TPTP Problem File: SLH0180^1.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_00532_022707__13125180_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1414 ( 448 unt; 146 typ; 0 def)
% Number of atoms : 4485 (1101 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 12170 ( 331 ~; 94 |; 164 &;9591 @)
% ( 0 <=>;1990 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 1224 (1224 >; 0 *; 0 +; 0 <<)
% Number of symbols : 139 ( 138 usr; 18 con; 0-4 aty)
% Number of variables : 3694 ( 128 ^;3499 !; 67 ?;3694 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:31:15.223
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $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 (138)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
archim6058952711729229775r_real: real > int ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Real__Oreal,type,
comp_c2117349707075585901x_real: ( complex > complex ) > ( real > complex ) > real > complex ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Int__Oint_001t__Nat__Onat,type,
comp_complex_int_nat: ( complex > int ) > ( nat > complex ) > nat > int ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Nat__Onat_001t__Nat__Onat,type,
comp_complex_nat_nat: ( complex > nat ) > ( nat > complex ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Nat__Onat_001t__Real__Oreal,type,
comp_c3423117485846644111t_real: ( complex > nat ) > ( real > complex ) > real > nat ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Nat__Onat,type,
comp_c7990426058975542799al_nat: ( complex > real ) > ( nat > complex ) > nat > real ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Real__Oreal,type,
comp_c3333796836230738283l_real: ( complex > real ) > ( real > complex ) > real > real ).
thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Complex__Ocomplex_001t__Nat__Onat,type,
comp_int_complex_nat: ( int > complex ) > ( nat > int ) > nat > complex ).
thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Nat__Onat,type,
comp_int_int_nat: ( int > int ) > ( nat > int ) > nat > int ).
thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
comp_int_nat_nat: ( int > nat ) > ( nat > int ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Real__Oreal_001t__Nat__Onat,type,
comp_int_real_nat: ( int > real ) > ( nat > int ) > nat > real ).
thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Real__Oreal_001t__Real__Oreal,type,
comp_int_real_real: ( int > real ) > ( real > int ) > real > real ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Complex__Ocomplex_001t__Real__Oreal,type,
comp_n4215249288434654095x_real: ( nat > complex ) > ( real > nat ) > real > complex ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Int__Oint_001t__Real__Oreal,type,
comp_nat_int_real: ( nat > int ) > ( real > nat ) > real > int ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
comp_nat_nat_real: ( nat > nat ) > ( real > nat ) > real > nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Real__Oreal,type,
comp_nat_real_real: ( nat > real ) > ( real > nat ) > real > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Nat__Onat,type,
comp_r1225911664865567631ex_nat: ( real > complex ) > ( nat > real ) > nat > complex ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Real__Oreal,type,
comp_r1968866223832618731x_real: ( real > complex ) > ( real > real ) > real > complex ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Int__Oint_001t__Nat__Onat,type,
comp_real_int_nat: ( real > int ) > ( nat > real ) > nat > int ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Int__Oint_001t__Real__Oreal,type,
comp_real_int_real: ( real > int ) > ( real > real ) > real > int ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Nat__Onat_001t__Nat__Onat,type,
comp_real_nat_nat: ( real > nat ) > ( nat > real ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Nat__Onat_001t__Real__Oreal,type,
comp_real_nat_real: ( real > nat ) > ( real > real ) > real > nat ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Nat__Onat,type,
comp_real_real_nat: ( real > real ) > ( nat > real ) > nat > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
comp_real_real_real: ( real > real ) > ( real > real ) > real > real ).
thf(sy_c_Fun_Omonotone__on_001t__Complex__Ocomplex_001t__Int__Oint,type,
monoto2404022921142102083ex_int: set_complex > ( complex > complex > $o ) > ( int > int > $o ) > ( complex > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Complex__Ocomplex_001t__Nat__Onat,type,
monoto2406513391651152359ex_nat: set_complex > ( complex > complex > $o ) > ( nat > nat > $o ) > ( complex > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Complex__Ocomplex_001t__Real__Oreal,type,
monoto7363281639122250051x_real: set_complex > ( complex > complex > $o ) > ( real > real > $o ) > ( complex > real ) > $o ).
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__Complex__Ocomplex,type,
monoto8010102104071190503omplex: set_nat > ( nat > nat > $o ) > ( complex > complex > $o ) > ( nat > complex ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Int__Oint,type,
monotone_on_nat_int: set_nat > ( nat > nat > $o ) > ( int > int > $o ) > ( nat > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
monotone_on_nat_nat: set_nat > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Real__Oreal,type,
monotone_on_nat_real: set_nat > ( nat > nat > $o ) > ( real > real > $o ) > ( nat > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Int__Oint,type,
monotone_on_real_int: set_real > ( real > real > $o ) > ( int > int > $o ) > ( real > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Nat__Onat,type,
monotone_on_real_nat: set_real > ( real > real > $o ) > ( nat > nat > $o ) > ( real > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Real__Oreal,type,
monoto4017252874604999745l_real: set_real > ( real > real > $o ) > ( real > real > $o ) > ( real > real ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
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_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_Henstock__Kurzweil__Integration_Ointegrable__on_001t__Real__Oreal_001t__Real__Oreal,type,
hensto5963834015518849588l_real: ( real > real ) > set_real > $o ).
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_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_Oord__class_Oless_001t__Complex__Ocomplex,type,
ord_less_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_less_set_complex: set_complex > set_complex > $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__Complex__Ocomplex,type,
ord_less_eq_complex: complex > complex > $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__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Complex__Ocomplex_J,type,
top_top_set_complex: set_complex ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
top_top_set_int: set_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
image_1468599708987790691omplex: ( complex > complex ) > set_complex > set_complex ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Int__Oint,type,
image_complex_int: ( complex > int ) > set_complex > set_int ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Nat__Onat,type,
image_complex_nat: ( complex > nat ) > set_complex > set_nat ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Real__Oreal,type,
image_complex_real: ( complex > real ) > set_complex > set_real ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Complex__Ocomplex,type,
image_int_complex: ( int > complex ) > set_int > set_complex ).
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__Int__Oint_001t__Real__Oreal,type,
image_int_real: ( int > real ) > set_int > set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Complex__Ocomplex,type,
image_nat_complex: ( nat > complex ) > set_nat > set_complex ).
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__Complex__Ocomplex,type,
image_real_complex: ( real > complex ) > set_real > set_complex ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Int__Oint,type,
image_real_int: ( real > int ) > set_real > set_int ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Nat__Onat,type,
image_real_nat: ( real > nat ) > set_real > set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Int__Oint,type,
set_ord_atLeast_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
set_ord_atLeast_real: real > set_real ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
topolo9015423870875150044omplex: set_complex > ( complex > complex ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Complex__Ocomplex_001t__Nat__Onat,type,
topolo3759945079839938046ex_nat: set_complex > ( complex > nat ) > $o ).
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__Complex__Ocomplex,type,
topolo140161755405200382omplex: set_nat > ( nat > complex ) > $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_Topological__Spaces_Ouniformly__continuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
topolo8845477368217174713l_real: set_real > ( real > real ) > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $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__092_060delta_062____,type,
delta: real ).
thf(sy_v__092_060epsilon_062____,type,
epsilon: real ).
thf(sy_v_a,type,
a: real ).
thf(sy_v_a__seg____,type,
a_seg: real > real ).
thf(sy_v_b,type,
b: real ).
thf(sy_v_del____,type,
del: real > real ).
thf(sy_v_f,type,
f: real > real ).
thf(sy_v_f1____,type,
f1: real > real ).
thf(sy_v_f2____,type,
f2: real > real ).
thf(sy_v_g,type,
g: real > real ).
thf(sy_v_k____,type,
k: nat ).
thf(sy_v_lower____,type,
lower: real > real ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_upper____,type,
upper: real > real ).
thf(sy_v_x____,type,
x: real ).
thf(sy_v_y____,type,
y: real ).
thf(sy_v_yidx____,type,
yidx: real > nat ).
% Relevant facts (1267)
thf(fact_0_f_I1_J,axiom,
( ( f @ zero_zero_real )
= zero_zero_real ) ).
% f(1)
thf(fact_1__092_060open_062_092_060And_062s_Ar_O_A_092_060lbrakk_062r_A_092_060in_062_A_1230_O_O_125_059_As_A_092_060in_062_A_1230_O_O_125_059_Ar_A_060_As_092_060rbrakk_062_A_092_060Longrightarrow_062_Af_Ar_A_060_Af_As_092_060close_062,axiom,
! [R: real,S: real] :
( ( member_real @ R @ ( set_ord_atLeast_real @ zero_zero_real ) )
=> ( ( member_real @ S @ ( set_ord_atLeast_real @ zero_zero_real ) )
=> ( ( ord_less_real @ R @ S )
=> ( ord_less_real @ ( f @ R ) @ ( f @ S ) ) ) ) ) ).
% \<open>\<And>s r. \<lbrakk>r \<in> {0..}; s \<in> {0..}; r < s\<rbrakk> \<Longrightarrow> f r < f s\<close>
thf(fact_2__092_060open_062a__seg_A_Ireal_A_Iyidx_Ay_J_J_A_092_060le_062_Aa__seg_A_Ireal_Ak_J_092_060close_062,axiom,
ord_less_eq_real @ ( a_seg @ ( semiri5074537144036343181t_real @ ( yidx @ y ) ) ) @ ( a_seg @ ( semiri5074537144036343181t_real @ k ) ) ).
% \<open>a_seg (real (yidx y)) \<le> a_seg (real k)\<close>
thf(fact_3_x_I1_J,axiom,
( ( f @ x )
= y ) ).
% x(1)
thf(fact_4_f__lims_I1_J,axiom,
ord_less_eq_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ k ) ) ) @ y ).
% f_lims(1)
thf(fact_5__092_060open_0620_A_060_A_092_060epsilon_062_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ epsilon ).
% \<open>0 < \<epsilon>\<close>
thf(fact_6_a__seg__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( a_seg @ X ) @ ( a_seg @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% a_seg_le_iff
thf(fact_7_x__lims_I1_J,axiom,
ord_less_eq_real @ ( a_seg @ ( semiri5074537144036343181t_real @ k ) ) @ x ).
% x_lims(1)
thf(fact_8_f__lims_I2_J,axiom,
ord_less_real @ y @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ k ) ) ) ) ).
% f_lims(2)
thf(fact_9_a__seg__ge__0,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( a_seg @ X ) )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% a_seg_ge_0
thf(fact_10_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_11_a__seg__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( a_seg @ X ) @ ( a_seg @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% a_seg_less_iff
thf(fact_12_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_13_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_14_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_15__092_060open_062_092_060And_062thesis_O_A_I_092_060lbrakk_062f_A_Ia__seg_A_Ireal_Ak_J_J_A_092_060le_062_Ay_059_Ay_A_060_Af_A_Ia__seg_A_Ireal_A_ISuc_Ak_J_J_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ( ( ord_less_eq_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ k ) ) ) @ y )
=> ~ ( ord_less_real @ y @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ k ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<lbrakk>f (a_seg (real k)) \<le> y; y < f (a_seg (real (Suc k)))\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_16_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_17_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_18_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_19_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_20_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_21_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_22_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_23_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_24_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_25_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_26_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_27_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_28_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_29_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_30_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_31_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_32_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_33_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_34_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_35_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_36_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_37_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_38_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_39_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_40_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_41_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_42_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_43_x__lims_I2_J,axiom,
ord_less_real @ x @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ k ) ) ) ).
% x_lims(2)
thf(fact_44_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_45_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_46_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_47_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_48_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_49_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_50_cont,axiom,
topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ f ).
% cont
thf(fact_51__092_060open_0620_A_060_A_092_060delta_062_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ delta ).
% \<open>0 < \<delta>\<close>
thf(fact_52_del__gt0,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_real @ zero_zero_real @ ( del @ E ) ) ) ).
% del_gt0
thf(fact_53_sm,axiom,
monoto4017252874604999745l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ ord_less_real @ ord_less_real @ f ).
% sm
thf(fact_54_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_55_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_56_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_57_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_58_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_59_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_60_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_61_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_62_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_63_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_64_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_65_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_66_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_67_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_68_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_69_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_70_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_71_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_72_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_73_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_74_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_75_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_76_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_77_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_78_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_79_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_80_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_81_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_82_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_83_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_84_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_85_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_86_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_87_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_88_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_89_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_90_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_91_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_92_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_93_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_94_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_95_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N4: nat] :
( ( P3 @ N4 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N4 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_96_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: real] : ( P @ A3 @ A3 )
=> ( ! [A3: real,B2: real] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_97_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_98_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_99_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_100_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_101_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_102_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_103_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_104_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_105_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_106_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_107_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_108_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_109_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_110_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_111_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_112_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_113_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_114_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_115_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_116_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_117_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_118_Suc__le__D,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
=> ? [M4: nat] :
( M3
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_119_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_120_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_121_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_122_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_123_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_124_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_125_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_126_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_127_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_128_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_129_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 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_130_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_131_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_132_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_133_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_134_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_135_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_136_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_137_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z )
=> ( R2 @ X3 @ Z ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_138_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_139_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_140_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_141_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_142_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_143_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_144_order__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_145_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_146_order__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_147_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_148_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_149_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_150_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_151_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_152_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_153_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_154_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_155_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_156_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_157_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_158_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_159_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_160_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_161_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_162_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_163_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_164_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_165_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_166_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_167_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_168_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_169_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_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_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_182_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_183_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_184_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_185_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_186_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_187_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_188_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_189_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_190_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_191_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_192_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_193_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_194_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_195_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_196_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_197_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_198_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_199_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_200_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_201_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_202_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_203_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_204_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_205_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_206_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_207_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_208_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_209_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_210_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_211_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_212_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_213_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_214_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_215_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_216_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_217_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_218_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_219_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_220_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_221_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_222_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_223_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_224_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_225_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_226_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_227_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_228_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_229_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_230_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_231_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_232_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_233_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_234_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_235_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_236_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_237_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_238_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_239_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_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_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_252_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_253_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_254_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_255_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_256_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_257_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_258_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_259_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_260_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_261_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_262_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_263_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_264_order__less__le__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_265_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_266_order__less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_267_order__le__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_268_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_269_order__le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_270_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_271_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_272_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_273_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_274_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_275_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_276_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_277_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_278_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_279_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_280_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_281_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_282_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_283_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_284_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_285_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_286_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_287_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y5: int] :
( ( ord_less_eq_int @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_288_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_real @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_289_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_290_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y5: int] :
( ( ord_less_int @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_291_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_292_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_293_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_294_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_295_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_296_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_297_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_298_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_299_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_300_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_301_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_302_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_303_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_304_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_305_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_306_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_307_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_308_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_309_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_real @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_310_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_311_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_int @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_312_dense__le__bounded,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_313_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_314_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_315_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_316_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_317_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_318_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_319_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_320_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_321_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_322_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_323_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_324_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_325_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_326_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_327_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_328_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_329_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_330_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_331_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_332_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_333_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_334_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y5: int] :
( ( ord_less_eq_int @ X4 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_335_dense__le,axiom,
! [Y: real,Z2: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_le
thf(fact_336_dense__ge,axiom,
! [Z2: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_337_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_338_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_339_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_340_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_341_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_342_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_343_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_344_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_345_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_346_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_347_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_348_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_349_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_350_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_351_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_352_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_353_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_354_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_355_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_356_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_357_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_358_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_359_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_360_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_361_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_362_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_363_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_364_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_365_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_366_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_367_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_368_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_369_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_370_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_371_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_372_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_373_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_374_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_375_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_376_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_377_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_378_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_379_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_380_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_381_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_382_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_383_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_384_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_385_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_386_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_387_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_388_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_389_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_390_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_391_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_392_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_393_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_394_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_395_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_396_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_397_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_398_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_399_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_400_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_401_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_402_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_403_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_404_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_405_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_406_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_407_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_408_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_409_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_410_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_411_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_412_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_413_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_414_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_415_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: real,B2: real] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_416_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_417_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_418_order__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_419_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_420_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_421_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_422_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_423_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_424_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_425_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_426_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_427_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_428_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_429_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_430_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_431_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_432_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_433_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ( ord_less_eq_real @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_434_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_435_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
= ( ^ [X4: int,Y5: int] :
( ( ord_less_eq_int @ X4 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_436_le__cases3,axiom,
! [X: real,Y: real,Z2: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_437_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_438_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_439_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_440_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_441_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_442_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_443_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_444_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_445__092_060open_0620_A_092_060le_062_Ab_092_060close_062,axiom,
ord_less_eq_real @ zero_zero_real @ b ).
% \<open>0 \<le> b\<close>
thf(fact_446_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_447_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_448_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_449_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_450_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_451_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_452_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_453_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_454_a,axiom,
ord_less_eq_real @ zero_zero_real @ a ).
% a
thf(fact_455_seq__mono__lemma,axiom,
! [M: nat,D: nat > real,E: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_real @ ( D @ N3 ) @ ( E @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_real @ ( E @ N3 ) @ ( E @ M ) ) )
=> ! [N5: nat] :
( ( ord_less_eq_nat @ M @ N5 )
=> ( ord_less_real @ ( D @ N5 ) @ ( E @ M ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_456_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_457_atLeast__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atLeast_nat @ X )
= ( set_ord_atLeast_nat @ Y ) )
= ( X = Y ) ) ).
% atLeast_eq_iff
thf(fact_458_False,axiom,
a != zero_zero_real ).
% False
thf(fact_459_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_real @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_460_f_I2_J,axiom,
( ( f @ a )
= b ) ).
% f(2)
thf(fact_461__092_060open_0620_A_060_Aa_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ a ).
% \<open>0 < a\<close>
thf(fact_462__092_060open_062_092_060delta_062_A_092_060le_062_Aa_092_060close_062,axiom,
ord_less_eq_real @ delta @ a ).
% \<open>\<delta> \<le> a\<close>
thf(fact_463_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_464_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_465_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_466_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_467_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_468_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_469_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_470_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_471_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_472_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_473_x_I2_J,axiom,
member_real @ x @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ).
% x(2)
thf(fact_474_that,axiom,
member_real @ y @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) ).
% that
thf(fact_475_g,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ a )
=> ( ( g @ ( f @ X ) )
= X ) ) ) ).
% g
thf(fact_476_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_477_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_478_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_479_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_480_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_481_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_482_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_483_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_484_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_485_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_486_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_487_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_488_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_489_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_490_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_491_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_492_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_493_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_494_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_495_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_496_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_497_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_498_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_499_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_500_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P @ I4 @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I4 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_501_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_502_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_503_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_504_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_505_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_506_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_507_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_508_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_509_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_510_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_511_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_512_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_513_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_514_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_515_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_516_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_517_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_518_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
| ( M2 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_519_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_520_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
& ( M2 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_521_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_522_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_523_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_524_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_525_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_526_diff__induct,axiom,
! [P: nat > nat > $o,M: 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 @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_527_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_528_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_529_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_530_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_531_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_532_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_533_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_534_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_535_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_536_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_537_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_538_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_539_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_540_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_541_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_542_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_543_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_544_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_545_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_546_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_547_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_548_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_549_complete__real,axiom,
! [S2: set_real] :
( ? [X6: real] : ( member_real @ X6 @ S2 )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y3: real] :
( ! [X6: real] :
( ( member_real @ X6 @ S2 )
=> ( ord_less_eq_real @ X6 @ Y3 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_550_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M7 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_551_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_552_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_553_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_554_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_555_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_556_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_557_f1__lower,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ a )
=> ( ord_less_eq_real @ ( f1 @ X ) @ ( f @ X ) ) ) ) ).
% f1_lower
thf(fact_558_f2__upper,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ a )
=> ( ord_less_eq_real @ ( f @ X ) @ ( f2 @ X ) ) ) ) ).
% f2_upper
thf(fact_559_sm__0a,axiom,
monoto4017252874604999745l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) @ ord_less_real @ ord_less_real @ f ).
% sm_0a
thf(fact_560__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_A_092_060lbrakk_062f_Ax_A_061_Ay_059_Ax_A_092_060in_062_A_1230_O_Oa_125_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X3: real] :
( ( ( f @ X3 )
= y )
=> ~ ( member_real @ X3 @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ) ) ).
% \<open>\<And>thesis. (\<And>x. \<lbrakk>f x = y; x \<in> {0..a}\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_561_cont__0a,axiom,
topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) @ f ).
% cont_0a
thf(fact_562_fa__eq__b,axiom,
( ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ n ) ) )
= b ) ).
% fa_eq_b
thf(fact_563__092_060open_062strict__mono_Aa__seg_092_060close_062,axiom,
monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ a_seg ).
% \<open>strict_mono a_seg\<close>
thf(fact_564_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_565_ord_Omono__on__subset,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > nat,B4: set_nat] :
( ( monotone_on_nat_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( monotone_on_nat_nat @ B4 @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_566__092_060open_0620_A_060_An_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% \<open>0 < n\<close>
thf(fact_567_a__seg__eq__a__iff,axiom,
! [X: real] :
( ( ( a_seg @ X )
= a )
= ( X
= ( semiri5074537144036343181t_real @ n ) ) ) ).
% a_seg_eq_a_iff
thf(fact_568_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_569_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_570_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_571_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_572_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_573_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_574__092_060open_062continuous__on_A_1230_O_Ob_125_Ag_092_060close_062,axiom,
topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) @ g ).
% \<open>continuous_on {0..b} g\<close>
thf(fact_575_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_576_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_577_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_578_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_579_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_580_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_581_a__seg__le__a,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( a_seg @ X ) @ a )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ n ) ) ) ).
% a_seg_le_a
thf(fact_582_not__UNIV__le__Icc,axiom,
! [L: real,H: real] :
~ ( ord_less_eq_set_real @ top_top_set_real @ ( set_or1222579329274155063t_real @ L @ H ) ) ).
% not_UNIV_le_Icc
thf(fact_583_not__UNIV__le__Icc,axiom,
! [L: nat,H: nat] :
~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) ) ).
% not_UNIV_le_Icc
thf(fact_584_not__UNIV__le__Icc,axiom,
! [L: int,H: int] :
~ ( ord_less_eq_set_int @ top_top_set_int @ ( set_or1266510415728281911st_int @ L @ H ) ) ).
% not_UNIV_le_Icc
thf(fact_585_monotoneI,axiom,
! [Orda: real > real > $o,Ordb: real > real > $o,F: real > real] :
( ! [X3: real,Y3: real] :
( ( Orda @ X3 @ Y3 )
=> ( Ordb @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monoto4017252874604999745l_real @ top_top_set_real @ Orda @ Ordb @ F ) ) ).
% monotoneI
thf(fact_586_monotoneI,axiom,
! [Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat] :
( ! [X3: nat,Y3: nat] :
( ( Orda @ X3 @ Y3 )
=> ( Ordb @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ Orda @ Ordb @ F ) ) ).
% monotoneI
thf(fact_587_monotoneD,axiom,
! [Orda: real > real > $o,Ordb: real > real > $o,F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ Orda @ Ordb @ F )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monotoneD
thf(fact_588_monotoneD,axiom,
! [Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ Orda @ Ordb @ F )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monotoneD
thf(fact_589_not__UNIV__eq__Icc,axiom,
! [L2: real,H2: real] :
( top_top_set_real
!= ( set_or1222579329274155063t_real @ L2 @ H2 ) ) ).
% not_UNIV_eq_Icc
thf(fact_590_not__UNIV__eq__Icc,axiom,
! [L2: nat,H2: nat] :
( top_top_set_nat
!= ( set_or1269000886237332187st_nat @ L2 @ H2 ) ) ).
% not_UNIV_eq_Icc
thf(fact_591_not__UNIV__eq__Icc,axiom,
! [L2: int,H2: int] :
( top_top_set_int
!= ( set_or1266510415728281911st_int @ L2 @ H2 ) ) ).
% not_UNIV_eq_Icc
thf(fact_592_top_Oextremum__uniqueI,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A )
=> ( A = top_top_set_real ) ) ).
% top.extremum_uniqueI
thf(fact_593_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_594_top_Oextremum__uniqueI,axiom,
! [A: set_complex] :
( ( ord_le211207098394363844omplex @ top_top_set_complex @ A )
=> ( A = top_top_set_complex ) ) ).
% top.extremum_uniqueI
thf(fact_595_top_Oextremum__unique,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A )
= ( A = top_top_set_real ) ) ).
% top.extremum_unique
thf(fact_596_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_597_top_Oextremum__unique,axiom,
! [A: set_complex] :
( ( ord_le211207098394363844omplex @ top_top_set_complex @ A )
= ( A = top_top_set_complex ) ) ).
% top.extremum_unique
thf(fact_598_top__greatest,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).
% top_greatest
thf(fact_599_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_600_top__greatest,axiom,
! [A: set_complex] : ( ord_le211207098394363844omplex @ A @ top_top_set_complex ) ).
% top_greatest
thf(fact_601_top_Oextremum__strict,axiom,
! [A: set_real] :
~ ( ord_less_set_real @ top_top_set_real @ A ) ).
% top.extremum_strict
thf(fact_602_top_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A ) ).
% top.extremum_strict
thf(fact_603_top_Oextremum__strict,axiom,
! [A: set_complex] :
~ ( ord_less_set_complex @ top_top_set_complex @ A ) ).
% top.extremum_strict
thf(fact_604_top_Onot__eq__extremum,axiom,
! [A: set_real] :
( ( A != top_top_set_real )
= ( ord_less_set_real @ A @ top_top_set_real ) ) ).
% top.not_eq_extremum
thf(fact_605_top_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != top_top_set_nat )
= ( ord_less_set_nat @ A @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_606_top_Onot__eq__extremum,axiom,
! [A: set_complex] :
( ( A != top_top_set_complex )
= ( ord_less_set_complex @ A @ top_top_set_complex ) ) ).
% top.not_eq_extremum
thf(fact_607_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_608_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_609_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_610_not__UNIV__eq__Ici,axiom,
! [L2: real] :
( top_top_set_real
!= ( set_ord_atLeast_real @ L2 ) ) ).
% not_UNIV_eq_Ici
thf(fact_611_mono__imp__mono__on,axiom,
! [F: complex > real,A2: set_complex] :
( ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_real @ F )
=> ( monoto7363281639122250051x_real @ A2 @ ord_less_eq_complex @ ord_less_eq_real @ F ) ) ).
% mono_imp_mono_on
thf(fact_612_mono__imp__mono__on,axiom,
! [F: complex > nat,A2: set_complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_nat @ F )
=> ( monoto2406513391651152359ex_nat @ A2 @ ord_less_eq_complex @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_613_mono__imp__mono__on,axiom,
! [F: complex > int,A2: set_complex] :
( ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_int @ F )
=> ( monoto2404022921142102083ex_int @ A2 @ ord_less_eq_complex @ ord_less_eq_int @ F ) ) ).
% mono_imp_mono_on
thf(fact_614_mono__imp__mono__on,axiom,
! [F: real > real,A2: set_real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( monoto4017252874604999745l_real @ A2 @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% mono_imp_mono_on
thf(fact_615_mono__imp__mono__on,axiom,
! [F: real > nat,A2: set_real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( monotone_on_real_nat @ A2 @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_616_mono__imp__mono__on,axiom,
! [F: real > int,A2: set_real] :
( ( monotone_on_real_int @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_int @ F )
=> ( monotone_on_real_int @ A2 @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ).
% mono_imp_mono_on
thf(fact_617_mono__imp__mono__on,axiom,
! [F: nat > real,A2: set_nat] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_real @ F )
=> ( monotone_on_nat_real @ A2 @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ).
% mono_imp_mono_on
thf(fact_618_mono__imp__mono__on,axiom,
! [F: nat > nat,A2: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_619_mono__imp__mono__on,axiom,
! [F: nat > int,A2: set_nat] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_int @ F )
=> ( monotone_on_nat_int @ A2 @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ).
% mono_imp_mono_on
thf(fact_620_mono__imp__mono__on,axiom,
! [F: int > real,A2: set_int] :
( ( monotone_on_int_real @ top_top_set_int @ ord_less_eq_int @ ord_less_eq_real @ F )
=> ( monotone_on_int_real @ A2 @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ).
% mono_imp_mono_on
thf(fact_621_monoI,axiom,
! [F: complex > real] :
( ! [X3: complex,Y3: complex] :
( ( ord_less_eq_complex @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_real @ F ) ) ).
% monoI
thf(fact_622_monoI,axiom,
! [F: complex > nat] :
( ! [X3: complex,Y3: complex] :
( ( ord_less_eq_complex @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_623_monoI,axiom,
! [F: complex > int] :
( ! [X3: complex,Y3: complex] :
( ( ord_less_eq_complex @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_int @ F ) ) ).
% monoI
thf(fact_624_monoI,axiom,
! [F: real > real] :
( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% monoI
thf(fact_625_monoI,axiom,
! [F: real > nat] :
( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_real_nat @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_626_monoI,axiom,
! [F: real > int] :
( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_real_int @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ).
% monoI
thf(fact_627_monoI,axiom,
! [F: nat > real] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ).
% monoI
thf(fact_628_monoI,axiom,
! [F: nat > nat] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_629_monoI,axiom,
! [F: nat > int] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_int @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ).
% monoI
thf(fact_630_monoI,axiom,
! [F: int > real] :
( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_int_real @ top_top_set_int @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ).
% monoI
thf(fact_631_monoE,axiom,
! [F: complex > real,X: complex,Y: complex] :
( ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_complex @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_632_monoE,axiom,
! [F: complex > nat,X: complex,Y: complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_complex @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_633_monoE,axiom,
! [F: complex > int,X: complex,Y: complex] :
( ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_complex @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_634_monoE,axiom,
! [F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_635_monoE,axiom,
! [F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_636_monoE,axiom,
! [F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_637_monoE,axiom,
! [F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_638_monoE,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_639_monoE,axiom,
! [F: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_640_monoE,axiom,
! [F: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ top_top_set_int @ ord_less_eq_int @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_641_monoD,axiom,
! [F: complex > real,X: complex,Y: complex] :
( ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_complex @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_642_monoD,axiom,
! [F: complex > nat,X: complex,Y: complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_complex @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_643_monoD,axiom,
! [F: complex > int,X: complex,Y: complex] :
( ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_complex @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_644_monoD,axiom,
! [F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_645_monoD,axiom,
! [F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_646_monoD,axiom,
! [F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_647_monoD,axiom,
! [F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_648_monoD,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_649_monoD,axiom,
! [F: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_650_monoD,axiom,
! [F: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ top_top_set_int @ ord_less_eq_int @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_651_strict__mono__less,axiom,
! [F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ F )
=> ( ( ord_less_real @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_652_strict__mono__less,axiom,
! [F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_real @ ord_less_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_653_strict__mono__less,axiom,
! [F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ top_top_set_real @ ord_less_real @ ord_less_int @ F )
=> ( ( ord_less_int @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_654_strict__mono__less,axiom,
! [F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_nat @ ord_less_real @ F )
=> ( ( ord_less_real @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_655_strict__mono__less,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_656_strict__mono__less,axiom,
! [F: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_nat @ ord_less_int @ F )
=> ( ( ord_less_int @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_657_strict__mono__less,axiom,
! [F: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ top_top_set_int @ ord_less_int @ ord_less_real @ F )
=> ( ( ord_less_real @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_658_strict__mono__less,axiom,
! [F: int > nat,X: int,Y: int] :
( ( monotone_on_int_nat @ top_top_set_int @ ord_less_int @ ord_less_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_659_strict__mono__less,axiom,
! [F: int > int,X: int,Y: int] :
( ( monotone_on_int_int @ top_top_set_int @ ord_less_int @ ord_less_int @ F )
=> ( ( ord_less_int @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_660_strict__mono__eq,axiom,
! [F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_661_strict__mono__eq,axiom,
! [F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_real @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_662_strict__mono__eq,axiom,
! [F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ top_top_set_real @ ord_less_real @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_663_strict__mono__eq,axiom,
! [F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_nat @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_664_strict__mono__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_665_strict__mono__eq,axiom,
! [F: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_nat @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_666_strict__mono__eq,axiom,
! [F: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ top_top_set_int @ ord_less_int @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_667_strict__mono__eq,axiom,
! [F: int > nat,X: int,Y: int] :
( ( monotone_on_int_nat @ top_top_set_int @ ord_less_int @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_668_strict__mono__eq,axiom,
! [F: int > int,X: int,Y: int] :
( ( monotone_on_int_int @ top_top_set_int @ ord_less_int @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_669_strict__monoI,axiom,
! [F: complex > real] :
( ! [X3: complex,Y3: complex] :
( ( ord_less_complex @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_complex @ ord_less_real @ F ) ) ).
% strict_monoI
thf(fact_670_strict__monoI,axiom,
! [F: complex > nat] :
( ! [X3: complex,Y3: complex] :
( ( ord_less_complex @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_complex @ ord_less_nat @ F ) ) ).
% strict_monoI
thf(fact_671_strict__monoI,axiom,
! [F: complex > int] :
( ! [X3: complex,Y3: complex] :
( ( ord_less_complex @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_complex @ ord_less_int @ F ) ) ).
% strict_monoI
thf(fact_672_strict__monoI,axiom,
! [F: real > real] :
( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ F ) ) ).
% strict_monoI
thf(fact_673_strict__monoI,axiom,
! [F: real > nat] :
( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_real_nat @ top_top_set_real @ ord_less_real @ ord_less_nat @ F ) ) ).
% strict_monoI
thf(fact_674_strict__monoI,axiom,
! [F: real > int] :
( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_real_int @ top_top_set_real @ ord_less_real @ ord_less_int @ F ) ) ).
% strict_monoI
thf(fact_675_strict__monoI,axiom,
! [F: nat > real] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_real @ top_top_set_nat @ ord_less_nat @ ord_less_real @ F ) ) ).
% strict_monoI
thf(fact_676_strict__monoI,axiom,
! [F: nat > nat] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_monoI
thf(fact_677_strict__monoI,axiom,
! [F: nat > int] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_int @ top_top_set_nat @ ord_less_nat @ ord_less_int @ F ) ) ).
% strict_monoI
thf(fact_678_strict__monoI,axiom,
! [F: int > real] :
( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_int_real @ top_top_set_int @ ord_less_int @ ord_less_real @ F ) ) ).
% strict_monoI
thf(fact_679_strict__monoD,axiom,
! [F: complex > real,X: complex,Y: complex] :
( ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_complex @ ord_less_real @ F )
=> ( ( ord_less_complex @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_680_strict__monoD,axiom,
! [F: complex > nat,X: complex,Y: complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_complex @ ord_less_nat @ F )
=> ( ( ord_less_complex @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_681_strict__monoD,axiom,
! [F: complex > int,X: complex,Y: complex] :
( ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_complex @ ord_less_int @ F )
=> ( ( ord_less_complex @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_682_strict__monoD,axiom,
! [F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ F )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_683_strict__monoD,axiom,
! [F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_real @ ord_less_nat @ F )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_684_strict__monoD,axiom,
! [F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ top_top_set_real @ ord_less_real @ ord_less_int @ F )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_685_strict__monoD,axiom,
! [F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_nat @ ord_less_real @ F )
=> ( ( ord_less_nat @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_686_strict__monoD,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_687_strict__monoD,axiom,
! [F: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_nat @ ord_less_int @ F )
=> ( ( ord_less_nat @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_688_strict__monoD,axiom,
! [F: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ top_top_set_int @ ord_less_int @ ord_less_real @ F )
=> ( ( ord_less_int @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_689_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_690_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_691_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_692_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_693_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_694_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_695_mono__invE,axiom,
! [F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( ord_less_real @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_real @ X @ Y ) ) ) ).
% mono_invE
thf(fact_696_mono__invE,axiom,
! [F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_real @ X @ Y ) ) ) ).
% mono_invE
thf(fact_697_mono__invE,axiom,
! [F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_int @ F )
=> ( ( ord_less_int @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_real @ X @ Y ) ) ) ).
% mono_invE
thf(fact_698_mono__invE,axiom,
! [F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_real @ F )
=> ( ( ord_less_real @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% mono_invE
thf(fact_699_mono__invE,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% mono_invE
thf(fact_700_mono__invE,axiom,
! [F: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_int @ F )
=> ( ( ord_less_int @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% mono_invE
thf(fact_701_mono__invE,axiom,
! [F: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ top_top_set_int @ ord_less_eq_int @ ord_less_eq_real @ F )
=> ( ( ord_less_real @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_int @ X @ Y ) ) ) ).
% mono_invE
thf(fact_702_mono__invE,axiom,
! [F: int > nat,X: int,Y: int] :
( ( monotone_on_int_nat @ top_top_set_int @ ord_less_eq_int @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_int @ X @ Y ) ) ) ).
% mono_invE
thf(fact_703_mono__invE,axiom,
! [F: int > int,X: int,Y: int] :
( ( monotone_on_int_int @ top_top_set_int @ ord_less_eq_int @ ord_less_eq_int @ F )
=> ( ( ord_less_int @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_int @ X @ Y ) ) ) ).
% mono_invE
thf(fact_704_strict__mono__mono,axiom,
! [F: complex > real] :
( ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_complex @ ord_less_real @ F )
=> ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_real @ F ) ) ).
% strict_mono_mono
thf(fact_705_strict__mono__mono,axiom,
! [F: complex > nat] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_complex @ ord_less_nat @ F )
=> ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_nat @ F ) ) ).
% strict_mono_mono
thf(fact_706_strict__mono__mono,axiom,
! [F: complex > int] :
( ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_complex @ ord_less_int @ F )
=> ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_int @ F ) ) ).
% strict_mono_mono
thf(fact_707_strict__mono__mono,axiom,
! [F: real > real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ F )
=> ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% strict_mono_mono
thf(fact_708_strict__mono__mono,axiom,
! [F: real > nat] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_real @ ord_less_nat @ F )
=> ( monotone_on_real_nat @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ).
% strict_mono_mono
thf(fact_709_strict__mono__mono,axiom,
! [F: real > int] :
( ( monotone_on_real_int @ top_top_set_real @ ord_less_real @ ord_less_int @ F )
=> ( monotone_on_real_int @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ).
% strict_mono_mono
thf(fact_710_strict__mono__mono,axiom,
! [F: nat > real] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_nat @ ord_less_real @ F )
=> ( monotone_on_nat_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ).
% strict_mono_mono
thf(fact_711_strict__mono__mono,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_mono
thf(fact_712_strict__mono__mono,axiom,
! [F: nat > int] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_nat @ ord_less_int @ F )
=> ( monotone_on_nat_int @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ).
% strict_mono_mono
thf(fact_713_strict__mono__mono,axiom,
! [F: int > real] :
( ( monotone_on_int_real @ top_top_set_int @ ord_less_int @ ord_less_real @ F )
=> ( monotone_on_int_real @ top_top_set_int @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ).
% strict_mono_mono
thf(fact_714_mono__strict__invE,axiom,
! [F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( ord_less_real @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ) ).
% mono_strict_invE
thf(fact_715_mono__strict__invE,axiom,
! [F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ) ).
% mono_strict_invE
thf(fact_716_mono__strict__invE,axiom,
! [F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_int @ F )
=> ( ( ord_less_int @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ) ).
% mono_strict_invE
thf(fact_717_mono__strict__invE,axiom,
! [F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_real @ F )
=> ( ( ord_less_real @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_nat @ X @ Y ) ) ) ).
% mono_strict_invE
thf(fact_718_mono__strict__invE,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_nat @ X @ Y ) ) ) ).
% mono_strict_invE
thf(fact_719_mono__strict__invE,axiom,
! [F: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_int @ F )
=> ( ( ord_less_int @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_nat @ X @ Y ) ) ) ).
% mono_strict_invE
thf(fact_720_mono__strict__invE,axiom,
! [F: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ top_top_set_int @ ord_less_eq_int @ ord_less_eq_real @ F )
=> ( ( ord_less_real @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_int @ X @ Y ) ) ) ).
% mono_strict_invE
thf(fact_721_mono__strict__invE,axiom,
! [F: int > nat,X: int,Y: int] :
( ( monotone_on_int_nat @ top_top_set_int @ ord_less_eq_int @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_int @ X @ Y ) ) ) ).
% mono_strict_invE
thf(fact_722_mono__strict__invE,axiom,
! [F: int > int,X: int,Y: int] :
( ( monotone_on_int_int @ top_top_set_int @ ord_less_eq_int @ ord_less_eq_int @ F )
=> ( ( ord_less_int @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_int @ X @ Y ) ) ) ).
% mono_strict_invE
thf(fact_723_strict__mono__less__eq,axiom,
! [F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ F )
=> ( ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_724_strict__mono__less__eq,axiom,
! [F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_nat @ ord_less_real @ F )
=> ( ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_725_strict__mono__less__eq,axiom,
! [F: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ top_top_set_int @ ord_less_int @ ord_less_real @ F )
=> ( ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_726_strict__mono__less__eq,axiom,
! [F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_real @ ord_less_nat @ F )
=> ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_727_strict__mono__less__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_728_strict__mono__less__eq,axiom,
! [F: int > nat,X: int,Y: int] :
( ( monotone_on_int_nat @ top_top_set_int @ ord_less_int @ ord_less_nat @ F )
=> ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_729_strict__mono__less__eq,axiom,
! [F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ top_top_set_real @ ord_less_real @ ord_less_int @ F )
=> ( ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_730_strict__mono__less__eq,axiom,
! [F: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_nat @ ord_less_int @ F )
=> ( ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_731_strict__mono__less__eq,axiom,
! [F: int > int,X: int,Y: int] :
( ( monotone_on_int_int @ top_top_set_int @ ord_less_int @ ord_less_int @ F )
=> ( ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_732_not__UNIV__le__Ici,axiom,
! [L: real] :
~ ( ord_less_eq_set_real @ top_top_set_real @ ( set_ord_atLeast_real @ L ) ) ).
% not_UNIV_le_Ici
thf(fact_733_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_734_monotone__onD,axiom,
! [A2: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A2 @ Orda @ Ordb @ F )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% monotone_onD
thf(fact_735_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_736_monotone__onI,axiom,
! [A2: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat] :
( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( Orda @ X3 @ Y3 )
=> ( Ordb @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ Orda @ Ordb @ F ) ) ).
% monotone_onI
thf(fact_737_monotone__on__def,axiom,
( monoto4017252874604999745l_real
= ( ^ [A5: set_real,Orda2: real > real > $o,Ordb2: real > real > $o,F2: real > real] :
! [X4: real] :
( ( member_real @ X4 @ A5 )
=> ! [Y5: real] :
( ( member_real @ Y5 @ A5 )
=> ( ( Orda2 @ X4 @ Y5 )
=> ( Ordb2 @ ( F2 @ X4 ) @ ( F2 @ Y5 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_738_monotone__on__def,axiom,
( monotone_on_nat_nat
= ( ^ [A5: set_nat,Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F2: nat > nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A5 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ A5 )
=> ( ( Orda2 @ X4 @ Y5 )
=> ( Ordb2 @ ( F2 @ X4 ) @ ( F2 @ Y5 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_739_mono__onI,axiom,
! [A2: set_real,F: real > real] :
( ! [R3: real,S3: real] :
( ( member_real @ R3 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( ord_less_eq_real @ R3 @ S3 )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A2 @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_740_mono__onI,axiom,
! [A2: set_real,F: real > nat] :
( ! [R3: real,S3: real] :
( ( member_real @ R3 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( ord_less_eq_real @ R3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A2 @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_741_mono__onI,axiom,
! [A2: set_real,F: real > int] :
( ! [R3: real,S3: real] :
( ( member_real @ R3 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( ord_less_eq_real @ R3 @ S3 )
=> ( ord_less_eq_int @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_int @ A2 @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_742_mono__onI,axiom,
! [A2: set_nat,F: nat > real] :
( ! [R3: nat,S3: nat] :
( ( member_nat @ R3 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_eq_nat @ R3 @ S3 )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_real @ A2 @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_743_mono__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [R3: nat,S3: nat] :
( ( member_nat @ R3 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_eq_nat @ R3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_744_mono__onI,axiom,
! [A2: set_nat,F: nat > int] :
( ! [R3: nat,S3: nat] :
( ( member_nat @ R3 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_eq_nat @ R3 @ S3 )
=> ( ord_less_eq_int @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_int @ A2 @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_745_mono__onI,axiom,
! [A2: set_int,F: int > real] :
( ! [R3: int,S3: int] :
( ( member_int @ R3 @ A2 )
=> ( ( member_int @ S3 @ A2 )
=> ( ( ord_less_eq_int @ R3 @ S3 )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_real @ A2 @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_746_mono__onI,axiom,
! [A2: set_int,F: int > nat] :
( ! [R3: int,S3: int] :
( ( member_int @ R3 @ A2 )
=> ( ( member_int @ S3 @ A2 )
=> ( ( ord_less_eq_int @ R3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_nat @ A2 @ ord_less_eq_int @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_747_mono__onI,axiom,
! [A2: set_int,F: int > int] :
( ! [R3: int,S3: int] :
( ( member_int @ R3 @ A2 )
=> ( ( member_int @ S3 @ A2 )
=> ( ( ord_less_eq_int @ R3 @ S3 )
=> ( ord_less_eq_int @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_int @ A2 @ ord_less_eq_int @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_748_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_749_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_750_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_751_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_752_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_753_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_754_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_755_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_756_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_757_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 )
= ( ! [R4: nat,S4: nat] :
( ( ( member_nat @ R4 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less_eq @ R4 @ S4 ) )
=> ( ord_less_eq_real @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_758_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 )
= ( ! [R4: real,S4: real] :
( ( ( member_real @ R4 @ A2 )
& ( member_real @ S4 @ A2 )
& ( Less_eq @ R4 @ S4 ) )
=> ( ord_less_eq_real @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_759_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 )
= ( ! [R4: real,S4: real] :
( ( ( member_real @ R4 @ A2 )
& ( member_real @ S4 @ A2 )
& ( Less_eq @ R4 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_760_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 )
= ( ! [R4: nat,S4: nat] :
( ( ( member_nat @ R4 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less_eq @ R4 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_761_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 )
= ( ! [R4: real,S4: real] :
( ( ( member_real @ R4 @ A2 )
& ( member_real @ S4 @ A2 )
& ( Less_eq @ R4 @ S4 ) )
=> ( ord_less_eq_int @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_762_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 )
= ( ! [R4: nat,S4: nat] :
( ( ( member_nat @ R4 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less_eq @ R4 @ S4 ) )
=> ( ord_less_eq_int @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_763_ord_Omono__onI,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > real] :
( ! [R3: nat,S3: nat] :
( ( member_nat @ R3 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less_eq @ R3 @ S3 )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_real @ A2 @ Less_eq @ ord_less_eq_real @ F ) ) ).
% ord.mono_onI
thf(fact_764_ord_Omono__onI,axiom,
! [A2: set_real,Less_eq: real > real > $o,F: real > real] :
( ! [R3: real,S3: real] :
( ( member_real @ R3 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( Less_eq @ R3 @ S3 )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A2 @ Less_eq @ ord_less_eq_real @ F ) ) ).
% ord.mono_onI
thf(fact_765_ord_Omono__onI,axiom,
! [A2: set_real,Less_eq: real > real > $o,F: real > nat] :
( ! [R3: real,S3: real] :
( ( member_real @ R3 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( Less_eq @ R3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_766_ord_Omono__onI,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ! [R3: nat,S3: nat] :
( ( member_nat @ R3 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less_eq @ R3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_767_ord_Omono__onI,axiom,
! [A2: set_real,Less_eq: real > real > $o,F: real > int] :
( ! [R3: real,S3: real] :
( ( member_real @ R3 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( Less_eq @ R3 @ S3 )
=> ( ord_less_eq_int @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_int @ A2 @ Less_eq @ ord_less_eq_int @ F ) ) ).
% ord.mono_onI
thf(fact_768_ord_Omono__onI,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > int] :
( ! [R3: nat,S3: nat] :
( ( member_nat @ R3 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less_eq @ R3 @ S3 )
=> ( ord_less_eq_int @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_int @ A2 @ Less_eq @ ord_less_eq_int @ F ) ) ).
% ord.mono_onI
thf(fact_769_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_770_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_771_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_772_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_773_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_774_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_775_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_776_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_777_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_778_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_779_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_780_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_781_ord_Ostrict__mono__onI,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > real] :
( ! [R3: nat,S3: nat] :
( ( member_nat @ R3 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less @ R3 @ S3 )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_real @ A2 @ Less @ ord_less_real @ F ) ) ).
% ord.strict_mono_onI
thf(fact_782_ord_Ostrict__mono__onI,axiom,
! [A2: set_real,Less: real > real > $o,F: real > real] :
( ! [R3: real,S3: real] :
( ( member_real @ R3 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( Less @ R3 @ S3 )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A2 @ Less @ ord_less_real @ F ) ) ).
% ord.strict_mono_onI
thf(fact_783_ord_Ostrict__mono__onI,axiom,
! [A2: set_real,Less: real > real > $o,F: real > nat] :
( ! [R3: real,S3: real] :
( ( member_real @ R3 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( Less @ R3 @ S3 )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A2 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_784_ord_Ostrict__mono__onI,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ! [R3: nat,S3: nat] :
( ( member_nat @ R3 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less @ R3 @ S3 )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_785_ord_Ostrict__mono__onI,axiom,
! [A2: set_real,Less: real > real > $o,F: real > int] :
( ! [R3: real,S3: real] :
( ( member_real @ R3 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( Less @ R3 @ S3 )
=> ( ord_less_int @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_int @ A2 @ Less @ ord_less_int @ F ) ) ).
% ord.strict_mono_onI
thf(fact_786_ord_Ostrict__mono__onI,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > int] :
( ! [R3: nat,S3: nat] :
( ( member_nat @ R3 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less @ R3 @ S3 )
=> ( ord_less_int @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_int @ A2 @ Less @ ord_less_int @ F ) ) ).
% ord.strict_mono_onI
thf(fact_787_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 )
= ( ! [R4: nat,S4: nat] :
( ( ( member_nat @ R4 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less @ R4 @ S4 ) )
=> ( ord_less_real @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_788_ord_Ostrict__mono__on__def,axiom,
! [A2: set_real,Less: real > real > $o,F: real > real] :
( ( monoto4017252874604999745l_real @ A2 @ Less @ ord_less_real @ F )
= ( ! [R4: real,S4: real] :
( ( ( member_real @ R4 @ A2 )
& ( member_real @ S4 @ A2 )
& ( Less @ R4 @ S4 ) )
=> ( ord_less_real @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_789_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 )
= ( ! [R4: real,S4: real] :
( ( ( member_real @ R4 @ A2 )
& ( member_real @ S4 @ A2 )
& ( Less @ R4 @ S4 ) )
=> ( ord_less_nat @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_790_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 )
= ( ! [R4: nat,S4: nat] :
( ( ( member_nat @ R4 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less @ R4 @ S4 ) )
=> ( ord_less_nat @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_791_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 )
= ( ! [R4: real,S4: real] :
( ( ( member_real @ R4 @ A2 )
& ( member_real @ S4 @ A2 )
& ( Less @ R4 @ S4 ) )
=> ( ord_less_int @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_792_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 )
= ( ! [R4: nat,S4: nat] :
( ( ( member_nat @ R4 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less @ R4 @ S4 ) )
=> ( ord_less_int @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_793_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_794_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_795_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_796_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_797_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_798_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_799_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_800_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_801_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_802_strict__mono__onI,axiom,
! [A2: set_real,F: real > real] :
( ! [R3: real,S3: real] :
( ( member_real @ R3 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( ord_less_real @ R3 @ S3 )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A2 @ ord_less_real @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_803_strict__mono__onI,axiom,
! [A2: set_real,F: real > nat] :
( ! [R3: real,S3: real] :
( ( member_real @ R3 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( ord_less_real @ R3 @ S3 )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A2 @ ord_less_real @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_804_strict__mono__onI,axiom,
! [A2: set_real,F: real > int] :
( ! [R3: real,S3: real] :
( ( member_real @ R3 @ A2 )
=> ( ( member_real @ S3 @ A2 )
=> ( ( ord_less_real @ R3 @ S3 )
=> ( ord_less_int @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_int @ A2 @ ord_less_real @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_805_strict__mono__onI,axiom,
! [A2: set_nat,F: nat > real] :
( ! [R3: nat,S3: nat] :
( ( member_nat @ R3 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_nat @ R3 @ S3 )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_real @ A2 @ ord_less_nat @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_806_strict__mono__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [R3: nat,S3: nat] :
( ( member_nat @ R3 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_nat @ R3 @ S3 )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_807_strict__mono__onI,axiom,
! [A2: set_nat,F: nat > int] :
( ! [R3: nat,S3: nat] :
( ( member_nat @ R3 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_nat @ R3 @ S3 )
=> ( ord_less_int @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_int @ A2 @ ord_less_nat @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_808_strict__mono__onI,axiom,
! [A2: set_int,F: int > real] :
( ! [R3: int,S3: int] :
( ( member_int @ R3 @ A2 )
=> ( ( member_int @ S3 @ A2 )
=> ( ( ord_less_int @ R3 @ S3 )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_real @ A2 @ ord_less_int @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_809_strict__mono__onI,axiom,
! [A2: set_int,F: int > nat] :
( ! [R3: int,S3: int] :
( ( member_int @ R3 @ A2 )
=> ( ( member_int @ S3 @ A2 )
=> ( ( ord_less_int @ R3 @ S3 )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_nat @ A2 @ ord_less_int @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_810_strict__mono__onI,axiom,
! [A2: set_int,F: int > int] :
( ! [R3: int,S3: int] :
( ( member_int @ R3 @ A2 )
=> ( ( member_int @ S3 @ A2 )
=> ( ( ord_less_int @ R3 @ S3 )
=> ( ord_less_int @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_int @ A2 @ ord_less_int @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_811_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_812_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_813_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_814_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_815_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_816_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_817_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_818_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_819_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_820_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_821_monotone__on__subset,axiom,
! [A2: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,B4: set_nat] :
( ( monotone_on_nat_nat @ A2 @ Orda @ Ordb @ F )
=> ( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( monotone_on_nat_nat @ B4 @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_822_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_823_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_824_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_825_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_826_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_827_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_828_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_829_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_830_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_831_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_832_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_833_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_834_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_835_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_836_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_837_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_838_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_839_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_840_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_841_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_842_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_843_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_844_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_845_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_846_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_847_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_848_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_849_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_850_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_851_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_852_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_853_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_854_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_855_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_856_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_857_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_858_an__less__del,axiom,
ord_less_real @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) @ ( del @ ( divide_divide_real @ epsilon @ a ) ) ).
% an_less_del
thf(fact_859_fim,axiom,
( ( image_real_real @ f @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
= ( set_or1222579329274155063t_real @ zero_zero_real @ b ) ) ).
% fim
thf(fact_860_f2__def,axiom,
( f2
= ( comp_real_real_real @ f @ upper ) ) ).
% f2_def
thf(fact_861_f1__def,axiom,
( f1
= ( comp_real_real_real @ f @ lower ) ) ).
% f1_def
thf(fact_862_intgb__g,axiom,
hensto5963834015518849588l_real @ g @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) ).
% intgb_g
thf(fact_863__092_060open_062uniformly__continuous__on_A_1230_O_Oa_125_Af_092_060close_062,axiom,
topolo8845477368217174713l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) @ f ).
% \<open>uniformly_continuous_on {0..a} f\<close>
thf(fact_864_atLeast__0,axiom,
( ( set_ord_atLeast_nat @ zero_zero_nat )
= top_top_set_nat ) ).
% atLeast_0
thf(fact_865_comp__apply,axiom,
( comp_real_real_real
= ( ^ [F2: real > real,G2: real > real,X4: real] : ( F2 @ ( G2 @ X4 ) ) ) ) ).
% comp_apply
thf(fact_866_intgb__f,axiom,
hensto5963834015518849588l_real @ f @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ).
% intgb_f
thf(fact_867_a__seg__def,axiom,
( a_seg
= ( ^ [U2: real] : ( divide_divide_real @ ( times_times_real @ U2 @ a ) @ ( semiri5074537144036343181t_real @ n ) ) ) ) ).
% a_seg_def
thf(fact_868_comp__def,axiom,
( comp_real_real_real
= ( ^ [F2: real > real,G2: real > real,X4: real] : ( F2 @ ( G2 @ X4 ) ) ) ) ).
% comp_def
thf(fact_869_comp__surj,axiom,
! [F: real > real,G: real > real] :
( ( ( image_real_real @ F @ top_top_set_real )
= top_top_set_real )
=> ( ( ( image_real_real @ G @ top_top_set_real )
= top_top_set_real )
=> ( ( image_real_real @ ( comp_real_real_real @ G @ F ) @ top_top_set_real )
= top_top_set_real ) ) ) ).
% comp_surj
thf(fact_870_comp__surj,axiom,
! [F: real > real,G: real > nat] :
( ( ( image_real_real @ F @ top_top_set_real )
= top_top_set_real )
=> ( ( ( image_real_nat @ G @ top_top_set_real )
= top_top_set_nat )
=> ( ( image_real_nat @ ( comp_real_nat_real @ G @ F ) @ top_top_set_real )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_871_comp__surj,axiom,
! [F: real > real,G: real > complex] :
( ( ( image_real_real @ F @ top_top_set_real )
= top_top_set_real )
=> ( ( ( image_real_complex @ G @ top_top_set_real )
= top_top_set_complex )
=> ( ( image_real_complex @ ( comp_r1968866223832618731x_real @ G @ F ) @ top_top_set_real )
= top_top_set_complex ) ) ) ).
% comp_surj
thf(fact_872_comp__surj,axiom,
! [F: real > nat,G: nat > int] :
( ( ( image_real_nat @ F @ top_top_set_real )
= top_top_set_nat )
=> ( ( ( image_nat_int @ G @ top_top_set_nat )
= top_top_set_int )
=> ( ( image_real_int @ ( comp_nat_int_real @ G @ F ) @ top_top_set_real )
= top_top_set_int ) ) ) ).
% comp_surj
thf(fact_873_comp__surj,axiom,
! [F: real > nat,G: nat > real] :
( ( ( image_real_nat @ F @ top_top_set_real )
= top_top_set_nat )
=> ( ( ( image_nat_real @ G @ top_top_set_nat )
= top_top_set_real )
=> ( ( image_real_real @ ( comp_nat_real_real @ G @ F ) @ top_top_set_real )
= top_top_set_real ) ) ) ).
% comp_surj
thf(fact_874_comp__surj,axiom,
! [F: real > nat,G: nat > nat] :
( ( ( image_real_nat @ F @ top_top_set_real )
= top_top_set_nat )
=> ( ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_real_nat @ ( comp_nat_nat_real @ G @ F ) @ top_top_set_real )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_875_comp__surj,axiom,
! [F: real > nat,G: nat > complex] :
( ( ( image_real_nat @ F @ top_top_set_real )
= top_top_set_nat )
=> ( ( ( image_nat_complex @ G @ top_top_set_nat )
= top_top_set_complex )
=> ( ( image_real_complex @ ( comp_n4215249288434654095x_real @ G @ F ) @ top_top_set_real )
= top_top_set_complex ) ) ) ).
% comp_surj
thf(fact_876_comp__surj,axiom,
! [F: real > complex,G: complex > real] :
( ( ( image_real_complex @ F @ top_top_set_real )
= top_top_set_complex )
=> ( ( ( image_complex_real @ G @ top_top_set_complex )
= top_top_set_real )
=> ( ( image_real_real @ ( comp_c3333796836230738283l_real @ G @ F ) @ top_top_set_real )
= top_top_set_real ) ) ) ).
% comp_surj
thf(fact_877_comp__surj,axiom,
! [F: real > complex,G: complex > nat] :
( ( ( image_real_complex @ F @ top_top_set_real )
= top_top_set_complex )
=> ( ( ( image_complex_nat @ G @ top_top_set_complex )
= top_top_set_nat )
=> ( ( image_real_nat @ ( comp_c3423117485846644111t_real @ G @ F ) @ top_top_set_real )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_878_comp__surj,axiom,
! [F: real > complex,G: complex > complex] :
( ( ( image_real_complex @ F @ top_top_set_real )
= top_top_set_complex )
=> ( ( ( image_1468599708987790691omplex @ G @ top_top_set_complex )
= top_top_set_complex )
=> ( ( image_real_complex @ ( comp_c2117349707075585901x_real @ G @ F ) @ top_top_set_real )
= top_top_set_complex ) ) ) ).
% comp_surj
thf(fact_879_comp__assoc,axiom,
! [F: real > real,G: real > real,H: real > real] :
( ( comp_real_real_real @ ( comp_real_real_real @ F @ G ) @ H )
= ( comp_real_real_real @ F @ ( comp_real_real_real @ G @ H ) ) ) ).
% comp_assoc
thf(fact_880_image__comp,axiom,
! [F: int > nat,G: nat > int,R: set_nat] :
( ( image_int_nat @ F @ ( image_nat_int @ G @ R ) )
= ( image_nat_nat @ ( comp_int_nat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_881_image__comp,axiom,
! [F: int > int,G: nat > int,R: set_nat] :
( ( image_int_int @ F @ ( image_nat_int @ G @ R ) )
= ( image_nat_int @ ( comp_int_int_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_882_image__comp,axiom,
! [F: int > real,G: nat > int,R: set_nat] :
( ( image_int_real @ F @ ( image_nat_int @ G @ R ) )
= ( image_nat_real @ ( comp_int_real_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_883_image__comp,axiom,
! [F: int > complex,G: nat > int,R: set_nat] :
( ( image_int_complex @ F @ ( image_nat_int @ G @ R ) )
= ( image_nat_complex @ ( comp_int_complex_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_884_image__comp,axiom,
! [F: real > nat,G: nat > real,R: set_nat] :
( ( image_real_nat @ F @ ( image_nat_real @ G @ R ) )
= ( image_nat_nat @ ( comp_real_nat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_885_image__comp,axiom,
! [F: real > int,G: nat > real,R: set_nat] :
( ( image_real_int @ F @ ( image_nat_real @ G @ R ) )
= ( image_nat_int @ ( comp_real_int_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_886_image__comp,axiom,
! [F: real > complex,G: nat > real,R: set_nat] :
( ( image_real_complex @ F @ ( image_nat_real @ G @ R ) )
= ( image_nat_complex @ ( comp_r1225911664865567631ex_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_887_image__comp,axiom,
! [F: complex > nat,G: nat > complex,R: set_nat] :
( ( image_complex_nat @ F @ ( image_nat_complex @ G @ R ) )
= ( image_nat_nat @ ( comp_complex_nat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_888_image__comp,axiom,
! [F: complex > int,G: nat > complex,R: set_nat] :
( ( image_complex_int @ F @ ( image_nat_complex @ G @ R ) )
= ( image_nat_int @ ( comp_complex_int_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_889_image__comp,axiom,
! [F: complex > real,G: nat > complex,R: set_nat] :
( ( image_complex_real @ F @ ( image_nat_complex @ G @ R ) )
= ( image_nat_real @ ( comp_c7990426058975542799al_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_890_comp__eq__dest,axiom,
! [A: real > real,B: real > real,C: real > real,D: real > real,V: real] :
( ( ( comp_real_real_real @ A @ B )
= ( comp_real_real_real @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_891_comp__eq__elim,axiom,
! [A: real > real,B: real > real,C: real > real,D: real > real] :
( ( ( comp_real_real_real @ A @ B )
= ( comp_real_real_real @ C @ D ) )
=> ! [V2: real] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_892_comp__eq__dest__lhs,axiom,
! [A: real > real,B: real > real,C: real > real,V: real] :
( ( ( comp_real_real_real @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_893_image__eq__imp__comp,axiom,
! [F: real > nat,A2: set_real,G: nat > nat,B4: set_nat,H: nat > real] :
( ( ( image_real_nat @ F @ A2 )
= ( image_nat_nat @ G @ B4 ) )
=> ( ( image_real_real @ ( comp_nat_real_real @ H @ F ) @ A2 )
= ( image_nat_real @ ( comp_nat_real_nat @ H @ G ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_894_image__eq__imp__comp,axiom,
! [F: real > int,A2: set_real,G: nat > int,B4: set_nat,H: int > real] :
( ( ( image_real_int @ F @ A2 )
= ( image_nat_int @ G @ B4 ) )
=> ( ( image_real_real @ ( comp_int_real_real @ H @ F ) @ A2 )
= ( image_nat_real @ ( comp_int_real_nat @ H @ G ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_895_image__eq__imp__comp,axiom,
! [F: real > complex,A2: set_real,G: nat > complex,B4: set_nat,H: complex > real] :
( ( ( image_real_complex @ F @ A2 )
= ( image_nat_complex @ G @ B4 ) )
=> ( ( image_real_real @ ( comp_c3333796836230738283l_real @ H @ F ) @ A2 )
= ( image_nat_real @ ( comp_c7990426058975542799al_nat @ H @ G ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_896_image__eq__imp__comp,axiom,
! [F: real > real,A2: set_real,G: real > real,B4: set_real,H: real > real] :
( ( ( image_real_real @ F @ A2 )
= ( image_real_real @ G @ B4 ) )
=> ( ( image_real_real @ ( comp_real_real_real @ H @ F ) @ A2 )
= ( image_real_real @ ( comp_real_real_real @ H @ G ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_897_image__eq__imp__comp,axiom,
! [F: real > real,A2: set_real,G: nat > real,B4: set_nat,H: real > nat] :
( ( ( image_real_real @ F @ A2 )
= ( image_nat_real @ G @ B4 ) )
=> ( ( image_real_nat @ ( comp_real_nat_real @ H @ F ) @ A2 )
= ( image_nat_nat @ ( comp_real_nat_nat @ H @ G ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_898_image__eq__imp__comp,axiom,
! [F: real > real,A2: set_real,G: nat > real,B4: set_nat,H: real > int] :
( ( ( image_real_real @ F @ A2 )
= ( image_nat_real @ G @ B4 ) )
=> ( ( image_real_int @ ( comp_real_int_real @ H @ F ) @ A2 )
= ( image_nat_int @ ( comp_real_int_nat @ H @ G ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_899_image__eq__imp__comp,axiom,
! [F: real > real,A2: set_real,G: nat > real,B4: set_nat,H: real > complex] :
( ( ( image_real_real @ F @ A2 )
= ( image_nat_real @ G @ B4 ) )
=> ( ( image_real_complex @ ( comp_r1968866223832618731x_real @ H @ F ) @ A2 )
= ( image_nat_complex @ ( comp_r1225911664865567631ex_nat @ H @ G ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_900_image__eq__imp__comp,axiom,
! [F: real > real,A2: set_real,G: nat > real,B4: set_nat,H: real > real] :
( ( ( image_real_real @ F @ A2 )
= ( image_nat_real @ G @ B4 ) )
=> ( ( image_real_real @ ( comp_real_real_real @ H @ F ) @ A2 )
= ( image_nat_real @ ( comp_real_real_nat @ H @ G ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_901_image__eq__imp__comp,axiom,
! [F: nat > nat,A2: set_nat,G: real > nat,B4: set_real,H: nat > real] :
( ( ( image_nat_nat @ F @ A2 )
= ( image_real_nat @ G @ B4 ) )
=> ( ( image_nat_real @ ( comp_nat_real_nat @ H @ F ) @ A2 )
= ( image_real_real @ ( comp_nat_real_real @ H @ G ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_902_image__eq__imp__comp,axiom,
! [F: nat > nat,A2: set_nat,G: nat > nat,B4: set_nat,H: nat > nat] :
( ( ( image_nat_nat @ F @ A2 )
= ( image_nat_nat @ G @ B4 ) )
=> ( ( image_nat_nat @ ( comp_nat_nat_nat @ H @ F ) @ A2 )
= ( image_nat_nat @ ( comp_nat_nat_nat @ H @ G ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_903_monotone__on__o,axiom,
! [A2: set_int,Orda: int > int > $o,Ordb: nat > nat > $o,F: int > nat,B4: set_nat,Ordc: nat > nat > $o,G: nat > int] :
( ( monotone_on_int_nat @ A2 @ Orda @ Ordb @ F )
=> ( ( monotone_on_nat_int @ B4 @ Ordc @ Orda @ G )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ G @ B4 ) @ A2 )
=> ( monotone_on_nat_nat @ B4 @ Ordc @ Ordb @ ( comp_int_nat_nat @ F @ G ) ) ) ) ) ).
% monotone_on_o
thf(fact_904_monotone__on__o,axiom,
! [A2: set_real,Orda: real > real > $o,Ordb: nat > nat > $o,F: real > nat,B4: set_nat,Ordc: nat > nat > $o,G: nat > real] :
( ( monotone_on_real_nat @ A2 @ Orda @ Ordb @ F )
=> ( ( monotone_on_nat_real @ B4 @ Ordc @ Orda @ G )
=> ( ( ord_less_eq_set_real @ ( image_nat_real @ G @ B4 ) @ A2 )
=> ( monotone_on_nat_nat @ B4 @ Ordc @ Ordb @ ( comp_real_nat_nat @ F @ G ) ) ) ) ) ).
% monotone_on_o
thf(fact_905_monotone__on__o,axiom,
! [A2: set_complex,Orda: complex > complex > $o,Ordb: nat > nat > $o,F: complex > nat,B4: set_nat,Ordc: nat > nat > $o,G: nat > complex] :
( ( monoto2406513391651152359ex_nat @ A2 @ Orda @ Ordb @ F )
=> ( ( monoto8010102104071190503omplex @ B4 @ Ordc @ Orda @ G )
=> ( ( ord_le211207098394363844omplex @ ( image_nat_complex @ G @ B4 ) @ A2 )
=> ( monotone_on_nat_nat @ B4 @ Ordc @ Ordb @ ( comp_complex_nat_nat @ F @ G ) ) ) ) ) ).
% monotone_on_o
thf(fact_906_monotone__on__o,axiom,
! [A2: set_real,Orda: real > real > $o,Ordb: real > real > $o,F: real > real,B4: set_nat,Ordc: nat > nat > $o,G: nat > real] :
( ( monoto4017252874604999745l_real @ A2 @ Orda @ Ordb @ F )
=> ( ( monotone_on_nat_real @ B4 @ Ordc @ Orda @ G )
=> ( ( ord_less_eq_set_real @ ( image_nat_real @ G @ B4 ) @ A2 )
=> ( monotone_on_nat_real @ B4 @ Ordc @ Ordb @ ( comp_real_real_nat @ F @ G ) ) ) ) ) ).
% monotone_on_o
thf(fact_907_monotone__on__o,axiom,
! [A2: set_real,Orda: real > real > $o,Ordb: real > real > $o,F: real > real,B4: set_real,Ordc: real > real > $o,G: real > real] :
( ( monoto4017252874604999745l_real @ A2 @ Orda @ Ordb @ F )
=> ( ( monoto4017252874604999745l_real @ B4 @ Ordc @ Orda @ G )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ G @ B4 ) @ A2 )
=> ( monoto4017252874604999745l_real @ B4 @ Ordc @ Ordb @ ( comp_real_real_real @ F @ G ) ) ) ) ) ).
% monotone_on_o
thf(fact_908_monotone__on__o,axiom,
! [A2: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,B4: set_nat,Ordc: nat > nat > $o,G: nat > nat] :
( ( monotone_on_nat_nat @ A2 @ Orda @ Ordb @ F )
=> ( ( monotone_on_nat_nat @ B4 @ Ordc @ Orda @ G )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ B4 ) @ A2 )
=> ( monotone_on_nat_nat @ B4 @ Ordc @ Ordb @ ( comp_nat_nat_nat @ F @ G ) ) ) ) ) ).
% monotone_on_o
thf(fact_909_surj__def,axiom,
! [F: real > real] :
( ( ( image_real_real @ F @ top_top_set_real )
= top_top_set_real )
= ( ! [Y5: real] :
? [X4: real] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_910_surj__def,axiom,
! [F: real > nat] :
( ( ( image_real_nat @ F @ top_top_set_real )
= top_top_set_nat )
= ( ! [Y5: nat] :
? [X4: real] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_911_surj__def,axiom,
! [F: real > complex] :
( ( ( image_real_complex @ F @ top_top_set_real )
= top_top_set_complex )
= ( ! [Y5: complex] :
? [X4: real] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_912_surj__def,axiom,
! [F: nat > int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
= ( ! [Y5: int] :
? [X4: nat] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_913_surj__def,axiom,
! [F: nat > real] :
( ( ( image_nat_real @ F @ top_top_set_nat )
= top_top_set_real )
= ( ! [Y5: real] :
? [X4: nat] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_914_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y5: nat] :
? [X4: nat] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_915_surj__def,axiom,
! [F: nat > complex] :
( ( ( image_nat_complex @ F @ top_top_set_nat )
= top_top_set_complex )
= ( ! [Y5: complex] :
? [X4: nat] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_916_surj__def,axiom,
! [F: complex > real] :
( ( ( image_complex_real @ F @ top_top_set_complex )
= top_top_set_real )
= ( ! [Y5: real] :
? [X4: complex] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_917_surj__def,axiom,
! [F: complex > nat] :
( ( ( image_complex_nat @ F @ top_top_set_complex )
= top_top_set_nat )
= ( ! [Y5: nat] :
? [X4: complex] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_918_surj__def,axiom,
! [F: complex > complex] :
( ( ( image_1468599708987790691omplex @ F @ top_top_set_complex )
= top_top_set_complex )
= ( ! [Y5: complex] :
? [X4: complex] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_919_surjI,axiom,
! [G: real > real,F: real > real] :
( ! [X3: real] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_real_real @ G @ top_top_set_real )
= top_top_set_real ) ) ).
% surjI
thf(fact_920_surjI,axiom,
! [G: real > nat,F: nat > real] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_real_nat @ G @ top_top_set_real )
= top_top_set_nat ) ) ).
% surjI
thf(fact_921_surjI,axiom,
! [G: real > complex,F: complex > real] :
( ! [X3: complex] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_real_complex @ G @ top_top_set_real )
= top_top_set_complex ) ) ).
% surjI
thf(fact_922_surjI,axiom,
! [G: nat > int,F: int > nat] :
( ! [X3: int] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_int @ G @ top_top_set_nat )
= top_top_set_int ) ) ).
% surjI
thf(fact_923_surjI,axiom,
! [G: nat > real,F: real > nat] :
( ! [X3: real] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_real @ G @ top_top_set_nat )
= top_top_set_real ) ) ).
% surjI
thf(fact_924_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_925_surjI,axiom,
! [G: nat > complex,F: complex > nat] :
( ! [X3: complex] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_complex @ G @ top_top_set_nat )
= top_top_set_complex ) ) ).
% surjI
thf(fact_926_surjI,axiom,
! [G: complex > real,F: real > complex] :
( ! [X3: real] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_complex_real @ G @ top_top_set_complex )
= top_top_set_real ) ) ).
% surjI
thf(fact_927_surjI,axiom,
! [G: complex > nat,F: nat > complex] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_complex_nat @ G @ top_top_set_complex )
= top_top_set_nat ) ) ).
% surjI
thf(fact_928_surjI,axiom,
! [G: complex > complex,F: complex > complex] :
( ! [X3: complex] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_1468599708987790691omplex @ G @ top_top_set_complex )
= top_top_set_complex ) ) ).
% surjI
thf(fact_929_surjE,axiom,
! [F: real > real,Y: real] :
( ( ( image_real_real @ F @ top_top_set_real )
= top_top_set_real )
=> ~ ! [X3: real] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_930_surjE,axiom,
! [F: real > nat,Y: nat] :
( ( ( image_real_nat @ F @ top_top_set_real )
= top_top_set_nat )
=> ~ ! [X3: real] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_931_surjE,axiom,
! [F: real > complex,Y: complex] :
( ( ( image_real_complex @ F @ top_top_set_real )
= top_top_set_complex )
=> ~ ! [X3: real] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_932_surjE,axiom,
! [F: nat > int,Y: int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ~ ! [X3: nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_933_surjE,axiom,
! [F: nat > real,Y: real] :
( ( ( image_nat_real @ F @ top_top_set_nat )
= top_top_set_real )
=> ~ ! [X3: nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_934_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X3: nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_935_surjE,axiom,
! [F: nat > complex,Y: complex] :
( ( ( image_nat_complex @ F @ top_top_set_nat )
= top_top_set_complex )
=> ~ ! [X3: nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_936_surjE,axiom,
! [F: complex > real,Y: real] :
( ( ( image_complex_real @ F @ top_top_set_complex )
= top_top_set_real )
=> ~ ! [X3: complex] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_937_surjE,axiom,
! [F: complex > nat,Y: nat] :
( ( ( image_complex_nat @ F @ top_top_set_complex )
= top_top_set_nat )
=> ~ ! [X3: complex] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_938_surjE,axiom,
! [F: complex > complex,Y: complex] :
( ( ( image_1468599708987790691omplex @ F @ top_top_set_complex )
= top_top_set_complex )
=> ~ ! [X3: complex] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_939_surjD,axiom,
! [F: real > real,Y: real] :
( ( ( image_real_real @ F @ top_top_set_real )
= top_top_set_real )
=> ? [X3: real] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_940_surjD,axiom,
! [F: real > nat,Y: nat] :
( ( ( image_real_nat @ F @ top_top_set_real )
= top_top_set_nat )
=> ? [X3: real] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_941_surjD,axiom,
! [F: real > complex,Y: complex] :
( ( ( image_real_complex @ F @ top_top_set_real )
= top_top_set_complex )
=> ? [X3: real] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_942_surjD,axiom,
! [F: nat > int,Y: int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_943_surjD,axiom,
! [F: nat > real,Y: real] :
( ( ( image_nat_real @ F @ top_top_set_nat )
= top_top_set_real )
=> ? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_944_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_945_surjD,axiom,
! [F: nat > complex,Y: complex] :
( ( ( image_nat_complex @ F @ top_top_set_nat )
= top_top_set_complex )
=> ? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_946_surjD,axiom,
! [F: complex > real,Y: real] :
( ( ( image_complex_real @ F @ top_top_set_complex )
= top_top_set_real )
=> ? [X3: complex] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_947_surjD,axiom,
! [F: complex > nat,Y: nat] :
( ( ( image_complex_nat @ F @ top_top_set_complex )
= top_top_set_nat )
=> ? [X3: complex] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_948_surjD,axiom,
! [F: complex > complex,Y: complex] :
( ( ( image_1468599708987790691omplex @ F @ top_top_set_complex )
= top_top_set_complex )
=> ? [X3: complex] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_949_invertible__fixpoint__property,axiom,
! [T2: set_int,I: int > nat,S2: set_nat,R: nat > int,G: int > int] :
( ( topolo2181401217840723324nt_nat @ T2 @ I )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ I @ T2 ) @ S2 )
=> ( ( topolo1179557035430618492at_int @ S2 @ R )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ R @ S2 ) @ T2 )
=> ( ! [Y3: int] :
( ( member_int @ Y3 @ T2 )
=> ( ( R @ ( I @ Y3 ) )
= Y3 ) )
=> ( ! [F3: nat > nat] :
( ( topolo1182047505939668768at_nat @ S2 @ F3 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ S2 ) @ S2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ S2 )
& ( ( F3 @ X6 )
= X6 ) ) ) )
=> ( ( topolo2178910747331673048nt_int @ T2 @ G )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ G @ T2 ) @ T2 )
=> ~ ! [Y3: int] :
( ( member_int @ Y3 @ T2 )
=> ( ( G @ Y3 )
!= Y3 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_950_invertible__fixpoint__property,axiom,
! [T2: set_complex,I: complex > nat,S2: set_nat,R: nat > complex,G: complex > complex] :
( ( topolo3759945079839938046ex_nat @ T2 @ I )
=> ( ( ord_less_eq_set_nat @ ( image_complex_nat @ I @ T2 ) @ S2 )
=> ( ( topolo140161755405200382omplex @ S2 @ R )
=> ( ( ord_le211207098394363844omplex @ ( image_nat_complex @ R @ S2 ) @ T2 )
=> ( ! [Y3: complex] :
( ( member_complex @ Y3 @ T2 )
=> ( ( R @ ( I @ Y3 ) )
= Y3 ) )
=> ( ! [F3: nat > nat] :
( ( topolo1182047505939668768at_nat @ S2 @ F3 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ S2 ) @ S2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ S2 )
& ( ( F3 @ X6 )
= X6 ) ) ) )
=> ( ( topolo9015423870875150044omplex @ T2 @ G )
=> ( ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ G @ T2 ) @ T2 )
=> ~ ! [Y3: complex] :
( ( member_complex @ Y3 @ T2 )
=> ( ( G @ Y3 )
!= Y3 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_951_invertible__fixpoint__property,axiom,
! [T2: set_nat,I: nat > nat,S2: set_nat,R: nat > nat,G: nat > nat] :
( ( topolo1182047505939668768at_nat @ T2 @ I )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ I @ T2 ) @ S2 )
=> ( ( topolo1182047505939668768at_nat @ S2 @ R )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ S2 ) @ T2 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ T2 )
=> ( ( R @ ( I @ Y3 ) )
= Y3 ) )
=> ( ! [F3: nat > nat] :
( ( topolo1182047505939668768at_nat @ S2 @ F3 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ S2 ) @ S2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ S2 )
& ( ( F3 @ X6 )
= X6 ) ) ) )
=> ( ( topolo1182047505939668768at_nat @ T2 @ G )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ T2 ) @ T2 )
=> ~ ! [Y3: nat] :
( ( member_nat @ Y3 @ T2 )
=> ( ( G @ Y3 )
!= Y3 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_952_invertible__fixpoint__property,axiom,
! [T2: set_nat,I: nat > int,S2: set_int,R: int > nat,G: nat > nat] :
( ( topolo1179557035430618492at_int @ T2 @ I )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ I @ T2 ) @ S2 )
=> ( ( topolo2181401217840723324nt_nat @ S2 @ R )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ R @ S2 ) @ T2 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ T2 )
=> ( ( R @ ( I @ Y3 ) )
= Y3 ) )
=> ( ! [F3: int > int] :
( ( topolo2178910747331673048nt_int @ S2 @ F3 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F3 @ S2 ) @ S2 )
=> ? [X6: int] :
( ( member_int @ X6 @ S2 )
& ( ( F3 @ X6 )
= X6 ) ) ) )
=> ( ( topolo1182047505939668768at_nat @ T2 @ G )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ T2 ) @ T2 )
=> ~ ! [Y3: nat] :
( ( member_nat @ Y3 @ T2 )
=> ( ( G @ Y3 )
!= Y3 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_953_invertible__fixpoint__property,axiom,
! [T2: set_nat,I: nat > complex,S2: set_complex,R: complex > nat,G: nat > nat] :
( ( topolo140161755405200382omplex @ T2 @ I )
=> ( ( ord_le211207098394363844omplex @ ( image_nat_complex @ I @ T2 ) @ S2 )
=> ( ( topolo3759945079839938046ex_nat @ S2 @ R )
=> ( ( ord_less_eq_set_nat @ ( image_complex_nat @ R @ S2 ) @ T2 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ T2 )
=> ( ( R @ ( I @ Y3 ) )
= Y3 ) )
=> ( ! [F3: complex > complex] :
( ( topolo9015423870875150044omplex @ S2 @ F3 )
=> ( ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ F3 @ S2 ) @ S2 )
=> ? [X6: complex] :
( ( member_complex @ X6 @ S2 )
& ( ( F3 @ X6 )
= X6 ) ) ) )
=> ( ( topolo1182047505939668768at_nat @ T2 @ G )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ T2 ) @ T2 )
=> ~ ! [Y3: nat] :
( ( member_nat @ Y3 @ T2 )
=> ( ( G @ Y3 )
!= Y3 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_954_invertible__fixpoint__property,axiom,
! [T2: set_real,I: real > nat,S2: set_nat,R: nat > real,G: real > real] :
( ( topolo2287203362918339196al_nat @ T2 @ I )
=> ( ( ord_less_eq_set_nat @ ( image_real_nat @ I @ T2 ) @ S2 )
=> ( ( topolo6943266826644216316t_real @ S2 @ R )
=> ( ( ord_less_eq_set_real @ ( image_nat_real @ R @ S2 ) @ T2 )
=> ( ! [Y3: real] :
( ( member_real @ Y3 @ T2 )
=> ( ( R @ ( I @ Y3 ) )
= Y3 ) )
=> ( ! [F3: nat > nat] :
( ( topolo1182047505939668768at_nat @ S2 @ F3 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ S2 ) @ S2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ S2 )
& ( ( F3 @ X6 )
= X6 ) ) ) )
=> ( ( topolo5044208981011980120l_real @ T2 @ G )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ G @ T2 ) @ T2 )
=> ~ ! [Y3: real] :
( ( member_real @ Y3 @ T2 )
=> ( ( G @ Y3 )
!= Y3 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_955_invertible__fixpoint__property,axiom,
! [T2: set_nat,I: nat > real,S2: set_real,R: real > nat,G: nat > nat] :
( ( topolo6943266826644216316t_real @ T2 @ I )
=> ( ( ord_less_eq_set_real @ ( image_nat_real @ I @ T2 ) @ S2 )
=> ( ( topolo2287203362918339196al_nat @ S2 @ R )
=> ( ( ord_less_eq_set_nat @ ( image_real_nat @ R @ S2 ) @ T2 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ T2 )
=> ( ( R @ ( I @ Y3 ) )
= Y3 ) )
=> ( ! [F3: real > real] :
( ( topolo5044208981011980120l_real @ S2 @ F3 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ F3 @ S2 ) @ S2 )
=> ? [X6: real] :
( ( member_real @ X6 @ S2 )
& ( ( F3 @ X6 )
= X6 ) ) ) )
=> ( ( topolo1182047505939668768at_nat @ T2 @ G )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ T2 ) @ T2 )
=> ~ ! [Y3: nat] :
( ( member_nat @ Y3 @ T2 )
=> ( ( G @ Y3 )
!= Y3 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_956_invertible__fixpoint__property,axiom,
! [T2: set_real,I: real > real,S2: set_real,R: real > real,G: real > real] :
( ( topolo5044208981011980120l_real @ T2 @ I )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ I @ T2 ) @ S2 )
=> ( ( topolo5044208981011980120l_real @ S2 @ R )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ R @ S2 ) @ T2 )
=> ( ! [Y3: real] :
( ( member_real @ Y3 @ T2 )
=> ( ( R @ ( I @ Y3 ) )
= Y3 ) )
=> ( ! [F3: real > real] :
( ( topolo5044208981011980120l_real @ S2 @ F3 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ F3 @ S2 ) @ S2 )
=> ? [X6: real] :
( ( member_real @ X6 @ S2 )
& ( ( F3 @ X6 )
= X6 ) ) ) )
=> ( ( topolo5044208981011980120l_real @ T2 @ G )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ G @ T2 ) @ T2 )
=> ~ ! [Y3: real] :
( ( member_real @ Y3 @ T2 )
=> ( ( G @ Y3 )
!= Y3 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_957_mono__Suc,axiom,
monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).
% mono_Suc
thf(fact_958_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_959_strict__mono__inv,axiom,
! [F: real > real,G: real > real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ F )
=> ( ( ( image_real_real @ F @ top_top_set_real )
= top_top_set_real )
=> ( ! [X3: real] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_960_strict__mono__inv,axiom,
! [F: real > nat,G: nat > real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_real @ ord_less_nat @ F )
=> ( ( ( image_real_nat @ F @ top_top_set_real )
= top_top_set_nat )
=> ( ! [X3: real] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( monotone_on_nat_real @ top_top_set_nat @ ord_less_nat @ ord_less_real @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_961_strict__mono__inv,axiom,
! [F: real > int,G: int > real] :
( ( monotone_on_real_int @ top_top_set_real @ ord_less_real @ ord_less_int @ F )
=> ( ( ( image_real_int @ F @ top_top_set_real )
= top_top_set_int )
=> ( ! [X3: real] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( monotone_on_int_real @ top_top_set_int @ ord_less_int @ ord_less_real @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_962_strict__mono__inv,axiom,
! [F: nat > real,G: real > nat] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_nat @ ord_less_real @ F )
=> ( ( ( image_nat_real @ F @ top_top_set_nat )
= top_top_set_real )
=> ( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( monotone_on_real_nat @ top_top_set_real @ ord_less_real @ ord_less_nat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_963_strict__mono__inv,axiom,
! [F: nat > nat,G: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_964_strict__mono__inv,axiom,
! [F: nat > int,G: int > nat] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_nat @ ord_less_int @ F )
=> ( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( monotone_on_int_nat @ top_top_set_int @ ord_less_int @ ord_less_nat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_965_strict__mono__inv,axiom,
! [F: int > real,G: real > int] :
( ( monotone_on_int_real @ top_top_set_int @ ord_less_int @ ord_less_real @ F )
=> ( ( ( image_int_real @ F @ top_top_set_int )
= top_top_set_real )
=> ( ! [X3: int] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( monotone_on_real_int @ top_top_set_real @ ord_less_real @ ord_less_int @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_966_strict__mono__inv,axiom,
! [F: int > nat,G: nat > int] :
( ( monotone_on_int_nat @ top_top_set_int @ ord_less_int @ ord_less_nat @ F )
=> ( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
=> ( ! [X3: int] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( monotone_on_nat_int @ top_top_set_nat @ ord_less_nat @ ord_less_int @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_967_strict__mono__inv,axiom,
! [F: int > int,G: int > int] :
( ( monotone_on_int_int @ top_top_set_int @ ord_less_int @ ord_less_int @ F )
=> ( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ( ! [X3: int] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( monotone_on_int_int @ top_top_set_int @ ord_less_int @ ord_less_int @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_968_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_969_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_970_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_971_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_972_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_973_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_974_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_975_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_976_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_977_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_978_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_979_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_980_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_981_UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_I
thf(fact_982_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_983_UNIV__I,axiom,
! [X: complex] : ( member_complex @ X @ top_top_set_complex ) ).
% UNIV_I
thf(fact_984_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_985_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_986_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_987_image__eqI,axiom,
! [B: complex,F: nat > complex,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_complex @ B @ ( image_nat_complex @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_988_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_989_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_990_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_991_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_992_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_993_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_994_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_995_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_996_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_997_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_998_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_999_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_1000_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_1001_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_1002_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_1003_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1004_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1005_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_1006_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1007_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1008_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_1009_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_1010_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_1011_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_1012_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_1013_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_1014_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_1015_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_1016_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1017_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1018_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1019_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1020_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1021_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1022_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1023_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1024_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1025_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1026_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1027_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_1028_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_1029_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z2: real,W2: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z2 @ W2 ) )
= ( divide_divide_real @ ( times_times_real @ X @ W2 ) @ ( times_times_real @ Y @ Z2 ) ) ) ).
% divide_divide_times_eq
thf(fact_1030_times__divide__times__eq,axiom,
! [X: real,Y: real,Z2: real,W2: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z2 @ W2 ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ W2 ) ) ) ).
% times_divide_times_eq
thf(fact_1031_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_1032_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_1033_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_1034_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B3: real] : ( times_times_real @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_1035_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_1036_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B3: int] : ( times_times_int @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_1037_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_1038_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_1039_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_1040_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_1041_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_1042_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_1043_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_1044_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_1045_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_1046_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_1047_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_1048_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_1049_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_1050_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_1051_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_1052_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_1053_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_1054_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_1055_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_1056_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_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_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_1068_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_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_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_1079_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_1080_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_1081_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_1082_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_1083_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_1084_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_1085_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_1086_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_1087_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_1088_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_1089_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_1090_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_1091_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_1092_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_1093_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_1094_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_1095_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_1096_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y4: real] :
? [N3: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1097_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_1098_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_1099_real__divide__square__eq,axiom,
! [R: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
= ( divide_divide_real @ A @ R ) ) ).
% real_divide_square_eq
thf(fact_1100_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1101_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1102_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1103_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1104_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1105_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1106_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1107_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1108_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_1109_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1110_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1111_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1112_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1113_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1114_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_1115_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1116_zdiv__int,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_1117_div__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) ).
% div_mult2_eq
thf(fact_1118_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1119_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1120_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1121_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1122_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_1123_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_1124_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_1125_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1126_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1127_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1128_less__eq__div__iff__mult__less__eq,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1129_div__nat__eqI,axiom,
! [N: nat,Q: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q ) ) ) ).
% div_nat_eqI
thf(fact_1130_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_1131_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_1132_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1133_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1134_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
& ( P @ Q2 ) ) ) ) ).
% split_div'
thf(fact_1135_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1136_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1137_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1138_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1139_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1140_mono__times__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( times_times_nat @ N ) ) ) ).
% mono_times_nat
thf(fact_1141_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1142_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_1143_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1144_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1145_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1146_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1147_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_1148_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1149_zdiv__mono1,axiom,
! [A: int,A6: int,B: int] :
( ( ord_less_eq_int @ A @ A6 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A6 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1150_zdiv__mono2,axiom,
! [A: int,B5: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B5 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1151_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1152_zdiv__mono1__neg,axiom,
! [A: int,A6: int,B: int] :
( ( ord_less_eq_int @ A @ A6 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1153_zdiv__mono2__neg,axiom,
! [A: int,B5: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B5 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1154_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1155_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1156_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1157_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1158_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1159_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1160_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1161_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1162_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1163_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1164_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1165_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1166_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1167_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1168_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1169_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1170_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1171_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_1172_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_1173_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_1174_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1175_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1176_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1177_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_1178_Equivalence__Measurable__On__Borel_Ointegrable__on__mono__on,axiom,
! [A: real,B: real,F: real > real] :
( ( monoto4017252874604999745l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( hensto5963834015518849588l_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% Equivalence_Measurable_On_Borel.integrable_on_mono_on
thf(fact_1179_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,D3: real] :
( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ C2 @ D3 ) )
& ( ord_less_eq_real @ C2 @ D3 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_1180_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_1181_lower__def,axiom,
( lower
= ( ^ [X4: real] : ( a_seg @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ n ) @ X4 ) @ a ) ) ) ) ) ) ).
% lower_def
thf(fact_1182_n__def,axiom,
( n
= ( nat2 @ ( archim6058952711729229775r_real @ ( divide_divide_real @ a @ delta ) ) ) ) ).
% n_def
thf(fact_1183_k__def,axiom,
( k
= ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ ( divide_divide_real @ x @ a ) @ ( semiri5074537144036343181t_real @ n ) ) ) ) ) ).
% k_def
thf(fact_1184_upper__def,axiom,
( upper
= ( ^ [X4: real] : ( a_seg @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ n ) @ X4 ) @ a ) ) ) ) ) ) ).
% upper_def
thf(fact_1185_floor__divide__real__eq__div,axiom,
! [B: int,A: real] :
( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
= ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% floor_divide_real_eq_div
thf(fact_1186_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_1187_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_1188_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1189_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1190_zless__nat__conj,axiom,
! [W2: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W2 @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_1191_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_1192_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_1193_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1194_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_1195_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_1196_nat__mono__iff,axiom,
! [Z2: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_1197_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_1198_zless__nat__eq__int__zless,axiom,
! [M: nat,Z2: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1199_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( ( nat2 @ W2 )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1200_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1201_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N4: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N4 ) )
=> ( P @ N4 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1202_nat__le__eq__zle,axiom,
! [W2: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_1203_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_1204_nat__less__eq__zless,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_1205_nat__mult__distrib,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( nat2 @ ( times_times_int @ Z2 @ Z5 ) )
= ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1206_nat__floor__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= zero_zero_nat ) ) ).
% nat_floor_neg
thf(fact_1207_floor__eq3,axiom,
! [N: nat,X: real] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= N ) ) ) ).
% floor_eq3
thf(fact_1208_le__nat__floor,axiom,
! [X: nat,A: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
=> ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% le_nat_floor
thf(fact_1209_nat__div__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_1210_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_1211_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_1212_floor__eq4,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= N ) ) ) ).
% floor_eq4
thf(fact_1213_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1214_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1215_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1216_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_1217_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_1218_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_1219_real__non__denum,axiom,
~ ? [F4: nat > real] :
( ( image_nat_real @ F4 @ top_top_set_nat )
= top_top_set_real ) ).
% real_non_denum
thf(fact_1220_complex__non__denum,axiom,
~ ? [F4: nat > complex] :
( ( image_nat_complex @ F4 @ top_top_set_nat )
= top_top_set_complex ) ).
% complex_non_denum
thf(fact_1221_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1222_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_1223_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1224_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1225_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1226_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1227_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1228_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1229_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1230_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1231_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1232_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1233_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_1234_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1235_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1236_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q3: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I4: nat] :
( ( Q3 @ I4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I4 ) )
& ( ord_less_eq_real @ ( X3 @ I4 ) @ one_one_real ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X6: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L3 @ X6 @ I3 ) @ one_one_nat )
& ! [X6: nat > real,I3: nat] :
( ( ( P @ X6 )
& ( Q3 @ I3 )
& ( ( X6 @ I3 )
= zero_zero_real ) )
=> ( ( L3 @ X6 @ I3 )
= zero_zero_nat ) )
& ! [X6: nat > real,I3: nat] :
( ( ( P @ X6 )
& ( Q3 @ I3 )
& ( ( X6 @ I3 )
= one_one_real ) )
=> ( ( L3 @ X6 @ I3 )
= one_one_nat ) )
& ! [X6: nat > real,I3: nat] :
( ( ( P @ X6 )
& ( Q3 @ I3 )
& ( ( L3 @ X6 @ I3 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X6 @ I3 ) @ ( F @ X6 @ I3 ) ) )
& ! [X6: nat > real,I3: nat] :
( ( ( P @ X6 )
& ( Q3 @ I3 )
& ( ( L3 @ X6 @ I3 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X6 @ I3 ) @ ( X6 @ I3 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1237_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_1238_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1239_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1240_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1241_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1242_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1243_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1244_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1245_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1246_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1247_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1248_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1249_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1250_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1251_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1252_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1253_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1254_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1255_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1256_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1257_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1258_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1259_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1260_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1261_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1262_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1263_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1264_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1265_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1266_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( yidx @ y ) ) ) ) @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ k ) ) ) ).
%------------------------------------------------------------------------------