R4 @ Y @ Z4 )
=> ( R4 @ X4 @ Z4 ) ) )
=> ( ! [N3: nat] : ( R4 @ N3 @ ( suc @ N3 ) )
=> ( R4 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1173_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1174_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1175_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_1176_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1177_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1178_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1179_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1180_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X4: nat,Y: nat] :
( ( P @ X4 @ Y )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1181_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1182_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1183_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1184_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1185_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1186_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1187_zero__notin__Suc__image,axiom,
! [A3: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A3 ) ) ).
% zero_notin_Suc_image
thf(fact_1188_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1189_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1190_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1191_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1192_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1193_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1194_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1195_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1196_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1197_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1198_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1199_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1200_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1201_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1202_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1203_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1204_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1205_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1206_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1207_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1208_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1209_forall__pos__mono__1,axiom,
! [P: real > $o,E2: real] :
( ! [D5: real,E: real] :
( ( ord_less_real @ D5 @ E )
=> ( ( P @ D5 )
=> ( P @ E ) ) )
=> ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_1210_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_1211_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1212_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_1213_divide__real__def,axiom,
( divide_divide_real
= ( ^ [X2: real,Y2: real] : ( times_times_real @ X2 @ ( inverse_inverse_real @ Y2 ) ) ) ) ).
% divide_real_def
thf(fact_1214_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1215_real__of__int__div4,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).
% real_of_int_div4
thf(fact_1216_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1217_int__ops_I8_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_1218_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1219_verit__less__mono__div__int2,axiom,
! [A3: int,B5: int,N: int] :
( ( ord_less_eq_int @ A3 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A3 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1220_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_1221_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_1222_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_1223_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1224_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1225_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1226_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1227_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1228_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1229_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_1230_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1231_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1232_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1233_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1234_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1235_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1236_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1237_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1238_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1239_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1240_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1241_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1242_zdiv__mono1,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1243_zdiv__mono2,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1244_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_1245_zdiv__mono1__neg,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1246_zdiv__mono2__neg,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1247_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_1248_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1249_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1250_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_1251_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1252_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1253_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1254_less__eq__div__iff__mult__less__eq,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1255_div__nat__eqI,axiom,
! [N: nat,Q: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q ) ) ) ).
% div_nat_eqI
thf(fact_1256_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_1257_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_1258_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( divide_divide_int @ K @ L )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1259_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1260_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1261_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1262_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1263_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y3: int] :
( ( if_int @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y3: int] :
( ( if_int @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $true @ X @ Y3 )
= X ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y3: real] :
( ( if_real @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y3: real] :
( ( if_real @ $true @ X @ Y3 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_real @ zero_zero_real @ a ).
%------------------------------------------------------------------------------