TPTP Problem File: SLH0479^1.p
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%------------------------------------------------------------------------------
% 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_00790_034660__13263592_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1367 ( 447 unt; 99 typ; 0 def)
% Number of atoms : 4398 ( 992 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 12466 ( 372 ~; 122 |; 275 &;9586 @)
% ( 0 <=>;2111 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 803 ( 803 >; 0 *; 0 +; 0 <<)
% Number of symbols : 96 ( 93 usr; 12 con; 0-4 aty)
% Number of variables : 3922 ( 172 ^;3609 !; 141 ?;3922 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:32:32.316
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
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 (93)
thf(sy_c_Abstract__Topology__2_Oretraction_001t__Nat__Onat,type,
abstra7171991951520340845on_nat: set_nat > set_nat > ( nat > nat ) > $o ).
thf(sy_c_Abstract__Topology__2_Oretraction_001t__Real__Oreal,type,
abstra2606333701016485833n_real: set_real > set_real > ( real > real ) > $o ).
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
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_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_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_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__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
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_Lattices_Osup__class_Osup_001t__Int__Oint,type,
sup_sup_int: int > int > int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Real__Oreal,type,
sup_sup_real: real > real > real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Int__Oint_J,type,
sup_sup_set_int: set_int > set_int > set_int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Real__Oreal_J,type,
sup_sup_set_real: set_real > set_real > set_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_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_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_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
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_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
set_or6656581121297822940st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Real__Oreal,type,
set_or2392270231875598684t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
set_or1633881224788618240n_real: 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_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 ).
thf(sy_v_y____,type,
y: real ).
% Relevant facts (1264)
thf(fact_0_f_I2_J,axiom,
( ( f @ a )
!= b ) ).
% f(2)
thf(fact_1_True,axiom,
ord_less_real @ ( f @ a ) @ b ).
% True
thf(fact_2_f_I1_J,axiom,
( ( f @ zero_zero_real )
= zero_zero_real ) ).
% f(1)
thf(fact_3_calculation,axiom,
( a
= ( g @ ( f @ a ) ) ) ).
% calculation
thf(fact_4_assms_I3_J,axiom,
ord_less_real @ zero_zero_real @ a ).
% assms(3)
thf(fact_5_that,axiom,
member_real @ y @ ( set_or2392270231875598684t_real @ ( f @ a ) @ b ) ).
% that
thf(fact_6_g,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( g @ ( f @ X ) )
= X ) ) ).
% g
thf(fact_7__092_060open_0620_A_092_060le_062_Af_Aa_092_060close_062,axiom,
ord_less_eq_real @ zero_zero_real @ ( f @ a ) ).
% \<open>0 \<le> f a\<close>
thf(fact_8_minf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ~ ( ord_less_real @ T @ X2 ) ) ).
% minf(7)
thf(fact_9_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ~ ( ord_less_nat @ T @ X2 ) ) ).
% minf(7)
thf(fact_10_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ~ ( ord_less_int @ T @ X2 ) ) ).
% minf(7)
thf(fact_11_minf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ord_less_real @ X2 @ T ) ) ).
% minf(5)
thf(fact_12_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ord_less_nat @ X2 @ T ) ) ).
% minf(5)
thf(fact_13_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ord_less_int @ X2 @ T ) ) ).
% minf(5)
thf(fact_14_minf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_15_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_16_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_17_minf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_18_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_19_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_20_minf_I2_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_21_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_22_minf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z2 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z2 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_23_minf_I1_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_24_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_25_minf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z2 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z2 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_26_pinf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ord_less_real @ T @ X2 ) ) ).
% pinf(7)
thf(fact_27_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ord_less_nat @ T @ X2 ) ) ).
% pinf(7)
thf(fact_28_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ord_less_int @ T @ X2 ) ) ).
% pinf(7)
thf(fact_29_assms_I4_J,axiom,
ord_less_eq_real @ zero_zero_real @ b ).
% assms(4)
thf(fact_30_f__iff_I2_J,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( f @ X ) @ ( f @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% f_iff(2)
thf(fact_31_f__iff_I1_J,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( f @ X ) @ ( f @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% f_iff(1)
thf(fact_32_pinf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ T ) ) ).
% pinf(6)
thf(fact_33_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ T ) ) ).
% pinf(6)
thf(fact_34_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% pinf(6)
thf(fact_35_pinf_I8_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ord_less_eq_real @ T @ X2 ) ) ).
% pinf(8)
thf(fact_36_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ord_less_eq_nat @ T @ X2 ) ) ).
% pinf(8)
thf(fact_37_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ord_less_eq_int @ T @ X2 ) ) ).
% pinf(8)
thf(fact_38_minf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ord_less_eq_real @ X2 @ T ) ) ).
% minf(6)
thf(fact_39_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ord_less_eq_nat @ X2 @ T ) ) ).
% minf(6)
thf(fact_40_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ord_less_eq_int @ X2 @ T ) ) ).
% minf(6)
thf(fact_41_minf_I8_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ~ ( ord_less_eq_real @ T @ X2 ) ) ).
% minf(8)
thf(fact_42_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X2 ) ) ).
% minf(8)
thf(fact_43_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% minf(8)
thf(fact_44_pinf_I1_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_45_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_46_pinf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ Z2 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ Z2 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_47_pinf_I2_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_48_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_49_pinf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ Z2 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ Z2 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_50_pinf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_51_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_52_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_53_pinf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_54_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_55_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_56_pinf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ~ ( ord_less_real @ X2 @ T ) ) ).
% pinf(5)
thf(fact_57_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ~ ( ord_less_nat @ X2 @ T ) ) ).
% pinf(5)
thf(fact_58_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ~ ( ord_less_int @ X2 @ T ) ) ).
% pinf(5)
thf(fact_59_greaterThanAtMost__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or6656581121297822940st_int @ L @ U ) )
= ( ( ord_less_int @ L @ I )
& ( ord_less_eq_int @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_60_greaterThanAtMost__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or2392270231875598684t_real @ L @ U ) )
= ( ( ord_less_real @ L @ I )
& ( ord_less_eq_real @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_61_greaterThanAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_62_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_63_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_64_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_65_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_66_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_67_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_68_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_69_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_70_Ioc__inj,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or6656581121297822940st_int @ A @ B )
= ( set_or6656581121297822940st_int @ C @ D ) )
= ( ( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ D @ C ) )
| ( ( A = C )
& ( B = D ) ) ) ) ).
% Ioc_inj
thf(fact_71_Ioc__inj,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( set_or2392270231875598684t_real @ A @ B )
= ( set_or2392270231875598684t_real @ C @ D ) )
= ( ( ( ord_less_eq_real @ B @ A )
& ( ord_less_eq_real @ D @ C ) )
| ( ( A = C )
& ( B = D ) ) ) ) ).
% Ioc_inj
thf(fact_72_Ioc__inj,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or6659071591806873216st_nat @ A @ B )
= ( set_or6659071591806873216st_nat @ C @ D ) )
= ( ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ D @ C ) )
| ( ( A = C )
& ( B = D ) ) ) ) ).
% Ioc_inj
thf(fact_73_Ioc__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or6656581121297822940st_int @ A @ B ) @ ( set_or6656581121297822940st_int @ C @ D ) )
= ( ( ord_less_eq_int @ B @ A )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% Ioc_subset_iff
thf(fact_74_Ioc__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or2392270231875598684t_real @ A @ B ) @ ( set_or2392270231875598684t_real @ C @ D ) )
= ( ( ord_less_eq_real @ B @ A )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% Ioc_subset_iff
thf(fact_75_Ioc__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or6659071591806873216st_nat @ A @ B ) @ ( set_or6659071591806873216st_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% Ioc_subset_iff
thf(fact_76_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
| ( X4 = Y2 ) ) ) ) ).
% less_eq_real_def
thf(fact_77_bgauge__existence__lemma,axiom,
! [S: set_real,Q3: real > real > $o] :
( ( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ( Q3 @ D2 @ X4 ) ) ) )
= ( ! [X4: real] :
? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ( ( member_real @ X4 @ S )
=> ( Q3 @ D2 @ X4 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_78_bgauge__existence__lemma,axiom,
! [S: set_nat,Q3: real > nat > $o] :
( ( ! [X4: nat] :
( ( member_nat @ X4 @ S )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ( Q3 @ D2 @ X4 ) ) ) )
= ( ! [X4: nat] :
? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ( ( member_nat @ X4 @ S )
=> ( Q3 @ D2 @ X4 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_79_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_80_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_81_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_82_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_83_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_84_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_85_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_86_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_87_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_88_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_89_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_90_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_91_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_92_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_93_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_94_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_95_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_96_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_97_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_98_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_99_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_100_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_101_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_102_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_103_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_104_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_105_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_106_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_107_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_108_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_109_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_110_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_111_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_112_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_113_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_114_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_115_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_116_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_117_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_118_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_119_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_120_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_121_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_122_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_123_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_124_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_125_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_126_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_127_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_128_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_129_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_130_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_131_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_132_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_133_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_134_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_135_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_136_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_137_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_138_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_139_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_140_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_141_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_142_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_143_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_144_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_145_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_146_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_147_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_148_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_149_order__trans,axiom,
! [X: real,Y: real,Z4: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z4 )
=> ( ord_less_eq_real @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_150_order__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_eq_nat @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_151_order__trans,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z4 )
=> ( ord_less_eq_int @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_152_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_153_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_154_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_155_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_156_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_157_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_158_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_159_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_160_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_161_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_162_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_163_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_164_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
& ( ord_less_eq_real @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_165_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_166_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
& ( ord_less_eq_int @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_167_le__cases3,axiom,
! [X: real,Y: real,Z4: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z4 ) )
=> ( ( ( ord_less_eq_real @ X @ Z4 )
=> ~ ( ord_less_eq_real @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z4 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z4 )
=> ~ ( ord_less_eq_real @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z4 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_168_le__cases3,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z4 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z4 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_169_le__cases3,axiom,
! [X: int,Y: int,Z4: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z4 ) )
=> ( ( ( ord_less_eq_int @ X @ Z4 )
=> ~ ( ord_less_eq_int @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z4 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z4 )
=> ~ ( ord_less_eq_int @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z4 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_170_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_171_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_172_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_173_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_174_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_175_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_176_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_177_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_178_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_179_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_180_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_181_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z: real] :
( ( ord_less_real @ X @ Z )
& ( ord_less_real @ Z @ Y ) ) ) ).
% dense
thf(fact_182_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_183_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_184_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_185_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_186_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_187_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_188_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_189_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_190_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_191_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_192_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_193_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_194_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_195_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_196_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_197_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_198_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_199_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_200_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_201_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_202_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_203_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_204_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_205_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_206_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_207_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P4 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_208_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_209_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_210_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_211_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_212_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_213_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_214_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_215_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_216_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_217_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_218_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_219_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_220_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_221_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_222_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_223_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_224_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_225_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_226_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_227_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_228_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_229_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_230_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_231_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_232_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_233_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_234_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_235_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_236_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_237_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_238_order__less__trans,axiom,
! [X: real,Y: real,Z4: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z4 )
=> ( ord_less_real @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_239_order__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_240_order__less__trans,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z4 )
=> ( ord_less_int @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_241_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_242_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_243_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_244_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_245_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_246_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_247_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_248_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_249_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_250_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_251_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_252_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_253_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_254_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_255_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_256_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_257_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_258_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_259_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_260_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_261_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_262_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_263_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_264_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_265_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_266_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_267_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_268_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_269_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_270_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_271_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_272_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_273_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_274_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_275_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_276_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_277_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_278_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_279_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_280_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_281_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_282_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_283_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_284_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_285_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_286_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_287_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_288_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_289_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_290_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_291_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_292_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_293_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_294_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_295_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_296_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_297_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_298_complete__real,axiom,
! [S2: set_real] :
( ? [X2: real] : ( member_real @ X2 @ S2 )
=> ( ? [Z2: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ? [Y3: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ord_less_eq_real @ X2 @ Y3 ) )
& ! [Z2: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ord_less_eq_real @ Y3 @ Z2 ) ) ) ) ) ).
% complete_real
thf(fact_299_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_300_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_301_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_302_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_303_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_304_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_305_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_306_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_307_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_308_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_309_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_310_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_311_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_312_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_313_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_314_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_315_dense__ge,axiom,
! [Z4: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z4 ) ) ).
% dense_ge
thf(fact_316_dense__le,axiom,
! [Y: real,Z4: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y @ Z4 ) ) ).
% dense_le
thf(fact_317_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
& ~ ( ord_less_eq_real @ Y2 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_318_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_319_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
& ~ ( ord_less_eq_int @ Y2 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_320_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_321_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_322_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_323_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_324_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_325_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_326_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_327_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_328_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_329_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_330_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_331_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_332_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_333_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_334_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_335_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_336_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_337_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_338_dense__ge__bounded,axiom,
! [Z4: real,X: real,Y: real] :
( ( ord_less_real @ Z4 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z4 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z4 ) ) ) ).
% dense_ge_bounded
thf(fact_339_dense__le__bounded,axiom,
! [X: real,Y: real,Z4: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z4 ) ) )
=> ( ord_less_eq_real @ Y @ Z4 ) ) ) ).
% dense_le_bounded
thf(fact_340_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_real @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_341_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_342_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_int @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_343_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_344_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_345_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_346_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_347_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_348_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_349_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_350_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_351_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_352_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_353_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_354_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_355_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_356_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_357_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_358_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_359_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_360_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_361_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
| ( X4 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_362_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
| ( X4 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_363_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
| ( X4 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_364_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
& ( X4 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_365_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
& ( X4 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_366_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
& ( X4 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_367_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_368_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_369_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_370_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_371_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_372_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_373_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_374_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_375_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_376_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_377_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_378_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_379_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_380_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_381_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_382_order__le__less__trans,axiom,
! [X: real,Y: real,Z4: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z4 )
=> ( ord_less_real @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_383_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_384_order__le__less__trans,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z4 )
=> ( ord_less_int @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_385_order__less__le__trans,axiom,
! [X: real,Y: real,Z4: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z4 )
=> ( ord_less_real @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_386_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_387_order__less__le__trans,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z4 )
=> ( ord_less_int @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_388_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_389_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_390_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_391_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_392_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_393_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_394_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_395_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_396_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_397_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_398_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_399_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_400_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_401_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_402_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_403_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_404_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_405_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_406_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_407_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_408_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_409_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_410_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_411_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_412_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_413_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_414_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_415_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_416_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_417_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_418_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_419_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_420_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_421_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 )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_422_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 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_423_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 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_424_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_425_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_426_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_427_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_428_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_429_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_430_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_431_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_432_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_433_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_434_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_435_subsetI,axiom,
! [A2: set_real,B4: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ X3 @ B4 ) )
=> ( ord_less_eq_set_real @ A2 @ B4 ) ) ).
% subsetI
thf(fact_436_subsetI,axiom,
! [A2: set_nat,B4: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B4 ) )
=> ( ord_less_eq_set_nat @ A2 @ B4 ) ) ).
% subsetI
thf(fact_437_seq__mono__lemma,axiom,
! [M2: nat,D: nat > real,E2: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_real @ ( D @ N3 ) @ ( E2 @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_real @ ( E2 @ N3 ) @ ( E2 @ M2 ) ) )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_real @ ( D @ N4 ) @ ( E2 @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_438_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_439_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_440_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_441_verit__comp__simplify1_I3_J,axiom,
! [B5: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B5 @ A5 ) )
= ( ord_less_real @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_442_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_443_verit__comp__simplify1_I3_J,axiom,
! [B5: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
= ( ord_less_int @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_444_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
& ~ ( ord_less_eq_real @ Y2 @ X4 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_445_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B )
& ! [X2: real] :
( ( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_real @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D3: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_446_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_447_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X2: int] :
( ( ( ord_less_eq_int @ A @ X2 )
& ( ord_less_int @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D3: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_448_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_449_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_450_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_451_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1633881224788618240n_real @ A @ B ) @ ( set_or2392270231875598684t_real @ C @ D ) )
= ( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_452_sm,axiom,
monoto4017252874604999745l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ ord_less_real @ ord_less_real @ f ).
% sm
thf(fact_453_atLeast__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( set_ord_atLeast_real @ X )
= ( set_ord_atLeast_real @ Y ) )
= ( X = Y ) ) ).
% atLeast_eq_iff
thf(fact_454_sm__gx,axiom,
monoto4017252874604999745l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ ord_less_real @ ord_less_real @ g ).
% sm_gx
thf(fact_455_greaterThanLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_456_greaterThanLessThan__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or5832277885323065728an_int @ L @ U ) )
= ( ( ord_less_int @ L @ I )
& ( ord_less_int @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_457_greaterThanLessThan__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or1633881224788618240n_real @ L @ U ) )
= ( ( ord_less_real @ L @ I )
& ( ord_less_real @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_458_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_459_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_460_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_461_atLeast__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ X ) @ ( set_ord_atLeast_nat @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_462_atLeast__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_atLeast_int @ X ) @ ( set_ord_atLeast_int @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_463_atLeast__subset__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_ord_atLeast_real @ X ) @ ( set_ord_atLeast_real @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_464_fim,axiom,
( ( image_real_real @ f @ ( set_ord_atLeast_real @ zero_zero_real ) )
= ( set_ord_atLeast_real @ zero_zero_real ) ) ).
% fim
thf(fact_465_cont,axiom,
topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ f ).
% cont
thf(fact_466_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M3: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_467_psubsetD,axiom,
! [A2: set_real,B4: set_real,C: real] :
( ( ord_less_set_real @ A2 @ B4 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_468_psubsetD,axiom,
! [A2: set_nat,B4: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B4 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_469_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1633881224788618240n_real @ A @ B ) @ ( set_or1633881224788618240n_real @ C @ D ) )
= ( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_470_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_471_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_472_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_473_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_474_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_475_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_476_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_477_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_478_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_479_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_480_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [T2: real] :
( ( member_real @ T2 @ A6 )
=> ( member_real @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_481_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_482_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [X4: real] :
( ( member_real @ X4 @ A6 )
=> ( member_real @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_483_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( member_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_484_subsetD,axiom,
! [A2: set_real,B4: set_real,C: real] :
( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B4 ) ) ) ).
% subsetD
thf(fact_485_subsetD,axiom,
! [A2: set_nat,B4: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_486_in__mono,axiom,
! [A2: set_real,B4: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B4 ) ) ) ).
% in_mono
thf(fact_487_in__mono,axiom,
! [A2: set_nat,B4: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_488_ord_Omono__on__subset,axiom,
! [A2: set_real,Less_eq: real > real > $o,F: real > real,B4: set_real] :
( ( monoto4017252874604999745l_real @ A2 @ Less_eq @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_real @ B4 @ A2 )
=> ( monoto4017252874604999745l_real @ B4 @ Less_eq @ ord_less_eq_real @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_489_mono__on__subset,axiom,
! [A2: set_real,F: real > real,B4: set_real] :
( ( monoto4017252874604999745l_real @ A2 @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_real @ B4 @ A2 )
=> ( monoto4017252874604999745l_real @ B4 @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ) ).
% mono_on_subset
thf(fact_490_mono__on__subset,axiom,
! [A2: set_real,F: real > nat,B4: set_real] :
( ( monotone_on_real_nat @ A2 @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_real @ B4 @ A2 )
=> ( monotone_on_real_nat @ B4 @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_491_mono__on__subset,axiom,
! [A2: set_real,F: real > int,B4: set_real] :
( ( monotone_on_real_int @ A2 @ ord_less_eq_real @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_set_real @ B4 @ A2 )
=> ( monotone_on_real_int @ B4 @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ) ).
% mono_on_subset
thf(fact_492_mono__on__subset,axiom,
! [A2: set_nat,F: nat > real,B4: set_nat] :
( ( monotone_on_nat_real @ A2 @ ord_less_eq_nat @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( monotone_on_nat_real @ B4 @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ) ).
% mono_on_subset
thf(fact_493_mono__on__subset,axiom,
! [A2: set_nat,F: nat > nat,B4: set_nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( monotone_on_nat_nat @ B4 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_494_mono__on__subset,axiom,
! [A2: set_nat,F: nat > int,B4: set_nat] :
( ( monotone_on_nat_int @ A2 @ ord_less_eq_nat @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( monotone_on_nat_int @ B4 @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ) ).
% mono_on_subset
thf(fact_495_mono__on__subset,axiom,
! [A2: set_int,F: int > real,B4: set_int] :
( ( monotone_on_int_real @ A2 @ ord_less_eq_int @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_int @ B4 @ A2 )
=> ( monotone_on_int_real @ B4 @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ) ).
% mono_on_subset
thf(fact_496_mono__on__subset,axiom,
! [A2: set_int,F: int > nat,B4: set_int] :
( ( monotone_on_int_nat @ A2 @ ord_less_eq_int @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_int @ B4 @ A2 )
=> ( monotone_on_int_nat @ B4 @ ord_less_eq_int @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_497_mono__on__subset,axiom,
! [A2: set_int,F: int > int,B4: set_int] :
( ( monotone_on_int_int @ A2 @ ord_less_eq_int @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_set_int @ B4 @ A2 )
=> ( monotone_on_int_int @ B4 @ ord_less_eq_int @ ord_less_eq_int @ F ) ) ) ).
% mono_on_subset
thf(fact_498_mono__on__greaterD,axiom,
! [A2: set_real,G: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ A2 @ ord_less_eq_real @ ord_less_eq_real @ G )
=> ( ( member_real @ X @ A2 )
=> ( ( member_real @ Y @ A2 )
=> ( ( ord_less_real @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_real @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_499_mono__on__greaterD,axiom,
! [A2: set_real,G: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ A2 @ ord_less_eq_real @ ord_less_eq_nat @ G )
=> ( ( member_real @ X @ A2 )
=> ( ( member_real @ Y @ A2 )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_real @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_500_mono__on__greaterD,axiom,
! [A2: set_real,G: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ A2 @ ord_less_eq_real @ ord_less_eq_int @ G )
=> ( ( member_real @ X @ A2 )
=> ( ( member_real @ Y @ A2 )
=> ( ( ord_less_int @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_real @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_501_mono__on__greaterD,axiom,
! [A2: set_nat,G: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ A2 @ ord_less_eq_nat @ ord_less_eq_real @ G )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ord_less_real @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_nat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_502_mono__on__greaterD,axiom,
! [A2: set_nat,G: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ G )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_nat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_503_mono__on__greaterD,axiom,
! [A2: set_nat,G: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ A2 @ ord_less_eq_nat @ ord_less_eq_int @ G )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ord_less_int @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_nat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_504_mono__on__greaterD,axiom,
! [A2: set_int,G: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ A2 @ ord_less_eq_int @ ord_less_eq_real @ G )
=> ( ( member_int @ X @ A2 )
=> ( ( member_int @ Y @ A2 )
=> ( ( ord_less_real @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_int @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_505_mono__on__greaterD,axiom,
! [A2: set_int,G: int > nat,X: int,Y: int] :
( ( monotone_on_int_nat @ A2 @ ord_less_eq_int @ ord_less_eq_nat @ G )
=> ( ( member_int @ X @ A2 )
=> ( ( member_int @ Y @ A2 )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_int @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_506_mono__on__greaterD,axiom,
! [A2: set_int,G: int > int,X: int,Y: int] :
( ( monotone_on_int_int @ A2 @ ord_less_eq_int @ ord_less_eq_int @ G )
=> ( ( member_int @ X @ A2 )
=> ( ( member_int @ Y @ A2 )
=> ( ( ord_less_int @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_int @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_507_strict__mono__on__leD,axiom,
! [A2: set_real,F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ A2 @ ord_less_real @ ord_less_real @ F )
=> ( ( member_real @ X @ A2 )
=> ( ( member_real @ Y @ A2 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_508_strict__mono__on__leD,axiom,
! [A2: set_real,F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ A2 @ ord_less_real @ ord_less_nat @ F )
=> ( ( member_real @ X @ A2 )
=> ( ( member_real @ Y @ A2 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_509_strict__mono__on__leD,axiom,
! [A2: set_real,F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ A2 @ ord_less_real @ ord_less_int @ F )
=> ( ( member_real @ X @ A2 )
=> ( ( member_real @ Y @ A2 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_510_strict__mono__on__leD,axiom,
! [A2: set_nat,F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ A2 @ ord_less_nat @ ord_less_real @ F )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_511_strict__mono__on__leD,axiom,
! [A2: set_nat,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_512_strict__mono__on__leD,axiom,
! [A2: set_nat,F: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ A2 @ ord_less_nat @ ord_less_int @ F )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_513_strict__mono__on__leD,axiom,
! [A2: set_int,F: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ A2 @ ord_less_int @ ord_less_real @ F )
=> ( ( member_int @ X @ A2 )
=> ( ( member_int @ Y @ A2 )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_514_strict__mono__on__leD,axiom,
! [A2: set_int,F: int > nat,X: int,Y: int] :
( ( monotone_on_int_nat @ A2 @ ord_less_int @ ord_less_nat @ F )
=> ( ( member_int @ X @ A2 )
=> ( ( member_int @ Y @ A2 )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_515_strict__mono__on__leD,axiom,
! [A2: set_int,F: int > int,X: int,Y: int] :
( ( monotone_on_int_int @ A2 @ ord_less_int @ ord_less_int @ F )
=> ( ( member_int @ X @ A2 )
=> ( ( member_int @ Y @ A2 )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_516_strict__mono__on__imp__mono__on,axiom,
! [A2: set_real,F: real > real] :
( ( monoto4017252874604999745l_real @ A2 @ ord_less_real @ ord_less_real @ F )
=> ( monoto4017252874604999745l_real @ A2 @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_517_strict__mono__on__imp__mono__on,axiom,
! [A2: set_real,F: real > nat] :
( ( monotone_on_real_nat @ A2 @ ord_less_real @ ord_less_nat @ F )
=> ( monotone_on_real_nat @ A2 @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_518_strict__mono__on__imp__mono__on,axiom,
! [A2: set_real,F: real > int] :
( ( monotone_on_real_int @ A2 @ ord_less_real @ ord_less_int @ F )
=> ( monotone_on_real_int @ A2 @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_519_strict__mono__on__imp__mono__on,axiom,
! [A2: set_nat,F: nat > real] :
( ( monotone_on_nat_real @ A2 @ ord_less_nat @ ord_less_real @ F )
=> ( monotone_on_nat_real @ A2 @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_520_strict__mono__on__imp__mono__on,axiom,
! [A2: set_nat,F: nat > nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_521_strict__mono__on__imp__mono__on,axiom,
! [A2: set_nat,F: nat > int] :
( ( monotone_on_nat_int @ A2 @ ord_less_nat @ ord_less_int @ F )
=> ( monotone_on_nat_int @ A2 @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_522_strict__mono__on__imp__mono__on,axiom,
! [A2: set_int,F: int > real] :
( ( monotone_on_int_real @ A2 @ ord_less_int @ ord_less_real @ F )
=> ( monotone_on_int_real @ A2 @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_523_strict__mono__on__imp__mono__on,axiom,
! [A2: set_int,F: int > nat] :
( ( monotone_on_int_nat @ A2 @ ord_less_int @ ord_less_nat @ F )
=> ( monotone_on_int_nat @ A2 @ ord_less_eq_int @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_524_strict__mono__on__imp__mono__on,axiom,
! [A2: set_int,F: int > int] :
( ( monotone_on_int_int @ A2 @ ord_less_int @ ord_less_int @ F )
=> ( monotone_on_int_int @ A2 @ ord_less_eq_int @ ord_less_eq_int @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_525_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_526_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_527_monotone__on__subset,axiom,
! [A2: set_real,Orda: real > real > $o,Ordb: real > real > $o,F: real > real,B4: set_real] :
( ( monoto4017252874604999745l_real @ A2 @ Orda @ Ordb @ F )
=> ( ( ord_less_eq_set_real @ B4 @ A2 )
=> ( monoto4017252874604999745l_real @ B4 @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_528_image__eqI,axiom,
! [B: real,F: real > real,X: real,A2: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ B @ ( image_real_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_529_image__eqI,axiom,
! [B: nat,F: real > nat,X: real,A2: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A2 )
=> ( member_nat @ B @ ( image_real_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_530_image__eqI,axiom,
! [B: int,F: nat > int,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_int @ B @ ( image_nat_int @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_531_image__eqI,axiom,
! [B: real,F: nat > real,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_real @ B @ ( image_nat_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_532_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_533_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_534_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_535_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_536_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_537_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_538_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_539_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_540_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_541_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_542_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_543_imageI,axiom,
! [X: real,A2: set_real,F: real > real] :
( ( member_real @ X @ A2 )
=> ( member_real @ ( F @ X ) @ ( image_real_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_544_imageI,axiom,
! [X: real,A2: set_real,F: real > nat] :
( ( member_real @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_real_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_545_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > int] :
( ( member_nat @ X @ A2 )
=> ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ A2 ) ) ) ).
% imageI
thf(fact_546_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > real] :
( ( member_nat @ X @ A2 )
=> ( member_real @ ( F @ X ) @ ( image_nat_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_547_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_548_image__iff,axiom,
! [Z4: int,F: nat > int,A2: set_nat] :
( ( member_int @ Z4 @ ( image_nat_int @ F @ A2 ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( Z4
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_549_image__iff,axiom,
! [Z4: real,F: real > real,A2: set_real] :
( ( member_real @ Z4 @ ( image_real_real @ F @ A2 ) )
= ( ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( Z4
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_550_image__iff,axiom,
! [Z4: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z4 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( Z4
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_551_bex__imageD,axiom,
! [F: real > real,A2: set_real,P: real > $o] :
( ? [X2: real] :
( ( member_real @ X2 @ ( image_real_real @ F @ A2 ) )
& ( P @ X2 ) )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_552_bex__imageD,axiom,
! [F: nat > int,A2: set_nat,P: int > $o] :
( ? [X2: int] :
( ( member_int @ X2 @ ( image_nat_int @ F @ A2 ) )
& ( P @ X2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_553_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_554_image__cong,axiom,
! [M3: set_real,N5: set_real,F: real > real,G: real > real] :
( ( M3 = N5 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ N5 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_real_real @ F @ M3 )
= ( image_real_real @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_555_image__cong,axiom,
! [M3: set_nat,N5: set_nat,F: nat > int,G: nat > int] :
( ( M3 = N5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_int @ F @ M3 )
= ( image_nat_int @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_556_image__cong,axiom,
! [M3: set_nat,N5: set_nat,F: nat > nat,G: nat > nat] :
( ( M3 = N5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M3 )
= ( image_nat_nat @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_557_ball__imageD,axiom,
! [F: real > real,A2: set_real,P: real > $o] :
( ! [X3: real] :
( ( member_real @ X3 @ ( image_real_real @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_558_ball__imageD,axiom,
! [F: nat > int,A2: set_nat,P: int > $o] :
( ! [X3: int] :
( ( member_int @ X3 @ ( image_nat_int @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_559_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_560_rev__image__eqI,axiom,
! [X: real,A2: set_real,B: real,F: real > real] :
( ( member_real @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_real_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_561_rev__image__eqI,axiom,
! [X: real,A2: set_real,B: nat,F: real > nat] :
( ( member_real @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_real_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_562_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: int,F: nat > int] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_nat_int @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_563_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: real,F: nat > real] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_nat_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_564_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_565_invertible__fixpoint__property,axiom,
! [T3: set_int,I: int > nat,S2: set_nat,R: nat > int,G: int > int] :
( ( topolo2181401217840723324nt_nat @ T3 @ I )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ I @ T3 ) @ S2 )
=> ( ( topolo1179557035430618492at_int @ S2 @ R )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ R @ S2 ) @ T3 )
=> ( ! [Y3: int] :
( ( member_int @ Y3 @ T3 )
=> ( ( R @ ( I @ Y3 ) )
= Y3 ) )
=> ( ! [F2: nat > nat] :
( ( topolo1182047505939668768at_nat @ S2 @ F2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ S2 ) @ S2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ S2 )
& ( ( F2 @ X2 )
= X2 ) ) ) )
=> ( ( topolo2178910747331673048nt_int @ T3 @ G )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ G @ T3 ) @ T3 )
=> ~ ! [Y3: int] :
( ( member_int @ Y3 @ T3 )
=> ( ( G @ Y3 )
!= Y3 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_566_invertible__fixpoint__property,axiom,
! [T3: set_nat,I: nat > int,S2: set_int,R: int > nat,G: nat > nat] :
( ( topolo1179557035430618492at_int @ T3 @ I )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ I @ T3 ) @ S2 )
=> ( ( topolo2181401217840723324nt_nat @ S2 @ R )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ R @ S2 ) @ T3 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ T3 )
=> ( ( R @ ( I @ Y3 ) )
= Y3 ) )
=> ( ! [F2: int > int] :
( ( topolo2178910747331673048nt_int @ S2 @ F2 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F2 @ S2 ) @ S2 )
=> ? [X2: int] :
( ( member_int @ X2 @ S2 )
& ( ( F2 @ X2 )
= X2 ) ) ) )
=> ( ( topolo1182047505939668768at_nat @ T3 @ G )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ T3 ) @ T3 )
=> ~ ! [Y3: nat] :
( ( member_nat @ Y3 @ T3 )
=> ( ( G @ Y3 )
!= Y3 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_567_invertible__fixpoint__property,axiom,
! [T3: set_nat,I: nat > nat,S2: set_nat,R: nat > nat,G: nat > nat] :
( ( topolo1182047505939668768at_nat @ T3 @ I )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ I @ T3 ) @ S2 )
=> ( ( topolo1182047505939668768at_nat @ S2 @ R )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ S2 ) @ T3 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ T3 )
=> ( ( R @ ( I @ Y3 ) )
= Y3 ) )
=> ( ! [F2: nat > nat] :
( ( topolo1182047505939668768at_nat @ S2 @ F2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ S2 ) @ S2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ S2 )
& ( ( F2 @ X2 )
= X2 ) ) ) )
=> ( ( topolo1182047505939668768at_nat @ T3 @ G )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ T3 ) @ T3 )
=> ~ ! [Y3: nat] :
( ( member_nat @ Y3 @ T3 )
=> ( ( G @ Y3 )
!= Y3 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_568_invertible__fixpoint__property,axiom,
! [T3: set_real,I: real > nat,S2: set_nat,R: nat > real,G: real > real] :
( ( topolo2287203362918339196al_nat @ T3 @ I )
=> ( ( ord_less_eq_set_nat @ ( image_real_nat @ I @ T3 ) @ S2 )
=> ( ( topolo6943266826644216316t_real @ S2 @ R )
=> ( ( ord_less_eq_set_real @ ( image_nat_real @ R @ S2 ) @ T3 )
=> ( ! [Y3: real] :
( ( member_real @ Y3 @ T3 )
=> ( ( R @ ( I @ Y3 ) )
= Y3 ) )
=> ( ! [F2: nat > nat] :
( ( topolo1182047505939668768at_nat @ S2 @ F2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ S2 ) @ S2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ S2 )
& ( ( F2 @ X2 )
= X2 ) ) ) )
=> ( ( topolo5044208981011980120l_real @ T3 @ G )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ G @ T3 ) @ T3 )
=> ~ ! [Y3: real] :
( ( member_real @ Y3 @ T3 )
=> ( ( G @ Y3 )
!= Y3 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_569_invertible__fixpoint__property,axiom,
! [T3: set_nat,I: nat > real,S2: set_real,R: real > nat,G: nat > nat] :
( ( topolo6943266826644216316t_real @ T3 @ I )
=> ( ( ord_less_eq_set_real @ ( image_nat_real @ I @ T3 ) @ S2 )
=> ( ( topolo2287203362918339196al_nat @ S2 @ R )
=> ( ( ord_less_eq_set_nat @ ( image_real_nat @ R @ S2 ) @ T3 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ T3 )
=> ( ( R @ ( I @ Y3 ) )
= Y3 ) )
=> ( ! [F2: real > real] :
( ( topolo5044208981011980120l_real @ S2 @ F2 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ F2 @ S2 ) @ S2 )
=> ? [X2: real] :
( ( member_real @ X2 @ S2 )
& ( ( F2 @ X2 )
= X2 ) ) ) )
=> ( ( topolo1182047505939668768at_nat @ T3 @ G )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ T3 ) @ T3 )
=> ~ ! [Y3: nat] :
( ( member_nat @ Y3 @ T3 )
=> ( ( G @ Y3 )
!= Y3 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_570_invertible__fixpoint__property,axiom,
! [T3: set_real,I: real > real,S2: set_real,R: real > real,G: real > real] :
( ( topolo5044208981011980120l_real @ T3 @ I )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ I @ T3 ) @ S2 )
=> ( ( topolo5044208981011980120l_real @ S2 @ R )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ R @ S2 ) @ T3 )
=> ( ! [Y3: real] :
( ( member_real @ Y3 @ T3 )
=> ( ( R @ ( I @ Y3 ) )
= Y3 ) )
=> ( ! [F2: real > real] :
( ( topolo5044208981011980120l_real @ S2 @ F2 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ F2 @ S2 ) @ S2 )
=> ? [X2: real] :
( ( member_real @ X2 @ S2 )
& ( ( F2 @ X2 )
= X2 ) ) ) )
=> ( ( topolo5044208981011980120l_real @ T3 @ G )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ G @ T3 ) @ T3 )
=> ~ ! [Y3: real] :
( ( member_real @ Y3 @ T3 )
=> ( ( G @ Y3 )
!= Y3 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_571_image__mono,axiom,
! [A2: set_real,B4: set_real,F: real > real] :
( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A2 ) @ ( image_real_real @ F @ B4 ) ) ) ).
% image_mono
thf(fact_572_image__mono,axiom,
! [A2: set_nat,B4: set_nat,F: nat > int] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ ( image_nat_int @ F @ B4 ) ) ) ).
% image_mono
thf(fact_573_image__mono,axiom,
! [A2: set_nat,B4: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B4 ) ) ) ).
% image_mono
thf(fact_574_image__subsetI,axiom,
! [A2: set_real,F: real > real,B4: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_575_image__subsetI,axiom,
! [A2: set_real,F: real > nat,B4: set_nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_real_nat @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_576_image__subsetI,axiom,
! [A2: set_nat,F: nat > int,B4: set_int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_int @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_577_image__subsetI,axiom,
! [A2: set_nat,F: nat > real,B4: set_real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_real @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_578_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B4: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_579_subset__imageE,axiom,
! [B4: set_real,F: real > real,A2: set_real] :
( ( ord_less_eq_set_real @ B4 @ ( image_real_real @ F @ A2 ) )
=> ~ ! [C3: set_real] :
( ( ord_less_eq_set_real @ C3 @ A2 )
=> ( B4
!= ( image_real_real @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_580_subset__imageE,axiom,
! [B4: set_int,F: nat > int,A2: set_nat] :
( ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B4
!= ( image_nat_int @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_581_subset__imageE,axiom,
! [B4: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B4
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_582_image__subset__iff,axiom,
! [F: nat > int,A2: set_nat,B4: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ B4 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_int @ ( F @ X4 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_583_image__subset__iff,axiom,
! [F: real > real,A2: set_real,B4: set_real] :
( ( ord_less_eq_set_real @ ( image_real_real @ F @ A2 ) @ B4 )
= ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( member_real @ ( F @ X4 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_584_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B4 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_585_subset__image__iff,axiom,
! [B4: set_real,F: real > real,A2: set_real] :
( ( ord_less_eq_set_real @ B4 @ ( image_real_real @ F @ A2 ) )
= ( ? [AA: set_real] :
( ( ord_less_eq_set_real @ AA @ A2 )
& ( B4
= ( image_real_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_586_subset__image__iff,axiom,
! [B4: set_int,F: nat > int,A2: set_nat] :
( ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B4
= ( image_nat_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_587_subset__image__iff,axiom,
! [B4: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B4
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_588_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 ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ ( F @ A ) ) @ G ) ) ) ) ) ).
% strict_mono_continuous_invD
thf(fact_589_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_590_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_591_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_592_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_593_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_594_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_595_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_596_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_597_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_598_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_599_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_600_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_601_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_602_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_603_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_604_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_605_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_606_monotone__on__def,axiom,
( monoto4017252874604999745l_real
= ( ^ [A6: set_real,Orda2: real > real > $o,Ordb2: real > real > $o,F3: real > real] :
! [X4: real] :
( ( member_real @ X4 @ A6 )
=> ! [Y2: real] :
( ( member_real @ Y2 @ A6 )
=> ( ( Orda2 @ X4 @ Y2 )
=> ( Ordb2 @ ( F3 @ X4 ) @ ( F3 @ Y2 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_607_monotone__onI,axiom,
! [A2: set_real,Orda: real > real > $o,Ordb: real > real > $o,F: real > real] :
( ! [X3: real,Y3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_real @ Y3 @ A2 )
=> ( ( Orda @ X3 @ Y3 )
=> ( Ordb @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A2 @ Orda @ Ordb @ F ) ) ).
% monotone_onI
thf(fact_608_monotone__onD,axiom,
! [A2: set_real,Orda: real > real > $o,Ordb: real > real > $o,F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ A2 @ Orda @ Ordb @ F )
=> ( ( member_real @ X @ A2 )
=> ( ( member_real @ Y @ A2 )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% monotone_onD
thf(fact_609_mono__onI,axiom,
! [A2: set_real,F: real > real] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( ord_less_eq_real @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A2 @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_610_mono__onI,axiom,
! [A2: set_real,F: real > nat] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( ord_less_eq_real @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A2 @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_611_mono__onI,axiom,
! [A2: set_real,F: real > int] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( ord_less_eq_real @ R2 @ S3 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_int @ A2 @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_612_mono__onI,axiom,
! [A2: set_nat,F: nat > real] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_eq_nat @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_real @ A2 @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_613_mono__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_eq_nat @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_614_mono__onI,axiom,
! [A2: set_nat,F: nat > int] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_eq_nat @ R2 @ S3 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_int @ A2 @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_615_mono__onI,axiom,
! [A2: set_int,F: int > real] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A2 )
=> ( ( member_int @ S3 @ A2 )
=> ( ( ord_less_eq_int @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_real @ A2 @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_616_mono__onI,axiom,
! [A2: set_int,F: int > nat] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A2 )
=> ( ( member_int @ S3 @ A2 )
=> ( ( ord_less_eq_int @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_nat @ A2 @ ord_less_eq_int @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_617_mono__onI,axiom,
! [A2: set_int,F: int > int] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A2 )
=> ( ( member_int @ S3 @ A2 )
=> ( ( ord_less_eq_int @ R2 @ S3 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_int @ A2 @ ord_less_eq_int @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_618_mono__onD,axiom,
! [A2: set_real,F: real > real,R: real,S: real] :
( ( monoto4017252874604999745l_real @ A2 @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( member_real @ R @ A2 )
=> ( ( member_real @ S @ A2 )
=> ( ( ord_less_eq_real @ R @ S )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_619_mono__onD,axiom,
! [A2: set_real,F: real > nat,R: real,S: real] :
( ( monotone_on_real_nat @ A2 @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( member_real @ R @ A2 )
=> ( ( member_real @ S @ A2 )
=> ( ( ord_less_eq_real @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_620_mono__onD,axiom,
! [A2: set_real,F: real > int,R: real,S: real] :
( ( monotone_on_real_int @ A2 @ ord_less_eq_real @ ord_less_eq_int @ F )
=> ( ( member_real @ R @ A2 )
=> ( ( member_real @ S @ A2 )
=> ( ( ord_less_eq_real @ R @ S )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_621_mono__onD,axiom,
! [A2: set_nat,F: nat > real,R: nat,S: nat] :
( ( monotone_on_nat_real @ A2 @ ord_less_eq_nat @ ord_less_eq_real @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S @ A2 )
=> ( ( ord_less_eq_nat @ R @ S )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_622_mono__onD,axiom,
! [A2: set_nat,F: nat > nat,R: nat,S: nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S @ A2 )
=> ( ( ord_less_eq_nat @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_623_mono__onD,axiom,
! [A2: set_nat,F: nat > int,R: nat,S: nat] :
( ( monotone_on_nat_int @ A2 @ ord_less_eq_nat @ ord_less_eq_int @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S @ A2 )
=> ( ( ord_less_eq_nat @ R @ S )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_624_mono__onD,axiom,
! [A2: set_int,F: int > real,R: int,S: int] :
( ( monotone_on_int_real @ A2 @ ord_less_eq_int @ ord_less_eq_real @ F )
=> ( ( member_int @ R @ A2 )
=> ( ( member_int @ S @ A2 )
=> ( ( ord_less_eq_int @ R @ S )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_625_mono__onD,axiom,
! [A2: set_int,F: int > nat,R: int,S: int] :
( ( monotone_on_int_nat @ A2 @ ord_less_eq_int @ ord_less_eq_nat @ F )
=> ( ( member_int @ R @ A2 )
=> ( ( member_int @ S @ A2 )
=> ( ( ord_less_eq_int @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_626_mono__onD,axiom,
! [A2: set_int,F: int > int,R: int,S: int] :
( ( monotone_on_int_int @ A2 @ ord_less_eq_int @ ord_less_eq_int @ F )
=> ( ( member_int @ R @ A2 )
=> ( ( member_int @ S @ A2 )
=> ( ( ord_less_eq_int @ R @ S )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_627_ord_Omono__on__def,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > real] :
( ( monotone_on_nat_real @ A2 @ Less_eq @ ord_less_eq_real @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_628_ord_Omono__on__def,axiom,
! [A2: set_real,Less_eq: real > real > $o,F: real > real] :
( ( monoto4017252874604999745l_real @ A2 @ Less_eq @ ord_less_eq_real @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A2 )
& ( member_real @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_629_ord_Omono__on__def,axiom,
! [A2: set_real,Less_eq: real > real > $o,F: real > nat] :
( ( monotone_on_real_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A2 )
& ( member_real @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_630_ord_Omono__on__def,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_631_ord_Omono__on__def,axiom,
! [A2: set_real,Less_eq: real > real > $o,F: real > int] :
( ( monotone_on_real_int @ A2 @ Less_eq @ ord_less_eq_int @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A2 )
& ( member_real @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_int @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_632_ord_Omono__on__def,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > int] :
( ( monotone_on_nat_int @ A2 @ Less_eq @ ord_less_eq_int @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_int @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_633_ord_Omono__onI,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > real] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_real @ A2 @ Less_eq @ ord_less_eq_real @ F ) ) ).
% ord.mono_onI
thf(fact_634_ord_Omono__onI,axiom,
! [A2: set_real,Less_eq: real > real > $o,F: real > real] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A2 @ Less_eq @ ord_less_eq_real @ F ) ) ).
% ord.mono_onI
thf(fact_635_ord_Omono__onI,axiom,
! [A2: set_real,Less_eq: real > real > $o,F: real > nat] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_636_ord_Omono__onI,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_637_ord_Omono__onI,axiom,
! [A2: set_real,Less_eq: real > real > $o,F: real > int] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_int @ A2 @ Less_eq @ ord_less_eq_int @ F ) ) ).
% ord.mono_onI
thf(fact_638_ord_Omono__onI,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > int] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_int @ A2 @ Less_eq @ ord_less_eq_int @ F ) ) ).
% ord.mono_onI
thf(fact_639_ord_Omono__onD,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > real,R: nat,S: nat] :
( ( monotone_on_nat_real @ A2 @ Less_eq @ ord_less_eq_real @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S @ A2 )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_640_ord_Omono__onD,axiom,
! [A2: set_real,Less_eq: real > real > $o,F: real > real,R: real,S: real] :
( ( monoto4017252874604999745l_real @ A2 @ Less_eq @ ord_less_eq_real @ F )
=> ( ( member_real @ R @ A2 )
=> ( ( member_real @ S @ A2 )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_641_ord_Omono__onD,axiom,
! [A2: set_real,Less_eq: real > real > $o,F: real > nat,R: real,S: real] :
( ( monotone_on_real_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_real @ R @ A2 )
=> ( ( member_real @ S @ A2 )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_642_ord_Omono__onD,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > nat,R: nat,S: nat] :
( ( monotone_on_nat_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S @ A2 )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_643_ord_Omono__onD,axiom,
! [A2: set_real,Less_eq: real > real > $o,F: real > int,R: real,S: real] :
( ( monotone_on_real_int @ A2 @ Less_eq @ ord_less_eq_int @ F )
=> ( ( member_real @ R @ A2 )
=> ( ( member_real @ S @ A2 )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_644_ord_Omono__onD,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > int,R: nat,S: nat] :
( ( monotone_on_nat_int @ A2 @ Less_eq @ ord_less_eq_int @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S @ A2 )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_645_ord_Ostrict__mono__onD,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > real,R: nat,S: nat] :
( ( monotone_on_nat_real @ A2 @ Less @ ord_less_real @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S @ A2 )
=> ( ( Less @ R @ S )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_646_ord_Ostrict__mono__onD,axiom,
! [A2: set_real,Less: real > real > $o,F: real > real,R: real,S: real] :
( ( monoto4017252874604999745l_real @ A2 @ Less @ ord_less_real @ F )
=> ( ( member_real @ R @ A2 )
=> ( ( member_real @ S @ A2 )
=> ( ( Less @ R @ S )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_647_ord_Ostrict__mono__onD,axiom,
! [A2: set_real,Less: real > real > $o,F: real > nat,R: real,S: real] :
( ( monotone_on_real_nat @ A2 @ Less @ ord_less_nat @ F )
=> ( ( member_real @ R @ A2 )
=> ( ( member_real @ S @ A2 )
=> ( ( Less @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_648_ord_Ostrict__mono__onD,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > nat,R: nat,S: nat] :
( ( monotone_on_nat_nat @ A2 @ Less @ ord_less_nat @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S @ A2 )
=> ( ( Less @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_649_ord_Ostrict__mono__onD,axiom,
! [A2: set_real,Less: real > real > $o,F: real > int,R: real,S: real] :
( ( monotone_on_real_int @ A2 @ Less @ ord_less_int @ F )
=> ( ( member_real @ R @ A2 )
=> ( ( member_real @ S @ A2 )
=> ( ( Less @ R @ S )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_650_ord_Ostrict__mono__onD,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > int,R: nat,S: nat] :
( ( monotone_on_nat_int @ A2 @ Less @ ord_less_int @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S @ A2 )
=> ( ( Less @ R @ S )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_651_ord_Ostrict__mono__onI,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > real] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_real @ A2 @ Less @ ord_less_real @ F ) ) ).
% ord.strict_mono_onI
thf(fact_652_ord_Ostrict__mono__onI,axiom,
! [A2: set_real,Less: real > real > $o,F: real > real] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A2 @ Less @ ord_less_real @ F ) ) ).
% ord.strict_mono_onI
thf(fact_653_ord_Ostrict__mono__onI,axiom,
! [A2: set_real,Less: real > real > $o,F: real > nat] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A2 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_654_ord_Ostrict__mono__onI,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_655_ord_Ostrict__mono__onI,axiom,
! [A2: set_real,Less: real > real > $o,F: real > int] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_int @ A2 @ Less @ ord_less_int @ F ) ) ).
% ord.strict_mono_onI
thf(fact_656_ord_Ostrict__mono__onI,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > int] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_int @ A2 @ Less @ ord_less_int @ F ) ) ).
% ord.strict_mono_onI
thf(fact_657_ord_Ostrict__mono__on__def,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > real] :
( ( monotone_on_nat_real @ A2 @ Less @ ord_less_real @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_658_ord_Ostrict__mono__on__def,axiom,
! [A2: set_real,Less: real > real > $o,F: real > real] :
( ( monoto4017252874604999745l_real @ A2 @ Less @ ord_less_real @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A2 )
& ( member_real @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_659_ord_Ostrict__mono__on__def,axiom,
! [A2: set_real,Less: real > real > $o,F: real > nat] :
( ( monotone_on_real_nat @ A2 @ Less @ ord_less_nat @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A2 )
& ( member_real @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_660_ord_Ostrict__mono__on__def,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A2 @ Less @ ord_less_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_661_ord_Ostrict__mono__on__def,axiom,
! [A2: set_real,Less: real > real > $o,F: real > int] :
( ( monotone_on_real_int @ A2 @ Less @ ord_less_int @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A2 )
& ( member_real @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_int @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_662_ord_Ostrict__mono__on__def,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > int] :
( ( monotone_on_nat_int @ A2 @ Less @ ord_less_int @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_int @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_663_strict__mono__onD,axiom,
! [A2: set_real,F: real > real,R: real,S: real] :
( ( monoto4017252874604999745l_real @ A2 @ ord_less_real @ ord_less_real @ F )
=> ( ( member_real @ R @ A2 )
=> ( ( member_real @ S @ A2 )
=> ( ( ord_less_real @ R @ S )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_664_strict__mono__onD,axiom,
! [A2: set_real,F: real > nat,R: real,S: real] :
( ( monotone_on_real_nat @ A2 @ ord_less_real @ ord_less_nat @ F )
=> ( ( member_real @ R @ A2 )
=> ( ( member_real @ S @ A2 )
=> ( ( ord_less_real @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_665_strict__mono__onD,axiom,
! [A2: set_real,F: real > int,R: real,S: real] :
( ( monotone_on_real_int @ A2 @ ord_less_real @ ord_less_int @ F )
=> ( ( member_real @ R @ A2 )
=> ( ( member_real @ S @ A2 )
=> ( ( ord_less_real @ R @ S )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_666_strict__mono__onD,axiom,
! [A2: set_nat,F: nat > real,R: nat,S: nat] :
( ( monotone_on_nat_real @ A2 @ ord_less_nat @ ord_less_real @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S @ A2 )
=> ( ( ord_less_nat @ R @ S )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_667_strict__mono__onD,axiom,
! [A2: set_nat,F: nat > nat,R: nat,S: nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S @ A2 )
=> ( ( ord_less_nat @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_668_strict__mono__onD,axiom,
! [A2: set_nat,F: nat > int,R: nat,S: nat] :
( ( monotone_on_nat_int @ A2 @ ord_less_nat @ ord_less_int @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S @ A2 )
=> ( ( ord_less_nat @ R @ S )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_669_strict__mono__onD,axiom,
! [A2: set_int,F: int > real,R: int,S: int] :
( ( monotone_on_int_real @ A2 @ ord_less_int @ ord_less_real @ F )
=> ( ( member_int @ R @ A2 )
=> ( ( member_int @ S @ A2 )
=> ( ( ord_less_int @ R @ S )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_670_strict__mono__onD,axiom,
! [A2: set_int,F: int > nat,R: int,S: int] :
( ( monotone_on_int_nat @ A2 @ ord_less_int @ ord_less_nat @ F )
=> ( ( member_int @ R @ A2 )
=> ( ( member_int @ S @ A2 )
=> ( ( ord_less_int @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_671_strict__mono__onD,axiom,
! [A2: set_int,F: int > int,R: int,S: int] :
( ( monotone_on_int_int @ A2 @ ord_less_int @ ord_less_int @ F )
=> ( ( member_int @ R @ A2 )
=> ( ( member_int @ S @ A2 )
=> ( ( ord_less_int @ R @ S )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_672_strict__mono__onI,axiom,
! [A2: set_real,F: real > real] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( ord_less_real @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A2 @ ord_less_real @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_673_strict__mono__onI,axiom,
! [A2: set_real,F: real > nat] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( ord_less_real @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A2 @ ord_less_real @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_674_strict__mono__onI,axiom,
! [A2: set_real,F: real > int] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( ord_less_real @ R2 @ S3 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_int @ A2 @ ord_less_real @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_675_strict__mono__onI,axiom,
! [A2: set_nat,F: nat > real] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_nat @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_real @ A2 @ ord_less_nat @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_676_strict__mono__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_nat @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_677_strict__mono__onI,axiom,
! [A2: set_nat,F: nat > int] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_nat @ R2 @ S3 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_int @ A2 @ ord_less_nat @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_678_strict__mono__onI,axiom,
! [A2: set_int,F: int > real] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A2 )
=> ( ( member_int @ S3 @ A2 )
=> ( ( ord_less_int @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_real @ A2 @ ord_less_int @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_679_strict__mono__onI,axiom,
! [A2: set_int,F: int > nat] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A2 )
=> ( ( member_int @ S3 @ A2 )
=> ( ( ord_less_int @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_nat @ A2 @ ord_less_int @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_680_strict__mono__onI,axiom,
! [A2: set_int,F: int > int] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A2 )
=> ( ( member_int @ S3 @ A2 )
=> ( ( ord_less_int @ R2 @ S3 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_int @ A2 @ ord_less_int @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_681_strict__mono__on__eqD,axiom,
! [A2: set_real,F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ A2 @ ord_less_real @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_real @ X @ A2 )
=> ( ( member_real @ Y @ A2 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_682_strict__mono__on__eqD,axiom,
! [A2: set_real,F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ A2 @ ord_less_real @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_real @ X @ A2 )
=> ( ( member_real @ Y @ A2 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_683_strict__mono__on__eqD,axiom,
! [A2: set_real,F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ A2 @ ord_less_real @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_real @ X @ A2 )
=> ( ( member_real @ Y @ A2 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_684_strict__mono__on__eqD,axiom,
! [A2: set_nat,F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ A2 @ ord_less_nat @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_685_strict__mono__on__eqD,axiom,
! [A2: set_nat,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_686_strict__mono__on__eqD,axiom,
! [A2: set_nat,F: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ A2 @ ord_less_nat @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_687_strict__mono__on__eqD,axiom,
! [A2: set_int,F: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ A2 @ ord_less_int @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_int @ X @ A2 )
=> ( ( member_int @ Y @ A2 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_688_strict__mono__on__eqD,axiom,
! [A2: set_int,F: int > nat,X: int,Y: int] :
( ( monotone_on_int_nat @ A2 @ ord_less_int @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_int @ X @ A2 )
=> ( ( member_int @ Y @ A2 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_689_strict__mono__on__eqD,axiom,
! [A2: set_int,F: int > int,X: int,Y: int] :
( ( monotone_on_int_int @ A2 @ ord_less_int @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_int @ X @ A2 )
=> ( ( member_int @ Y @ A2 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_690_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_691_all__subset__image,axiom,
! [F: real > real,A2: set_real,P: set_real > $o] :
( ( ! [B6: set_real] :
( ( ord_less_eq_set_real @ B6 @ ( image_real_real @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_real] :
( ( ord_less_eq_set_real @ B6 @ A2 )
=> ( P @ ( image_real_real @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_692_all__subset__image,axiom,
! [F: nat > int,A2: set_nat,P: set_int > $o] :
( ( ! [B6: set_int] :
( ( ord_less_eq_set_int @ B6 @ ( image_nat_int @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( P @ ( image_nat_int @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_693_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_694_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_695_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_696_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_697_antimono__fun__sum__integral__diff_Ointro,axiom,
! [F: real > real] :
( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X3 ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ F )
=> ( integr4865894440751020556l_diff @ F ) ) ) ) ).
% antimono_fun_sum_integral_diff.intro
thf(fact_698_antimono__fun__sum__integral__diff__def,axiom,
( integr4865894440751020556l_diff
= ( ^ [F3: real > real] :
( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F3 @ Y2 ) @ ( F3 @ X4 ) ) ) )
& ! [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F3 @ X4 ) ) )
& ( topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ F3 ) ) ) ) ).
% antimono_fun_sum_integral_diff_def
thf(fact_699_retraction,axiom,
( abstra7171991951520340845on_nat
= ( ^ [S5: set_nat,T4: set_nat,R3: nat > nat] :
( ( ord_less_eq_set_nat @ T4 @ S5 )
& ( topolo1182047505939668768at_nat @ S5 @ R3 )
& ( ( image_nat_nat @ R3 @ S5 )
= T4 )
& ! [X4: nat] :
( ( member_nat @ X4 @ T4 )
=> ( ( R3 @ X4 )
= X4 ) ) ) ) ) ).
% retraction
thf(fact_700_retraction,axiom,
( abstra2606333701016485833n_real
= ( ^ [S5: set_real,T4: set_real,R3: real > real] :
( ( ord_less_eq_set_real @ T4 @ S5 )
& ( topolo5044208981011980120l_real @ S5 @ R3 )
& ( ( image_real_real @ R3 @ S5 )
= T4 )
& ! [X4: real] :
( ( member_real @ X4 @ T4 )
=> ( ( R3 @ X4 )
= X4 ) ) ) ) ) ).
% retraction
thf(fact_701_retractionE,axiom,
! [S2: set_nat,T3: set_nat,R: nat > nat] :
( ( abstra7171991951520340845on_nat @ S2 @ T3 @ R )
=> ~ ( ( T3
= ( image_nat_nat @ R @ S2 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ S2 ) @ S2 )
=> ( ( topolo1182047505939668768at_nat @ S2 @ R )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ( ( R @ ( R @ X2 ) )
= ( R @ X2 ) ) ) ) ) ) ) ).
% retractionE
thf(fact_702_retractionE,axiom,
! [S2: set_real,T3: set_real,R: real > real] :
( ( abstra2606333701016485833n_real @ S2 @ T3 @ R )
=> ~ ( ( T3
= ( image_real_real @ R @ S2 ) )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ R @ S2 ) @ S2 )
=> ( ( topolo5044208981011980120l_real @ S2 @ R )
=> ~ ! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ( R @ ( R @ X2 ) )
= ( R @ X2 ) ) ) ) ) ) ) ).
% retractionE
thf(fact_703_retraction__def,axiom,
( abstra7171991951520340845on_nat
= ( ^ [S5: set_nat,T4: set_nat,R3: nat > nat] :
( ( ord_less_eq_set_nat @ T4 @ S5 )
& ( topolo1182047505939668768at_nat @ S5 @ R3 )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ R3 @ S5 ) @ T4 )
& ! [X4: nat] :
( ( member_nat @ X4 @ T4 )
=> ( ( R3 @ X4 )
= X4 ) ) ) ) ) ).
% retraction_def
thf(fact_704_retraction__def,axiom,
( abstra2606333701016485833n_real
= ( ^ [S5: set_real,T4: set_real,R3: real > real] :
( ( ord_less_eq_set_real @ T4 @ S5 )
& ( topolo5044208981011980120l_real @ S5 @ R3 )
& ( ord_less_eq_set_real @ ( image_real_real @ R3 @ S5 ) @ T4 )
& ! [X4: real] :
( ( member_real @ X4 @ T4 )
=> ( ( R3 @ X4 )
= X4 ) ) ) ) ) ).
% retraction_def
thf(fact_705_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_706_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_707_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_708_atLeastAtMost__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_eq_real @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_709_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_710_atLeastAtMost__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_eq_int @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_711_atLeastatMost__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_712_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_713_atLeastatMost__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_714_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_715_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_716_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_717_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_718_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_719_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_720_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_721_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_722_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_723_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_724_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_725_retraction__idempotent,axiom,
! [S2: set_real,T3: set_real,R: real > real,X: real] :
( ( abstra2606333701016485833n_real @ S2 @ T3 @ R )
=> ( ( member_real @ X @ S2 )
=> ( ( R @ ( R @ X ) )
= ( R @ X ) ) ) ) ).
% retraction_idempotent
thf(fact_726_retraction__idempotent,axiom,
! [S2: set_nat,T3: set_nat,R: nat > nat,X: nat] :
( ( abstra7171991951520340845on_nat @ S2 @ T3 @ R )
=> ( ( member_nat @ X @ S2 )
=> ( ( R @ ( R @ X ) )
= ( R @ X ) ) ) ) ).
% retraction_idempotent
thf(fact_727_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_728_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_729_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_730_IVT2_H,axiom,
! [F: real > nat,B: real,Y: nat,A: real] :
( ( ord_less_eq_nat @ ( F @ B ) @ Y )
=> ( ( ord_less_eq_nat @ Y @ ( F @ A ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B )
& ( ( F @ X3 )
= Y ) ) ) ) ) ) ).
% IVT2'
thf(fact_731_IVT2_H,axiom,
! [F: real > int,B: real,Y: int,A: real] :
( ( ord_less_eq_int @ ( F @ B ) @ Y )
=> ( ( ord_less_eq_int @ Y @ ( F @ A ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2284712892409288920al_int @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B )
& ( ( F @ X3 )
= Y ) ) ) ) ) ) ).
% IVT2'
thf(fact_732_IVT2_H,axiom,
! [F: real > real,B: real,Y: real,A: real] :
( ( ord_less_eq_real @ ( F @ B ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( F @ A ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B )
& ( ( F @ X3 )
= Y ) ) ) ) ) ) ).
% IVT2'
thf(fact_733_IVT_H,axiom,
! [F: real > nat,A: real,Y: nat,B: real] :
( ( ord_less_eq_nat @ ( F @ A ) @ Y )
=> ( ( ord_less_eq_nat @ Y @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B )
& ( ( F @ X3 )
= Y ) ) ) ) ) ) ).
% IVT'
thf(fact_734_IVT_H,axiom,
! [F: real > int,A: real,Y: int,B: real] :
( ( ord_less_eq_int @ ( F @ A ) @ Y )
=> ( ( ord_less_eq_int @ Y @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2284712892409288920al_int @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B )
& ( ( F @ X3 )
= Y ) ) ) ) ) ) ).
% IVT'
thf(fact_735_IVT_H,axiom,
! [F: real > real,A: real,Y: real,B: real] :
( ( ord_less_eq_real @ ( F @ A ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B )
& ( ( F @ X3 )
= Y ) ) ) ) ) ) ).
% IVT'
thf(fact_736_antimono__fun__sum__integral__diff_Odec,axiom,
! [F: real > real,X: real,Y: real] :
( ( integr4865894440751020556l_diff @ F )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% antimono_fun_sum_integral_diff.dec
thf(fact_737_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_738_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_739_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_740_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_741_atLeastatMost__psubset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D )
& ( ( ord_less_real @ C @ A )
| ( ord_less_real @ B @ D ) ) ) )
& ( ord_less_eq_real @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_742_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_743_atLeastatMost__psubset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
= ( ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D )
& ( ( ord_less_int @ C @ A )
| ( ord_less_int @ B @ D ) ) ) )
& ( ord_less_eq_int @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_744_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_745_continuous__on__cong,axiom,
! [S: set_real,T: set_real,F: real > real,G: real > real] :
( ( S = T )
=> ( ! [X3: real] :
( ( member_real @ X3 @ T )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( topolo5044208981011980120l_real @ S @ F )
= ( topolo5044208981011980120l_real @ T @ G ) ) ) ) ).
% continuous_on_cong
thf(fact_746_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1633881224788618240n_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_747_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or2392270231875598684t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_748_idempotent__imp__retraction,axiom,
! [S2: set_nat,F: nat > nat] :
( ( topolo1182047505939668768at_nat @ S2 @ F )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ S2 ) @ S2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ( ( F @ ( F @ X3 ) )
= ( F @ X3 ) ) )
=> ( abstra7171991951520340845on_nat @ S2 @ ( image_nat_nat @ F @ S2 ) @ F ) ) ) ) ).
% idempotent_imp_retraction
thf(fact_749_idempotent__imp__retraction,axiom,
! [S2: set_real,F: real > real] :
( ( topolo5044208981011980120l_real @ S2 @ F )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ F @ S2 ) @ S2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ( F @ ( F @ X3 ) )
= ( F @ X3 ) ) )
=> ( abstra2606333701016485833n_real @ S2 @ ( image_real_real @ F @ S2 ) @ F ) ) ) ) ).
% idempotent_imp_retraction
thf(fact_750_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,D4: real] :
( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ C2 @ D4 ) )
& ( ord_less_eq_real @ C2 @ D4 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_751_continuous__ge__on__Ioo,axiom,
! [C: real,D: real,G: real > real,A: real,X: real] :
( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ C @ D ) @ G )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ C @ D ) )
=> ( ord_less_eq_real @ A @ ( G @ X3 ) ) )
=> ( ( ord_less_real @ C @ D )
=> ( ( member_real @ X @ ( set_or1222579329274155063t_real @ C @ D ) )
=> ( ord_less_eq_real @ A @ ( G @ X ) ) ) ) ) ) ).
% continuous_ge_on_Ioo
thf(fact_752_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: 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 @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_753_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( P @ M ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_754_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( P @ M ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_755_antimono__fun__sum__integral__diff_Osum__integral__diff__series__antimono,axiom,
! [F: real > real,M2: nat,N: nat] :
( ( integr4865894440751020556l_diff @ F )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_real @ ( integr3166334062659923703series @ F @ N ) @ ( integr3166334062659923703series @ F @ M2 ) ) ) ) ).
% antimono_fun_sum_integral_diff.sum_integral_diff_series_antimono
thf(fact_756_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_757_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_758_greaterThanLessThan__eq__iff,axiom,
! [R: real,S: real,T: real,U: real] :
( ( ( set_or1633881224788618240n_real @ R @ S )
= ( set_or1633881224788618240n_real @ T @ U ) )
= ( ( ( ord_less_eq_real @ S @ R )
& ( ord_less_eq_real @ U @ T ) )
| ( ( R = T )
& ( S = U ) ) ) ) ).
% greaterThanLessThan_eq_iff
thf(fact_759_ivl__disj__un__two_I5_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_int @ L @ M2 )
=> ( ( ord_less_eq_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or5832277885323065728an_int @ L @ M2 ) @ ( set_or1266510415728281911st_int @ M2 @ U ) )
= ( set_or6656581121297822940st_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_760_ivl__disj__un__two_I5_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ M2 ) @ ( set_or1269000886237332187st_nat @ M2 @ U ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_761_ivl__disj__un__two_I5_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_real @ L @ M2 )
=> ( ( ord_less_eq_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or1633881224788618240n_real @ L @ M2 ) @ ( set_or1222579329274155063t_real @ M2 @ U ) )
= ( set_or2392270231875598684t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_762_image__mult__atLeastAtMost,axiom,
! [D: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ( image_real_real @ ( times_times_real @ D ) @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ ( times_times_real @ D @ A ) @ ( times_times_real @ D @ B ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_763_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_764_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_765_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_766_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_767_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_768_mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_769_mult__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_770_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_771_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_772_Un__iff,axiom,
! [C: real,A2: set_real,B4: set_real] :
( ( member_real @ C @ ( sup_sup_set_real @ A2 @ B4 ) )
= ( ( member_real @ C @ A2 )
| ( member_real @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_773_Un__iff,axiom,
! [C: nat,A2: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B4 ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_774_UnCI,axiom,
! [C: real,B4: set_real,A2: set_real] :
( ( ~ ( member_real @ C @ B4 )
=> ( member_real @ C @ A2 ) )
=> ( member_real @ C @ ( sup_sup_set_real @ A2 @ B4 ) ) ) ).
% UnCI
thf(fact_775_UnCI,axiom,
! [C: nat,B4: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B4 )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B4 ) ) ) ).
% UnCI
thf(fact_776_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_777_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M2 @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_778_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_779_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_780_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_781_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_782_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_783_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_784_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_785_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_786_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_787_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_788_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_789_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_790_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_791_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_792_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_793_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_794_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_795_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_796_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_797_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_798_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_799_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_800_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_801_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_802_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_803_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_804_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_805_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_806_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_807_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_808_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_809_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_810_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_811_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_812_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_813_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_814_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_815_UnI2,axiom,
! [C: real,B4: set_real,A2: set_real] :
( ( member_real @ C @ B4 )
=> ( member_real @ C @ ( sup_sup_set_real @ A2 @ B4 ) ) ) ).
% UnI2
thf(fact_816_UnI2,axiom,
! [C: nat,B4: set_nat,A2: set_nat] :
( ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B4 ) ) ) ).
% UnI2
thf(fact_817_UnI1,axiom,
! [C: real,A2: set_real,B4: set_real] :
( ( member_real @ C @ A2 )
=> ( member_real @ C @ ( sup_sup_set_real @ A2 @ B4 ) ) ) ).
% UnI1
thf(fact_818_UnI1,axiom,
! [C: nat,A2: set_nat,B4: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B4 ) ) ) ).
% UnI1
thf(fact_819_UnE,axiom,
! [C: real,A2: set_real,B4: set_real] :
( ( member_real @ C @ ( sup_sup_set_real @ A2 @ B4 ) )
=> ( ~ ( member_real @ C @ A2 )
=> ( member_real @ C @ B4 ) ) ) ).
% UnE
thf(fact_820_UnE,axiom,
! [C: nat,A2: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B4 ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B4 ) ) ) ).
% UnE
thf(fact_821_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_822_image__Un,axiom,
! [F: real > real,A2: set_real,B4: set_real] :
( ( image_real_real @ F @ ( sup_sup_set_real @ A2 @ B4 ) )
= ( sup_sup_set_real @ ( image_real_real @ F @ A2 ) @ ( image_real_real @ F @ B4 ) ) ) ).
% image_Un
thf(fact_823_image__Un,axiom,
! [F: nat > int,A2: set_nat,B4: set_nat] :
( ( image_nat_int @ F @ ( sup_sup_set_nat @ A2 @ B4 ) )
= ( sup_sup_set_int @ ( image_nat_int @ F @ A2 ) @ ( image_nat_int @ F @ B4 ) ) ) ).
% image_Un
thf(fact_824_image__Un,axiom,
! [F: nat > nat,A2: set_nat,B4: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A2 @ B4 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B4 ) ) ) ).
% image_Un
thf(fact_825_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_826_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_827_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_828_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_829_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_830_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_831_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_832_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_833_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_834_continuous__on__mult__const,axiom,
! [S: set_real,C: real] : ( topolo5044208981011980120l_real @ S @ ( times_times_real @ C ) ) ).
% continuous_on_mult_const
thf(fact_835_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_836_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 )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_837_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_838_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_839_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_840_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_841_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_842_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_843_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_844_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_845_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_846_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_847_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_848_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_849_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_850_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_851_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_852_ivl__disj__un__two__touch_I4_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_eq_real @ L @ M2 )
=> ( ( ord_less_eq_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M2 ) @ ( set_or1222579329274155063t_real @ M2 @ U ) )
= ( set_or1222579329274155063t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_853_ivl__disj__un__two__touch_I4_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or1269000886237332187st_nat @ M2 @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_854_ivl__disj__un__two__touch_I4_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_eq_int @ L @ M2 )
=> ( ( ord_less_eq_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M2 ) @ ( set_or1266510415728281911st_int @ M2 @ U ) )
= ( set_or1266510415728281911st_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_855_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_856_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_857_ivl__disj__un__two_I6_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_eq_int @ L @ M2 )
=> ( ( ord_less_eq_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or6656581121297822940st_int @ L @ M2 ) @ ( set_or6656581121297822940st_int @ M2 @ U ) )
= ( set_or6656581121297822940st_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_858_ivl__disj__un__two_I6_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_eq_real @ L @ M2 )
=> ( ( ord_less_eq_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or2392270231875598684t_real @ L @ M2 ) @ ( set_or2392270231875598684t_real @ M2 @ U ) )
= ( set_or2392270231875598684t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_859_ivl__disj__un__two_I6_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M2 ) @ ( set_or6659071591806873216st_nat @ M2 @ U ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_860_ivl__disj__un__two_I8_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_eq_int @ L @ M2 )
=> ( ( ord_less_eq_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M2 ) @ ( set_or6656581121297822940st_int @ M2 @ U ) )
= ( set_or1266510415728281911st_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_861_ivl__disj__un__two_I8_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_eq_real @ L @ M2 )
=> ( ( ord_less_eq_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M2 ) @ ( set_or2392270231875598684t_real @ M2 @ U ) )
= ( set_or1222579329274155063t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_862_ivl__disj__un__two_I8_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or6659071591806873216st_nat @ M2 @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_863_ivl__disj__un__two__touch_I3_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_int @ L @ M2 )
=> ( ( ord_less_eq_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or6656581121297822940st_int @ L @ M2 ) @ ( set_or1266510415728281911st_int @ M2 @ U ) )
= ( set_or6656581121297822940st_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_864_ivl__disj__un__two__touch_I3_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_real @ L @ M2 )
=> ( ( ord_less_eq_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or2392270231875598684t_real @ L @ M2 ) @ ( set_or1222579329274155063t_real @ M2 @ U ) )
= ( set_or2392270231875598684t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_865_ivl__disj__un__two__touch_I3_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M2 ) @ ( set_or1269000886237332187st_nat @ M2 @ U ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_866_ivl__disj__un__two_I2_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_eq_int @ L @ M2 )
=> ( ( ord_less_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or6656581121297822940st_int @ L @ M2 ) @ ( set_or5832277885323065728an_int @ M2 @ U ) )
= ( set_or5832277885323065728an_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_867_ivl__disj__un__two_I2_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M2 ) @ ( set_or5834768355832116004an_nat @ M2 @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_868_ivl__disj__un__two_I2_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_eq_real @ L @ M2 )
=> ( ( ord_less_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or2392270231875598684t_real @ L @ M2 ) @ ( set_or1633881224788618240n_real @ M2 @ U ) )
= ( set_or1633881224788618240n_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_869_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_870_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_871_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_872_sup_Obounded__iff,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B @ C ) @ A )
= ( ( ord_less_eq_real @ B @ A )
& ( ord_less_eq_real @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_873_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_874_sup_Obounded__iff,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_875_le__sup__iff,axiom,
! [X: real,Y: real,Z4: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ X @ Y ) @ Z4 )
= ( ( ord_less_eq_real @ X @ Z4 )
& ( ord_less_eq_real @ Y @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_876_le__sup__iff,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z4 )
= ( ( ord_less_eq_nat @ X @ Z4 )
& ( ord_less_eq_nat @ Y @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_877_le__sup__iff,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z4 )
= ( ( ord_less_eq_int @ X @ Z4 )
& ( ord_less_eq_int @ Y @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_878_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_879_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_880_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_881_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_882_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_883_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_884_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_885_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_886_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_887_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_888_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_889_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_890_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_891_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_892_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_893_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_894_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_895_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_896_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_897_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_898_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_899_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_900_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_901_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_902_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_903_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_904_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_905_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_906_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_907_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_908_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_909_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_910_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_911_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_912_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_913_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_914_inf__sup__ord_I4_J,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_915_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_916_inf__sup__ord_I4_J,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_917_inf__sup__ord_I3_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sup_sup_real @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_918_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_919_inf__sup__ord_I3_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_920_le__supE,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_real @ A @ X )
=> ~ ( ord_less_eq_real @ B @ X ) ) ) ).
% le_supE
thf(fact_921_le__supE,axiom,
! [A: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_922_le__supE,axiom,
! [A: int,B: int,X: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_int @ A @ X )
=> ~ ( ord_less_eq_int @ B @ X ) ) ) ).
% le_supE
thf(fact_923_le__supI,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ B @ X )
=> ( ord_less_eq_real @ ( sup_sup_real @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_924_le__supI,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_925_le__supI,axiom,
! [A: int,X: int,B: int] :
( ( ord_less_eq_int @ A @ X )
=> ( ( ord_less_eq_int @ B @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_926_sup__ge1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sup_sup_real @ X @ Y ) ) ).
% sup_ge1
thf(fact_927_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_928_sup__ge1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge1
thf(fact_929_sup__ge2,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X @ Y ) ) ).
% sup_ge2
thf(fact_930_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_931_sup__ge2,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge2
thf(fact_932_le__supI1,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ord_less_eq_real @ X @ ( sup_sup_real @ A @ B ) ) ) ).
% le_supI1
thf(fact_933_le__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_934_le__supI1,axiom,
! [X: int,A: int,B: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).
% le_supI1
thf(fact_935_le__supI2,axiom,
! [X: real,B: real,A: real] :
( ( ord_less_eq_real @ X @ B )
=> ( ord_less_eq_real @ X @ ( sup_sup_real @ A @ B ) ) ) ).
% le_supI2
thf(fact_936_le__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_937_le__supI2,axiom,
! [X: int,B: int,A: int] :
( ( ord_less_eq_int @ X @ B )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).
% le_supI2
thf(fact_938_sup_Omono,axiom,
! [C: real,A: real,D: real,B: real] :
( ( ord_less_eq_real @ C @ A )
=> ( ( ord_less_eq_real @ D @ B )
=> ( ord_less_eq_real @ ( sup_sup_real @ C @ D ) @ ( sup_sup_real @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_939_sup_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_940_sup_Omono,axiom,
! [C: int,A: int,D: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ( ord_less_eq_int @ D @ B )
=> ( ord_less_eq_int @ ( sup_sup_int @ C @ D ) @ ( sup_sup_int @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_941_sup__mono,axiom,
! [A: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ B @ D )
=> ( ord_less_eq_real @ ( sup_sup_real @ A @ B ) @ ( sup_sup_real @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_942_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_943_sup__mono,axiom,
! [A: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ( ord_less_eq_int @ B @ D )
=> ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ ( sup_sup_int @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_944_sup__least,axiom,
! [Y: real,X: real,Z4: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ Z4 @ X )
=> ( ord_less_eq_real @ ( sup_sup_real @ Y @ Z4 ) @ X ) ) ) ).
% sup_least
thf(fact_945_sup__least,axiom,
! [Y: nat,X: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z4 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z4 ) @ X ) ) ) ).
% sup_least
thf(fact_946_sup__least,axiom,
! [Y: int,X: int,Z4: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ Z4 @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z4 ) @ X ) ) ) ).
% sup_least
thf(fact_947_le__iff__sup,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y2: real] :
( ( sup_sup_real @ X4 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_948_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y2: nat] :
( ( sup_sup_nat @ X4 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_949_le__iff__sup,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y2: int] :
( ( sup_sup_int @ X4 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_950_sup_OorderE,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( A
= ( sup_sup_real @ A @ B ) ) ) ).
% sup.orderE
thf(fact_951_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_952_sup_OorderE,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( A
= ( sup_sup_int @ A @ B ) ) ) ).
% sup.orderE
thf(fact_953_sup_OorderI,axiom,
! [A: real,B: real] :
( ( A
= ( sup_sup_real @ A @ B ) )
=> ( ord_less_eq_real @ B @ A ) ) ).
% sup.orderI
thf(fact_954_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_955_sup_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( sup_sup_int @ A @ B ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
% sup.orderI
thf(fact_956_sup__unique,axiom,
! [F: real > real > real,X: real,Y: real] :
( ! [X3: real,Y3: real] : ( ord_less_eq_real @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: real,Y3: real] : ( ord_less_eq_real @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: real,Y3: real,Z: real] :
( ( ord_less_eq_real @ Y3 @ X3 )
=> ( ( ord_less_eq_real @ Z @ X3 )
=> ( ord_less_eq_real @ ( F @ Y3 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_real @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_957_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y3 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_958_sup__unique,axiom,
! [F: int > int > int,X: int,Y: int] :
( ! [X3: int,Y3: int] : ( ord_less_eq_int @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: int,Y3: int,Z: int] :
( ( ord_less_eq_int @ Y3 @ X3 )
=> ( ( ord_less_eq_int @ Z @ X3 )
=> ( ord_less_eq_int @ ( F @ Y3 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_int @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_959_sup_Oabsorb1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( sup_sup_real @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_960_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_961_sup_Oabsorb1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( sup_sup_int @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_962_sup_Oabsorb2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( sup_sup_real @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_963_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_964_sup_Oabsorb2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( sup_sup_int @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_965_sup__absorb1,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( sup_sup_real @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_966_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_967_sup__absorb1,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( sup_sup_int @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_968_sup__absorb2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( sup_sup_real @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_969_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_970_sup__absorb2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( sup_sup_int @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_971_sup_OboundedE,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_real @ B @ A )
=> ~ ( ord_less_eq_real @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_972_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_973_sup_OboundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_int @ B @ A )
=> ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_974_sup_OboundedI,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ A )
=> ( ord_less_eq_real @ ( sup_sup_real @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_975_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_976_sup_OboundedI,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_977_sup_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] :
( A3
= ( sup_sup_real @ A3 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_978_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_979_sup_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( A3
= ( sup_sup_int @ A3 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_980_sup_Ocobounded1,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ A @ ( sup_sup_real @ A @ B ) ) ).
% sup.cobounded1
thf(fact_981_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_982_sup_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ A @ ( sup_sup_int @ A @ B ) ) ).
% sup.cobounded1
thf(fact_983_sup_Ocobounded2,axiom,
! [B: real,A: real] : ( ord_less_eq_real @ B @ ( sup_sup_real @ A @ B ) ) ).
% sup.cobounded2
thf(fact_984_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_985_sup_Ocobounded2,axiom,
! [B: int,A: int] : ( ord_less_eq_int @ B @ ( sup_sup_int @ A @ B ) ) ).
% sup.cobounded2
thf(fact_986_sup_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] :
( ( sup_sup_real @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_987_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_988_sup_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( ( sup_sup_int @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_989_sup_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] :
( ( sup_sup_real @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_990_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( sup_sup_nat @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_991_sup_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] :
( ( sup_sup_int @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_992_sup_OcoboundedI1,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ A )
=> ( ord_less_eq_real @ C @ ( sup_sup_real @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_993_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_994_sup_OcoboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_995_sup_OcoboundedI2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ ( sup_sup_real @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_996_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_997_sup_OcoboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_998_sup_Ostrict__coboundedI2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ ( sup_sup_real @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_999_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1000_sup_Ostrict__coboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1001_sup_Ostrict__coboundedI1,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ A )
=> ( ord_less_real @ C @ ( sup_sup_real @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1002_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1003_sup_Ostrict__coboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ A )
=> ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1004_sup_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( A3
= ( sup_sup_real @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1005_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1006_sup_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( A3
= ( sup_sup_int @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1007_sup_Ostrict__boundedE,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( sup_sup_real @ B @ C ) @ A )
=> ~ ( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_1008_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_1009_sup_Ostrict__boundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_int @ ( sup_sup_int @ B @ C ) @ A )
=> ~ ( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_1010_sup_Oabsorb4,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( sup_sup_real @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_1011_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_1012_sup_Oabsorb4,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( sup_sup_int @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_1013_sup_Oabsorb3,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( sup_sup_real @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_1014_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_1015_sup_Oabsorb3,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( sup_sup_int @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_1016_less__supI2,axiom,
! [X: real,B: real,A: real] :
( ( ord_less_real @ X @ B )
=> ( ord_less_real @ X @ ( sup_sup_real @ A @ B ) ) ) ).
% less_supI2
thf(fact_1017_less__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_nat @ X @ B )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_1018_less__supI2,axiom,
! [X: int,B: int,A: int] :
( ( ord_less_int @ X @ B )
=> ( ord_less_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).
% less_supI2
thf(fact_1019_less__supI1,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_real @ X @ A )
=> ( ord_less_real @ X @ ( sup_sup_real @ A @ B ) ) ) ).
% less_supI1
thf(fact_1020_less__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_nat @ X @ A )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_1021_less__supI1,axiom,
! [X: int,A: int,B: int] :
( ( ord_less_int @ X @ A )
=> ( ord_less_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).
% less_supI1
thf(fact_1022_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1023_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1024_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_1025_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_1026_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_1027_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_1028_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_1029_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_1030_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_1031_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_1032_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_1033_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_1034_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_1035_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_1036_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_1037_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_1038_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_1039_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_1040_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_1041_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_1042_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_1043_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_1044_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_1045_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_1046_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_1047_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_1048_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_1049_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_1050_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_1051_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_1052_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_1053_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_1054_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_1055_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_1056_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_1057_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_1058_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_1059_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_1060_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_1061_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_1062_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1063_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1064_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1065_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1066_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1067_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1068_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_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_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_1079_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_1080_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_1081_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_1082_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_1083_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_1084_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_1085_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_1086_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_1087_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_1088_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_1089_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_1090_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_1091_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_1092_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_1093_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_1094_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_1095_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_1096_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_1097_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_1098_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_1099_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_1100_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_1101_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_1102_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_1103_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_1104_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_1105_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_1106_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_1107_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_1108_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_1109_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_1110_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_1111_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_1112_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_1113_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_1114_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_1115_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1116_mult__less__le__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( 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 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1117_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1118_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1119_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1120_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_1121_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_1122_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1123_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_1124_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_1125_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_1126_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_1127_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1128_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1129_not__int__zless__negative,axiom,
! [N: nat,M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_1130_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1131_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1132_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_1133_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1134_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_1135_nat__less__iff,axiom,
! [W2: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M2 )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_1136_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1137_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1138_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1139_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1140_nat__le__0,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ Z4 @ zero_zero_int )
=> ( ( nat2 @ Z4 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1141_zless__nat__conj,axiom,
! [W2: int,Z4: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z4 ) )
= ( ( ord_less_int @ zero_zero_int @ Z4 )
& ( ord_less_int @ W2 @ Z4 ) ) ) ).
% zless_nat_conj
thf(fact_1142_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_1143_zero__less__nat__eq,axiom,
! [Z4: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z4 ) )
= ( ord_less_int @ zero_zero_int @ Z4 ) ) ).
% zero_less_nat_eq
thf(fact_1144_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1145_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1146_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1147_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1148_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times_nat @ M2 @ N ) )
=> ( ( N = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1149_nat__mono__iff,axiom,
! [Z4: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z4 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z4 ) )
= ( ord_less_int @ W2 @ Z4 ) ) ) ).
% nat_mono_iff
thf(fact_1150_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_1151_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z4: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z4 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z4 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1152_nat__eq__iff,axiom,
! [W2: int,M2: nat] :
( ( ( nat2 @ W2 )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1153_nat__eq__iff2,axiom,
! [M2: nat,W2: int] :
( ( M2
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1154_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_1155_nat__le__eq__zle,axiom,
! [W2: int,Z4: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z4 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z4 ) )
= ( ord_less_eq_int @ W2 @ Z4 ) ) ) ).
% nat_le_eq_zle
thf(fact_1156_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_1157_nat__less__eq__zless,axiom,
! [W2: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z4 ) )
= ( ord_less_int @ W2 @ Z4 ) ) ) ).
% nat_less_eq_zless
thf(fact_1158_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1159_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1160_int__one__le__iff__zero__less,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ one_one_int @ Z4 )
= ( ord_less_int @ zero_zero_int @ Z4 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1161_pos__zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1162_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_1163_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I2: nat] :
( ( Q @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I2 ) )
& ( ord_less_eq_real @ ( X3 @ I2 ) @ one_one_real ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X2: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L3 @ X2 @ I3 ) @ one_one_nat )
& ! [X2: nat > real,I3: nat] :
( ( ( P @ X2 )
& ( Q @ I3 )
& ( ( X2 @ I3 )
= zero_zero_real ) )
=> ( ( L3 @ X2 @ I3 )
= zero_zero_nat ) )
& ! [X2: nat > real,I3: nat] :
( ( ( P @ X2 )
& ( Q @ I3 )
& ( ( X2 @ I3 )
= one_one_real ) )
=> ( ( L3 @ X2 @ I3 )
= one_one_nat ) )
& ! [X2: nat > real,I3: nat] :
( ( ( P @ X2 )
& ( Q @ I3 )
& ( ( L3 @ X2 @ I3 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X2 @ I3 ) @ ( F @ X2 @ I3 ) ) )
& ! [X2: nat > real,I3: nat] :
( ( ( P @ X2 )
& ( Q @ I3 )
& ( ( L3 @ X2 @ I3 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X2 @ I3 ) @ ( X2 @ I3 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1164_one__less__nat__eq,axiom,
! [Z4: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z4 ) )
= ( ord_less_int @ one_one_int @ Z4 ) ) ).
% one_less_nat_eq
thf(fact_1165_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1166_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1167_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_1168_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1169_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1170_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_1171_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_1172_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1173_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1174_one__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1175_mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1176_one__le__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1177_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1178_negative__zless,axiom,
! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_1179_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1180_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1181_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1182_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_1183_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_1184_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_1185_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_1186_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1187_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_1188_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1189_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_1190_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_1191_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1192_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_1193_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1194_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_1195_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_1196_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_1197_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_1198_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_1199_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1200_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_1201_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1202_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1203_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M4: nat] :
( M7
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1204_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1205_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1206_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_1207_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1208_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1209_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R4: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X3: nat] : ( R4 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( R4 @ X3 @ Y3 )
=> ( ( R4 @ Y3 @ Z )
=> ( R4 @ X3 @ Z ) ) )
=> ( ! [N3: nat] : ( R4 @ N3 @ ( suc @ N3 ) )
=> ( R4 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1210_Suc__mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M2 )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M2 = N ) ) ).
% Suc_mult_cancel1
thf(fact_1211_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_1212_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1213_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1214_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1215_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1216_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1217_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_1218_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_1219_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_1220_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1221_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_1222_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_1223_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1224_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_1225_Icc__eq__insert__lb__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( set_or1269000886237332187st_nat @ M2 @ N )
= ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_1226_atLeastAtMostSuc__conv,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_1227_atLeastAtMost__insertL,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
= ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_1228_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_1229_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_1230_Suc__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1231_Suc__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1232_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1233_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_1234_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_1235_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_1236_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_1237_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1238_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1239_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1240_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1241_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_1242_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M: nat] :
( N
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1243_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_1244_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1245_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1246_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_1247_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_1248_one__less__mult,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% one_less_mult
thf(fact_1249_n__less__m__mult__n,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1250_n__less__n__mult__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1251_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_1252_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1253_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_1254_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_1255_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_1256_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_1257_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_1258_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_1259_nat__mult__div__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1260_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1261_verit__less__mono__div__int2,axiom,
! [A2: int,B4: int,N: int] :
( ( ord_less_eq_int @ A2 @ B4 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B4 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1262_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_1263_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
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_real @ ( g @ ( f @ a ) ) @ ( g @ y ) ).
%------------------------------------------------------------------------------