TPTP Problem File: SLH0526^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Youngs_Inequality/0000_Youngs/prob_00528_022455__13123254_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1402 ( 503 unt; 133 typ; 0 def)
% Number of atoms : 3975 (1049 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 11548 ( 334 ~; 111 |; 215 &;9162 @)
% ( 0 <=>;1726 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 789 ( 789 >; 0 *; 0 +; 0 <<)
% Number of symbols : 126 ( 125 usr; 20 con; 0-4 aty)
% Number of variables : 3506 ( 125 ^;3311 !; 70 ?;3506 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:31:12.780
%------------------------------------------------------------------------------
% 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 (125)
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__Int__Oint_001t__Complex__Ocomplex,type,
comp_c4623764063710898541omplex: ( complex > int ) > ( complex > complex ) > complex > int ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Int__Oint_001t__Real__Oreal,type,
comp_c5610039060539665899t_real: ( complex > int ) > ( real > complex ) > real > int ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Nat__Onat_001t__Complex__Ocomplex,type,
comp_c3648228598504721169omplex: ( complex > nat ) > ( complex > complex ) > complex > nat ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Nat__Onat_001t__Int__Oint,type,
comp_complex_nat_int: ( complex > nat ) > ( int > complex ) > int > nat ).
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__Complex__Ocomplex,type,
comp_c2063761206571265261omplex: ( complex > real ) > ( complex > complex ) > complex > real ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Int__Oint,type,
comp_c7987935588466492523al_int: ( complex > real ) > ( int > complex ) > int > real ).
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__Nat__Onat_001t__Nat__Onat,type,
comp_int_nat_nat: ( int > nat ) > ( nat > int ) > nat > nat ).
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__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__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__Complex__Ocomplex,type,
monoto2383616128001712325omplex: set_complex > ( complex > complex > $o ) > ( complex > complex > $o ) > ( complex > complex ) > $o ).
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__Complex__Ocomplex,type,
monoto8985637569277367875omplex: set_int > ( int > int > $o ) > ( complex > complex > $o ) > ( int > complex ) > $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__Complex__Ocomplex,type,
monoto7309693138617929411omplex: set_real > ( real > real > $o ) > ( complex > complex > $o ) > ( real > complex ) > $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__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
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__Complex__Ocomplex,type,
zero_zero_complex: complex ).
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_001_062_It__Complex__Ocomplex_M_Eo_J,type,
top_top_complex_o: complex > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Real__Oreal_M_Eo_J,type,
top_top_real_o: 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__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
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__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__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__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__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_thesis____,type,
thesis: $o ).
thf(sy_v_upper____,type,
upper: real > real ).
thf(sy_v_x____,type,
x: real ).
thf(sy_v_y____,type,
y: real ).
% Relevant facts (1267)
thf(fact_0_a,axiom,
ord_less_eq_real @ zero_zero_real @ a ).
% a
thf(fact_1_f_I1_J,axiom,
( ( f @ zero_zero_real )
= zero_zero_real ) ).
% f(1)
thf(fact_2_x_I1_J,axiom,
( ( f @ x )
= y ) ).
% x(1)
thf(fact_3_x__lims_I1_J,axiom,
ord_less_eq_real @ ( a_seg @ ( semiri5074537144036343181t_real @ k ) ) @ x ).
% x_lims(1)
thf(fact_4__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_5_x__lims_I2_J,axiom,
ord_less_real @ x @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ k ) ) ) ).
% x_lims(2)
thf(fact_6_f_I2_J,axiom,
( ( f @ a )
= b ) ).
% f(2)
thf(fact_7_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_8_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_9__092_060open_0620_A_060_Aa_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ a ).
% \<open>0 < a\<close>
thf(fact_10_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_11_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_12_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_13__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,
~ ! [X2: real] :
( ( ( f @ X2 )
= y )
=> ~ ( member_real @ X2 @ ( 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_14_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_15_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_16_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_17_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_18_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_19_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_20_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_21_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_22_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_23_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_24_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_25_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_26_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_27_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_28_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_29_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_30_False,axiom,
a != zero_zero_real ).
% False
thf(fact_31_x_I2_J,axiom,
member_real @ x @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ).
% x(2)
thf(fact_32__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_33_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_34_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_35_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_36_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_37_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_38_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_39_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_40_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_41_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
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_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_44_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_45_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_46_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_47_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_48_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_49_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_50_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_51_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_52__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_53_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_54__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_55_del__gt0,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_real @ zero_zero_real @ ( del @ E ) ) ) ).
% del_gt0
thf(fact_56_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_57_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_58_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_59_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_60_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_61_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_62_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_63_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_65_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_67_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_68_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( M2 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_69_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_70_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_71_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_72_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_73_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_74_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_75_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_76_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_77_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_78_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_79_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_80_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_81_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_82_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_83_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_84_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_85_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_86_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 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_87_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_88_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_89_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_90_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_91_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_92_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_93_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_94_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_95_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less_nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_96_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_97_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_98_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_99_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_100_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_101_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_102_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_103_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_104_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_105_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_106_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y3: nat,Z: nat] :
( ( R @ X2 @ Y3 )
=> ( ( R @ Y3 @ Z )
=> ( R @ X2 @ Z ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_107_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_108_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_109_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_110_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_111_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_112_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_113_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_114_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_115_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_116_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_117_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_118_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_119_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_120_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_121_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_122_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_123_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_124_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_125_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_126_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_127_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_128_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_129_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_130_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_131_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_132_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_133_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_134_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_135_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_136_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_137_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_138_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_139_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_140_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_141_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_142_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_143_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_144_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_145_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_146_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_147_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_148_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_149_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_150_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_151_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_152_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_153_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_154_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_155_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_156_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_157_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_158_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_159_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_160_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_161_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_162_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_163_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_164_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_165_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_166_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_167_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_168_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_169_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_170_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_171_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_172_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_173_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_174_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_175_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_176_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_177_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_178_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_179_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_180_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_181_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_182_order__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_eq_real @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_183_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_184_order__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_185_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_186_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_187_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_188_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_189_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_190_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_191_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_192_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_193_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_194_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_195_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_196_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_197_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [X3: real,Y6: real] :
( ( ord_less_eq_real @ X3 @ Y6 )
& ( ord_less_eq_real @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_198_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_199_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
& ( ord_less_eq_int @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_200_le__cases3,axiom,
! [X: real,Y: real,Z3: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_201_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_202_le__cases3,axiom,
! [X: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_203_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_204_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_205_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_206_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_207_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_208_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_209_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_210_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_211_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_212_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_213_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_214_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_215_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_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_225_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_226_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_227_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_228_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_229_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_230_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_231_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_232_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_233_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_234_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_235_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_236_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_237_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_238_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_239_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_240_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_241_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_242_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_243_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_244_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_245_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_246_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_247_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_248_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_249_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_250_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_251_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_252_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_253_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_254_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_255_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_256_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_257_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_258_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_259_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_260_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_261_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_262_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_263_order__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_264_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_265_order__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_266_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_267_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_268_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_269_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_270_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_271_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_272_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_273_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_274_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_275_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_276_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_277_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_278_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_279_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_280_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_281_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_282_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_283_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_284_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_285_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_286_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_287_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_288_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_289_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_290_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_291_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_292_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_293_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_294_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_295_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_296_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_297_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_298_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_299_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_300_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_301_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_302_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_303_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_304_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_305_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_306_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_307_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_308_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_309_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X2 )
=> ( P @ Y4 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_310_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_311_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_312_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_313_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_314_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_315_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_316_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_317_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_318_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_319_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_320_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_321_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_322_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_323_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_324_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_325_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_326_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_327_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_328_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_329_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_330_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_331_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_332_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_333_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_334_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_335_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_336_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_337_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_338_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_339_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_340_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_341_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_342_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_343_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_344_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_345_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_346_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_347_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_348_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_349_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_350_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_351_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_352_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_353_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_354_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_355_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_356_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_357_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_358_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_359_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_360_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_361_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_362_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_363_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_364_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_365_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_366_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_367_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_368_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_369_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_370_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_371_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_372_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_373_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_374_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_375_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_376_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_377_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_378_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 )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_379_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 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_380_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 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_381_order__less__le__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_382_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_383_order__less__le__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_384_order__le__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_385_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_386_order__le__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_387_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_388_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_389_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_390_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_391_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_392_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_393_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_394_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_395_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_396_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_397_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_398_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_399_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_400_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_401_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_402_order__less__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y6: real] :
( ( ord_less_eq_real @ X3 @ Y6 )
& ( X3 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_403_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ( X3 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_404_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
& ( X3 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_405_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y6: real] :
( ( ord_less_real @ X3 @ Y6 )
| ( X3 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_406_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_nat @ X3 @ Y6 )
| ( X3 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_407_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y6: int] :
( ( ord_less_int @ X3 @ Y6 )
| ( X3 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_408_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_409_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_410_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_411_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_412_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_413_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_414_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_415_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_416_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_417_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_418_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_419_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_420_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_421_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_422_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_423_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_424_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_425_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_426_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_real @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_427_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_428_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_int @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_429_dense__le__bounded,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z3 ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_430_dense__ge__bounded,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ Z3 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z3 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_431_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_432_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_433_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_434_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_435_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_436_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_437_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_438_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_439_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_440_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_441_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_442_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_443_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_444_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_445_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_446_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_447_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_448_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_449_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y6: real] :
( ( ord_less_eq_real @ X3 @ Y6 )
& ~ ( ord_less_eq_real @ Y6 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_450_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_451_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
& ~ ( ord_less_eq_int @ Y6 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_452_dense__le,axiom,
! [Y: real,Z3: real] :
( ! [X2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Z3 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_le
thf(fact_453_dense__ge,axiom,
! [Z3: real,Y: real] :
( ! [X2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( ord_less_eq_real @ Y @ X2 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_ge
thf(fact_454_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_455_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_456_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_457_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_458_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_459_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_460_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_461_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_462_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_463_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_464_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_465_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_466_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_467_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_468_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_469_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_470_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_471_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_472_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_473_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_474_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_475_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_476_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_477_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_478_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_479_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_480_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_481_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_482_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_483_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_484_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_485_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_486_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_487_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_488_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_489_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_490_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_491_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_492_fim,axiom,
( ( image_real_real @ f @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
= ( set_or1222579329274155063t_real @ zero_zero_real @ b ) ) ).
% fim
thf(fact_493_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_494_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_495_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_496_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_497_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_498_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_499_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_500_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_501_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_502_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_503_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_504_fa__eq__b,axiom,
( ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ n ) ) )
= b ) ).
% fa_eq_b
thf(fact_505__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_506_sm__0a,axiom,
monoto4017252874604999745l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) @ ord_less_real @ ord_less_real @ f ).
% sm_0a
thf(fact_507__092_060open_0620_A_060_An_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% \<open>0 < n\<close>
thf(fact_508_a__seg__eq__a__iff,axiom,
! [X: real] :
( ( ( a_seg @ X )
= a )
= ( X
= ( semiri5074537144036343181t_real @ n ) ) ) ).
% a_seg_eq_a_iff
thf(fact_509_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_510_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_511_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_512_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_513_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_514_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_515_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_516_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_517_sm,axiom,
monoto4017252874604999745l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ ord_less_real @ ord_less_real @ f ).
% sm
thf(fact_518_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_519_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_520_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_521_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_522_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_523_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_524_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] :
( ( P @ X2 @ Y3 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_525_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_526_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_527_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_528_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_529_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_530_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_531_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_532_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_533_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_534_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_535_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_536_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_537_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_538_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_539_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_540_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_541_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_542_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_543_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_544_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_545_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_546_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_547_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_548_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_549_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_550_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M7 ) )
=> ~ ! [M6: nat] :
( ( P @ M6 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M6 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_551_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_552_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_553_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_554_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_555_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_556_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_557_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_558_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_559_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_560_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_561__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_562_intgb__g,axiom,
hensto5963834015518849588l_real @ g @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) ).
% intgb_g
thf(fact_563_f1__def,axiom,
( f1
= ( comp_real_real_real @ f @ lower ) ) ).
% f1_def
thf(fact_564_f2__def,axiom,
( f2
= ( comp_real_real_real @ f @ upper ) ) ).
% f2_def
thf(fact_565__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_566_cont__0a,axiom,
topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) @ f ).
% cont_0a
thf(fact_567_cont,axiom,
topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ f ).
% cont
thf(fact_568_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_569_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_570_intgb__f,axiom,
hensto5963834015518849588l_real @ f @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ).
% intgb_f
thf(fact_571_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_572_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_573_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_574_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_575_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_576_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_577_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_578_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_579_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_580_a__seg__def,axiom,
( a_seg
= ( ^ [U2: real] : ( divide_divide_real @ ( times_times_real @ U2 @ a ) @ ( semiri5074537144036343181t_real @ n ) ) ) ) ).
% a_seg_def
thf(fact_581_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_582_not__UNIV__eq__Ici,axiom,
! [L2: real] :
( top_top_set_real
!= ( set_ord_atLeast_real @ L2 ) ) ).
% not_UNIV_eq_Ici
thf(fact_583_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_584_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_585_top_Oextremum__uniqueI,axiom,
! [A: set_complex] :
( ( ord_le211207098394363844omplex @ top_top_set_complex @ A )
=> ( A = top_top_set_complex ) ) ).
% top.extremum_uniqueI
thf(fact_586_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_587_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_588_top_Oextremum__unique,axiom,
! [A: set_complex] :
( ( ord_le211207098394363844omplex @ top_top_set_complex @ A )
= ( A = top_top_set_complex ) ) ).
% top.extremum_unique
thf(fact_589_top__greatest,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).
% top_greatest
thf(fact_590_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_591_top__greatest,axiom,
! [A: set_complex] : ( ord_le211207098394363844omplex @ A @ top_top_set_complex ) ).
% top_greatest
thf(fact_592_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_593_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_594_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_595_top_Oextremum__strict,axiom,
! [A: set_real] :
~ ( ord_less_set_real @ top_top_set_real @ A ) ).
% top.extremum_strict
thf(fact_596_top_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A ) ).
% top.extremum_strict
thf(fact_597_top_Oextremum__strict,axiom,
! [A: set_complex] :
~ ( ord_less_set_complex @ top_top_set_complex @ A ) ).
% top.extremum_strict
thf(fact_598_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_599_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_600_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_601_not__UNIV__eq__Icc,axiom,
! [L2: real,H2: real] :
( top_top_set_real
!= ( set_or1222579329274155063t_real @ L2 @ H2 ) ) ).
% not_UNIV_eq_Icc
thf(fact_602_not__UNIV__eq__Icc,axiom,
! [L2: nat,H2: nat] :
( top_top_set_nat
!= ( set_or1269000886237332187st_nat @ L2 @ H2 ) ) ).
% not_UNIV_eq_Icc
thf(fact_603_not__UNIV__eq__Icc,axiom,
! [L2: int,H2: int] :
( top_top_set_int
!= ( set_or1266510415728281911st_int @ L2 @ H2 ) ) ).
% not_UNIV_eq_Icc
thf(fact_604_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 ) ) )
=> ( ! [X2: real] :
( ( ord_less_eq_real @ A @ X2 )
=> ( ( G @ ( F @ X2 ) )
= X2 ) )
=> ( topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ ( F @ A ) ) @ G ) ) ) ) ) ).
% strict_mono_continuous_invD
thf(fact_605_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_606_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_607_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_608_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_609_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_610_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_611_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_612_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_613_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_614_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_615_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_616_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_617_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_618_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_619_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_620_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_621_UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_I
thf(fact_622_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_623_UNIV__I,axiom,
! [X: complex] : ( member_complex @ X @ top_top_set_complex ) ).
% UNIV_I
thf(fact_624_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_625_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_626_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_627_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_628_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_629_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_630_subsetI,axiom,
! [A2: set_real,B4: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( member_real @ X2 @ B4 ) )
=> ( ord_less_eq_set_real @ A2 @ B4 ) ) ).
% subsetI
thf(fact_631_subsetI,axiom,
! [A2: set_nat,B4: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B4 ) )
=> ( ord_less_eq_set_nat @ A2 @ B4 ) ) ).
% subsetI
thf(fact_632_atLeast__0,axiom,
( ( set_ord_atLeast_nat @ zero_zero_nat )
= top_top_set_nat ) ).
% atLeast_0
thf(fact_633_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_634_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_635_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_636_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_637_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_638_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_639_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_640_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_641_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_642_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_643_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_644_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_645_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_646_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_647_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_648_times__divide__times__eq,axiom,
! [X: real,Y: real,Z3: real,W2: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z3 @ W2 ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ Y @ W2 ) ) ) ).
% times_divide_times_eq
thf(fact_649_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z3: real,W2: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z3 @ W2 ) )
= ( divide_divide_real @ ( times_times_real @ X @ W2 ) @ ( times_times_real @ Y @ Z3 ) ) ) ).
% divide_divide_times_eq
thf(fact_650_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_651_top__set__def,axiom,
( top_top_set_real
= ( collect_real @ top_top_real_o ) ) ).
% top_set_def
thf(fact_652_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_653_top__set__def,axiom,
( top_top_set_complex
= ( collect_complex @ top_top_complex_o ) ) ).
% top_set_def
thf(fact_654_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_655_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_656_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_657_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_658_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_659_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_660_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_661_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_662_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_663_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_664_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_665_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_666_mono__Suc,axiom,
monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).
% mono_Suc
thf(fact_667_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_668_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_669_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_670_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_671_nonzero__eq__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_672_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A )
= ( B
= ( times_times_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_673_eq__divide__imp,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= B )
=> ( A
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_674_divide__eq__imp,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_675_eq__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_676_divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( divide_divide_real @ B @ C )
= A )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_677_frac__eq__eq,axiom,
! [Y: real,Z3: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z3 != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W2 @ Z3 ) )
= ( ( times_times_real @ X @ Z3 )
= ( times_times_real @ W2 @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_678_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_679_divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_680_less__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_681_neg__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_682_neg__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_683_pos__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_684_pos__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_685_mult__imp__div__pos__less,axiom,
! [Y: real,X: real,Z3: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z3 @ Y ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z3 ) ) ) ).
% mult_imp_div_pos_less
thf(fact_686_mult__imp__less__div__pos,axiom,
! [Y: real,Z3: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( times_times_real @ Z3 @ Y ) @ X )
=> ( ord_less_real @ Z3 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_687_divide__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_688_divide__strict__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_689_divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_690_le__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_691_divide__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono
thf(fact_692_neg__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_le_eq
thf(fact_693_neg__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_le_divide_eq
thf(fact_694_pos__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_le_eq
thf(fact_695_pos__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_le_divide_eq
thf(fact_696_mult__imp__div__pos__le,axiom,
! [Y: real,X: real,Z3: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ ( times_times_real @ Z3 @ Y ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z3 ) ) ) ).
% mult_imp_div_pos_le
thf(fact_697_mult__imp__le__div__pos,axiom,
! [Y: real,Z3: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z3 @ Y ) @ X )
=> ( ord_less_eq_real @ Z3 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_698_divide__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_699_linordered__field__no__lb,axiom,
! [X5: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X5 ) ).
% linordered_field_no_lb
thf(fact_700_linordered__field__no__ub,axiom,
! [X5: real] :
? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_701_UNIV__witness,axiom,
? [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_702_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_703_UNIV__witness,axiom,
? [X2: complex] : ( member_complex @ X2 @ top_top_set_complex ) ).
% UNIV_witness
thf(fact_704_UNIV__eq__I,axiom,
! [A2: set_real] :
( ! [X2: real] : ( member_real @ X2 @ A2 )
=> ( top_top_set_real = A2 ) ) ).
% UNIV_eq_I
thf(fact_705_UNIV__eq__I,axiom,
! [A2: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A2 )
=> ( top_top_set_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_706_UNIV__eq__I,axiom,
! [A2: set_complex] :
( ! [X2: complex] : ( member_complex @ X2 @ A2 )
=> ( top_top_set_complex = A2 ) ) ).
% UNIV_eq_I
thf(fact_707_rev__image__eqI,axiom,
! [X: real,A2: set_real,B: real,F: real > real] :
( ( member_real @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_real_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_708_rev__image__eqI,axiom,
! [X: real,A2: set_real,B: nat,F: real > nat] :
( ( member_real @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_real_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_709_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: int,F: nat > int] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_nat_int @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_710_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: complex,F: nat > complex] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_complex @ B @ ( image_nat_complex @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_711_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: real,F: nat > real] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_nat_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_712_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_713_ball__imageD,axiom,
! [F: real > real,A2: set_real,P: real > $o] :
( ! [X2: real] :
( ( member_real @ X2 @ ( image_real_real @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X5: real] :
( ( member_real @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_714_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_715_ball__imageD,axiom,
! [F: nat > int,A2: set_nat,P: int > $o] :
( ! [X2: int] :
( ( member_int @ X2 @ ( image_nat_int @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_716_ball__imageD,axiom,
! [F: nat > real,A2: set_nat,P: real > $o] :
( ! [X2: real] :
( ( member_real @ X2 @ ( image_nat_real @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_717_ball__imageD,axiom,
! [F: nat > complex,A2: set_nat,P: complex > $o] :
( ! [X2: complex] :
( ( member_complex @ X2 @ ( image_nat_complex @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_718_image__cong,axiom,
! [M7: set_real,N5: set_real,F: real > real,G: real > real] :
( ( M7 = N5 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ N5 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_real_real @ F @ M7 )
= ( image_real_real @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_719_image__cong,axiom,
! [M7: set_nat,N5: set_nat,F: nat > nat,G: nat > nat] :
( ( M7 = N5 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_nat @ F @ M7 )
= ( image_nat_nat @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_720_image__cong,axiom,
! [M7: set_nat,N5: set_nat,F: nat > int,G: nat > int] :
( ( M7 = N5 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_int @ F @ M7 )
= ( image_nat_int @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_721_image__cong,axiom,
! [M7: set_nat,N5: set_nat,F: nat > real,G: nat > real] :
( ( M7 = N5 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_real @ F @ M7 )
= ( image_nat_real @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_722_image__cong,axiom,
! [M7: set_nat,N5: set_nat,F: nat > complex,G: nat > complex] :
( ( M7 = N5 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_complex @ F @ M7 )
= ( image_nat_complex @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_723_bex__imageD,axiom,
! [F: real > real,A2: set_real,P: real > $o] :
( ? [X5: real] :
( ( member_real @ X5 @ ( image_real_real @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_724_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_725_bex__imageD,axiom,
! [F: nat > int,A2: set_nat,P: int > $o] :
( ? [X5: int] :
( ( member_int @ X5 @ ( image_nat_int @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_726_bex__imageD,axiom,
! [F: nat > real,A2: set_nat,P: real > $o] :
( ? [X5: real] :
( ( member_real @ X5 @ ( image_nat_real @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_727_bex__imageD,axiom,
! [F: nat > complex,A2: set_nat,P: complex > $o] :
( ? [X5: complex] :
( ( member_complex @ X5 @ ( image_nat_complex @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_728_image__iff,axiom,
! [Z3: int,F: nat > int,A2: set_nat] :
( ( member_int @ Z3 @ ( image_nat_int @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_729_image__iff,axiom,
! [Z3: complex,F: nat > complex,A2: set_nat] :
( ( member_complex @ Z3 @ ( image_nat_complex @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_730_image__iff,axiom,
! [Z3: real,F: real > real,A2: set_real] :
( ( member_real @ Z3 @ ( image_real_real @ F @ A2 ) )
= ( ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_731_image__iff,axiom,
! [Z3: real,F: nat > real,A2: set_nat] :
( ( member_real @ Z3 @ ( image_nat_real @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_732_image__iff,axiom,
! [Z3: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_733_imageI,axiom,
! [X: real,A2: set_real,F: real > real] :
( ( member_real @ X @ A2 )
=> ( member_real @ ( F @ X ) @ ( image_real_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_734_imageI,axiom,
! [X: real,A2: set_real,F: real > nat] :
( ( member_real @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_real_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_735_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > int] :
( ( member_nat @ X @ A2 )
=> ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ A2 ) ) ) ).
% imageI
thf(fact_736_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > complex] :
( ( member_nat @ X @ A2 )
=> ( member_complex @ ( F @ X ) @ ( image_nat_complex @ F @ A2 ) ) ) ).
% imageI
thf(fact_737_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > real] :
( ( member_nat @ X @ A2 )
=> ( member_real @ ( F @ X ) @ ( image_nat_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_738_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_739_in__mono,axiom,
! [A2: set_real,B4: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B4 ) ) ) ).
% in_mono
thf(fact_740_in__mono,axiom,
! [A2: set_nat,B4: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_741_subsetD,axiom,
! [A2: set_real,B4: set_real,C: real] :
( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B4 ) ) ) ).
% subsetD
thf(fact_742_subsetD,axiom,
! [A2: set_nat,B4: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_743_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [X3: real] :
( ( member_real @ X3 @ A5 )
=> ( member_real @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_744_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_745_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [T2: real] :
( ( member_real @ T2 @ A5 )
=> ( member_real @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_746_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_747_psubsetD,axiom,
! [A2: set_real,B4: set_real,C: real] :
( ( ord_less_set_real @ A2 @ B4 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_748_psubsetD,axiom,
! [A2: set_nat,B4: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B4 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_749_rangeI,axiom,
! [F: real > real,X: real] : ( member_real @ ( F @ X ) @ ( image_real_real @ F @ top_top_set_real ) ) ).
% rangeI
thf(fact_750_rangeI,axiom,
! [F: real > nat,X: real] : ( member_nat @ ( F @ X ) @ ( image_real_nat @ F @ top_top_set_real ) ) ).
% rangeI
thf(fact_751_rangeI,axiom,
! [F: nat > int,X: nat] : ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_752_rangeI,axiom,
! [F: nat > complex,X: nat] : ( member_complex @ ( F @ X ) @ ( image_nat_complex @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_753_rangeI,axiom,
! [F: nat > real,X: nat] : ( member_real @ ( F @ X ) @ ( image_nat_real @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_754_rangeI,axiom,
! [F: nat > nat,X: nat] : ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_755_rangeI,axiom,
! [F: complex > real,X: complex] : ( member_real @ ( F @ X ) @ ( image_complex_real @ F @ top_top_set_complex ) ) ).
% rangeI
thf(fact_756_rangeI,axiom,
! [F: complex > nat,X: complex] : ( member_nat @ ( F @ X ) @ ( image_complex_nat @ F @ top_top_set_complex ) ) ).
% rangeI
thf(fact_757_range__eqI,axiom,
! [B: real,F: real > real,X: real] :
( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_real_real @ F @ top_top_set_real ) ) ) ).
% range_eqI
thf(fact_758_range__eqI,axiom,
! [B: nat,F: real > nat,X: real] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_real_nat @ F @ top_top_set_real ) ) ) ).
% range_eqI
thf(fact_759_range__eqI,axiom,
! [B: int,F: nat > int,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_nat_int @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_760_range__eqI,axiom,
! [B: complex,F: nat > complex,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_complex @ B @ ( image_nat_complex @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_761_range__eqI,axiom,
! [B: real,F: nat > real,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_nat_real @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_762_range__eqI,axiom,
! [B: nat,F: nat > nat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_763_range__eqI,axiom,
! [B: real,F: complex > real,X: complex] :
( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_complex_real @ F @ top_top_set_complex ) ) ) ).
% range_eqI
thf(fact_764_range__eqI,axiom,
! [B: nat,F: complex > nat,X: complex] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_complex_nat @ F @ top_top_set_complex ) ) ) ).
% range_eqI
thf(fact_765_subset__UNIV,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ top_top_set_real ) ).
% subset_UNIV
thf(fact_766_subset__UNIV,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_767_subset__UNIV,axiom,
! [A2: set_complex] : ( ord_le211207098394363844omplex @ A2 @ top_top_set_complex ) ).
% subset_UNIV
thf(fact_768_image__mono,axiom,
! [A2: set_real,B4: set_real,F: real > real] :
( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A2 ) @ ( image_real_real @ F @ B4 ) ) ) ).
% image_mono
thf(fact_769_image__mono,axiom,
! [A2: set_nat,B4: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B4 ) ) ) ).
% image_mono
thf(fact_770_image__mono,axiom,
! [A2: set_nat,B4: set_nat,F: nat > int] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ ( image_nat_int @ F @ B4 ) ) ) ).
% image_mono
thf(fact_771_image__mono,axiom,
! [A2: set_nat,B4: set_nat,F: nat > real] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ A2 ) @ ( image_nat_real @ F @ B4 ) ) ) ).
% image_mono
thf(fact_772_image__mono,axiom,
! [A2: set_nat,B4: set_nat,F: nat > complex] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ord_le211207098394363844omplex @ ( image_nat_complex @ F @ A2 ) @ ( image_nat_complex @ F @ B4 ) ) ) ).
% image_mono
thf(fact_773_image__subsetI,axiom,
! [A2: set_real,F: real > real,B4: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( member_real @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_774_image__subsetI,axiom,
! [A2: set_real,F: real > nat,B4: set_nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_real_nat @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_775_image__subsetI,axiom,
! [A2: set_nat,F: nat > int,B4: set_int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_int @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_776_image__subsetI,axiom,
! [A2: set_nat,F: nat > complex,B4: set_complex] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_complex @ ( F @ X2 ) @ B4 ) )
=> ( ord_le211207098394363844omplex @ ( image_nat_complex @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_777_image__subsetI,axiom,
! [A2: set_nat,F: nat > real,B4: set_real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_real @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_778_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B4: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_779_subset__imageE,axiom,
! [B4: set_real,F: real > real,A2: set_real] :
( ( ord_less_eq_set_real @ B4 @ ( image_real_real @ F @ A2 ) )
=> ~ ! [C2: set_real] :
( ( ord_less_eq_set_real @ C2 @ A2 )
=> ( B4
!= ( image_real_real @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_780_subset__imageE,axiom,
! [B4: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B4
!= ( image_nat_nat @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_781_subset__imageE,axiom,
! [B4: set_int,F: nat > int,A2: set_nat] :
( ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B4
!= ( image_nat_int @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_782_subset__imageE,axiom,
! [B4: set_real,F: nat > real,A2: set_nat] :
( ( ord_less_eq_set_real @ B4 @ ( image_nat_real @ F @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B4
!= ( image_nat_real @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_783_subset__imageE,axiom,
! [B4: set_complex,F: nat > complex,A2: set_nat] :
( ( ord_le211207098394363844omplex @ B4 @ ( image_nat_complex @ F @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B4
!= ( image_nat_complex @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_784_image__subset__iff,axiom,
! [F: nat > int,A2: set_nat,B4: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ B4 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_int @ ( F @ X3 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_785_image__subset__iff,axiom,
! [F: nat > complex,A2: set_nat,B4: set_complex] :
( ( ord_le211207098394363844omplex @ ( image_nat_complex @ F @ A2 ) @ B4 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_complex @ ( F @ X3 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_786_image__subset__iff,axiom,
! [F: real > real,A2: set_real,B4: set_real] :
( ( ord_less_eq_set_real @ ( image_real_real @ F @ A2 ) @ B4 )
= ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ ( F @ X3 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_787_image__subset__iff,axiom,
! [F: nat > real,A2: set_nat,B4: set_real] :
( ( ord_less_eq_set_real @ ( image_nat_real @ F @ A2 ) @ B4 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_real @ ( F @ X3 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_788_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B4 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_789_subset__image__iff,axiom,
! [B4: set_real,F: real > real,A2: set_real] :
( ( ord_less_eq_set_real @ B4 @ ( image_real_real @ F @ A2 ) )
= ( ? [AA: set_real] :
( ( ord_less_eq_set_real @ AA @ A2 )
& ( B4
= ( image_real_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_790_subset__image__iff,axiom,
! [B4: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B4
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_791_subset__image__iff,axiom,
! [B4: set_int,F: nat > int,A2: set_nat] :
( ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B4
= ( image_nat_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_792_subset__image__iff,axiom,
! [B4: set_real,F: nat > real,A2: set_nat] :
( ( ord_less_eq_set_real @ B4 @ ( image_nat_real @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B4
= ( image_nat_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_793_subset__image__iff,axiom,
! [B4: set_complex,F: nat > complex,A2: set_nat] :
( ( ord_le211207098394363844omplex @ B4 @ ( image_nat_complex @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B4
= ( image_nat_complex @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_794_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_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 ) ) ) ) ).
% divide_le_0_iff
thf(fact_795_divide__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 @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_796_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_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_divide_iff
thf(fact_797_divide__nonneg__nonneg,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 @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_798_divide__nonneg__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_799_divide__nonpos__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_800_divide__nonpos__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_801_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_802_divide__neg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_803_divide__neg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_804_divide__pos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_805_divide__pos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_806_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_807_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_808_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_809_divide__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_810_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_811_range__subsetD,axiom,
! [F: real > real,B4: set_real,I: real] :
( ( ord_less_eq_set_real @ ( image_real_real @ F @ top_top_set_real ) @ B4 )
=> ( member_real @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_812_range__subsetD,axiom,
! [F: real > nat,B4: set_nat,I: real] :
( ( ord_less_eq_set_nat @ ( image_real_nat @ F @ top_top_set_real ) @ B4 )
=> ( member_nat @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_813_range__subsetD,axiom,
! [F: nat > int,B4: set_int,I: nat] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ top_top_set_nat ) @ B4 )
=> ( member_int @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_814_range__subsetD,axiom,
! [F: nat > complex,B4: set_complex,I: nat] :
( ( ord_le211207098394363844omplex @ ( image_nat_complex @ F @ top_top_set_nat ) @ B4 )
=> ( member_complex @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_815_range__subsetD,axiom,
! [F: nat > real,B4: set_real,I: nat] :
( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ B4 )
=> ( member_real @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_816_range__subsetD,axiom,
! [F: nat > nat,B4: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B4 )
=> ( member_nat @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_817_range__subsetD,axiom,
! [F: complex > real,B4: set_real,I: complex] :
( ( ord_less_eq_set_real @ ( image_complex_real @ F @ top_top_set_complex ) @ B4 )
=> ( member_real @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_818_range__subsetD,axiom,
! [F: complex > nat,B4: set_nat,I: complex] :
( ( ord_less_eq_set_nat @ ( image_complex_nat @ F @ top_top_set_complex ) @ B4 )
=> ( member_nat @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_819_frac__le,axiom,
! [Y: real,X: real,W2: real,Z3: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z3 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z3 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_le
thf(fact_820_frac__less,axiom,
! [X: real,Y: real,W2: real,Z3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z3 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z3 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_less
thf(fact_821_frac__less2,axiom,
! [X: real,Y: real,W2: real,Z3: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_real @ W2 @ Z3 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z3 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_less2
thf(fact_822_divide__le__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_823_divide__nonneg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_824_divide__nonneg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_pos
thf(fact_825_divide__nonpos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_neg
thf(fact_826_divide__nonpos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_827_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_828_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_829_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_830_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_831_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_832_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_833_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_834_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_835_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_836_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_837_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_838_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_839_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_840_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_841_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_842_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_843_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_844_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_845_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_846_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_847_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_848_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_849_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_850_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_851_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_852_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_853_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_854_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_855_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_856_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_857_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_858_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_859_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_860_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_861_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_862_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_863_real__divide__square__eq,axiom,
! [R2: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
= ( divide_divide_real @ A @ R2 ) ) ).
% real_divide_square_eq
thf(fact_864_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_865_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_866_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_867_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_868_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_869_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_870_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_871_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_872_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_873_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_874_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_875_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_876_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_877_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_878_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_879_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_880_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_881_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_882_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_883_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_884_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_885_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_886_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_887_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_888_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_889_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_890_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_891_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_892_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_893_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_894_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_895_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_896_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_897_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_898_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_899_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_900_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_901_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_902_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_903_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_904_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_905_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_906_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_907_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_908_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_909_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_910_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_911_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_912_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_913_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_914_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_915_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_916_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_917_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_918_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_919_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_920_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_921_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_922_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_923_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_924_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_925_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_926_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_927_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_928_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_929_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_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_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_938_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_939_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_940_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_941_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_942_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_943_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_944_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_945_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_946_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_947_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_948_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_949_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_950_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_951_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_952_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_953_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_954_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_955_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_956_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_957_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_958_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_959_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_960_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_961_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_962_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_963_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_964_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_965_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_966_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_967_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_968_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_969_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_970_mult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_971_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_972_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_973_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_974_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_975_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_976_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_977_mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_978_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_979_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_980_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_981_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_982_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_983_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_984_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_985_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_986_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_987_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_988_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_989_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_990_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_991_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_992_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_993_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_994_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_995_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_996_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_997_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_998_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_999_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1000_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_1001_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_1002_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_1003_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_1004_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_1005_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_1006_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_1007_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_1008_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_1009_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_1010_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_1011_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_1012_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_1013_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_1014_mult__less__le__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1015_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1016_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1017_mult__le__less__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1018_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1019_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1020_mult__right__le__imp__le,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1021_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1022_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1023_mult__left__le__imp__le,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1024_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1025_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1026_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1027_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1028_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1029_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1030_mult__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1031_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1032_mult__strict__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1033_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1034_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1035_mult__right__less__imp__less,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1036_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1037_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1038_mult__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1039_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1040_mult__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1041_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1042_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1043_mult__left__less__imp__less,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1044_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1045_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1046_mult__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1047_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1048_mult__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1049_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1050_Rings_Omono__mult,axiom,
! [A: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A )
=> ( monoto2383616128001712325omplex @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_complex @ ( times_times_complex @ A ) ) ) ).
% Rings.mono_mult
thf(fact_1051_Rings_Omono__mult,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ ( times_times_real @ A ) ) ) ).
% Rings.mono_mult
thf(fact_1052_Rings_Omono__mult,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( times_times_nat @ A ) ) ) ).
% Rings.mono_mult
thf(fact_1053_Rings_Omono__mult,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( monotone_on_int_int @ top_top_set_int @ ord_less_eq_int @ ord_less_eq_int @ ( times_times_int @ A ) ) ) ).
% Rings.mono_mult
thf(fact_1054_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_1055_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_1056_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 )
=> ( ! [M6: nat] :
( ( ord_less_nat @ zero_zero_nat @ M6 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M6 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1057_strict__mono__Suc__iff,axiom,
! [F: nat > real] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_nat @ ord_less_real @ F )
= ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) ) ) ) ).
% strict_mono_Suc_iff
thf(fact_1058_strict__mono__Suc__iff,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
= ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) ) ) ) ).
% strict_mono_Suc_iff
thf(fact_1059_strict__mono__Suc__iff,axiom,
! [F: nat > int] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_nat @ ord_less_int @ F )
= ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) ) ) ) ).
% strict_mono_Suc_iff
thf(fact_1060_strict__monoI__Suc,axiom,
! [F: nat > real] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( monotone_on_nat_real @ top_top_set_nat @ ord_less_nat @ ord_less_real @ F ) ) ).
% strict_monoI_Suc
thf(fact_1061_strict__monoI__Suc,axiom,
! [F: nat > nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_monoI_Suc
thf(fact_1062_strict__monoI__Suc,axiom,
! [F: nat > int] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( monotone_on_nat_int @ top_top_set_nat @ ord_less_nat @ ord_less_int @ F ) ) ).
% strict_monoI_Suc
thf(fact_1063_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_1064_complete__real,axiom,
! [S2: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S2 )
=> ( ? [Z4: real] :
! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ? [Y3: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S2 )
=> ( ord_less_eq_real @ X5 @ Y3 ) )
& ! [Z4: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_1065_continuous__on__cong,axiom,
! [S: set_real,T: set_real,F: real > real,G: real > real] :
( ( S = T )
=> ( ! [X2: real] :
( ( member_real @ X2 @ T )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( topolo5044208981011980120l_real @ S @ F )
= ( topolo5044208981011980120l_real @ T @ G ) ) ) ) ).
% continuous_on_cong
thf(fact_1066_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_1067_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y6: real] :
( ( ord_less_real @ X3 @ Y6 )
| ( X3 = Y6 ) ) ) ) ).
% less_eq_real_def
thf(fact_1068_continuous__on__subset,axiom,
! [S: set_real,F: real > real,T: set_real] :
( ( topolo5044208981011980120l_real @ S @ F )
=> ( ( ord_less_eq_set_real @ T @ S )
=> ( topolo5044208981011980120l_real @ T @ F ) ) ) ).
% continuous_on_subset
thf(fact_1069_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_1070_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_1071_uniformly__continuous__imp__continuous,axiom,
! [S: set_real,F: real > real] :
( ( topolo8845477368217174713l_real @ S @ F )
=> ( topolo5044208981011980120l_real @ S @ F ) ) ).
% uniformly_continuous_imp_continuous
thf(fact_1072_IVT_H,axiom,
! [F: real > nat,A: real,Y: nat,B: real] :
( ( ord_less_eq_nat @ ( F @ A ) @ Y )
=> ( ( ord_less_eq_nat @ Y @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X2: real] :
( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_eq_real @ X2 @ B )
& ( ( F @ X2 )
= Y ) ) ) ) ) ) ).
% IVT'
thf(fact_1073_IVT_H,axiom,
! [F: real > int,A: real,Y: int,B: real] :
( ( ord_less_eq_int @ ( F @ A ) @ Y )
=> ( ( ord_less_eq_int @ Y @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2284712892409288920al_int @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X2: real] :
( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_eq_real @ X2 @ B )
& ( ( F @ X2 )
= Y ) ) ) ) ) ) ).
% IVT'
thf(fact_1074_IVT_H,axiom,
! [F: real > real,A: real,Y: real,B: real] :
( ( ord_less_eq_real @ ( F @ A ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X2: real] :
( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_eq_real @ X2 @ B )
& ( ( F @ X2 )
= Y ) ) ) ) ) ) ).
% IVT'
thf(fact_1075_IVT2_H,axiom,
! [F: real > nat,B: real,Y: nat,A: real] :
( ( ord_less_eq_nat @ ( F @ B ) @ Y )
=> ( ( ord_less_eq_nat @ Y @ ( F @ A ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X2: real] :
( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_eq_real @ X2 @ B )
& ( ( F @ X2 )
= Y ) ) ) ) ) ) ).
% IVT2'
thf(fact_1076_IVT2_H,axiom,
! [F: real > int,B: real,Y: int,A: real] :
( ( ord_less_eq_int @ ( F @ B ) @ Y )
=> ( ( ord_less_eq_int @ Y @ ( F @ A ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2284712892409288920al_int @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X2: real] :
( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_eq_real @ X2 @ B )
& ( ( F @ X2 )
= Y ) ) ) ) ) ) ).
% IVT2'
thf(fact_1077_IVT2_H,axiom,
! [F: real > real,B: real,Y: real,A: real] :
( ( ord_less_eq_real @ ( F @ B ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( F @ A ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X2: real] :
( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_eq_real @ X2 @ B )
& ( ( F @ X2 )
= Y ) ) ) ) ) ) ).
% IVT2'
thf(fact_1078_continuous__on__compose,axiom,
! [S: set_nat,F: nat > real,G: real > real] :
( ( topolo6943266826644216316t_real @ S @ F )
=> ( ( topolo5044208981011980120l_real @ ( image_nat_real @ F @ S ) @ G )
=> ( topolo6943266826644216316t_real @ S @ ( comp_real_real_nat @ G @ F ) ) ) ) ).
% continuous_on_compose
thf(fact_1079_continuous__on__compose,axiom,
! [S: set_real,F: real > real,G: real > real] :
( ( topolo5044208981011980120l_real @ S @ F )
=> ( ( topolo5044208981011980120l_real @ ( image_real_real @ F @ S ) @ G )
=> ( topolo5044208981011980120l_real @ S @ ( comp_real_real_real @ G @ F ) ) ) ) ).
% continuous_on_compose
thf(fact_1080_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_1081_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_1082_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_1083_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_1084_strict__mono__leD,axiom,
! [R2: complex > real,M: complex,N: complex] :
( ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_complex @ ord_less_real @ R2 )
=> ( ( ord_less_eq_complex @ M @ N )
=> ( ord_less_eq_real @ ( R2 @ M ) @ ( R2 @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_1085_strict__mono__leD,axiom,
! [R2: complex > nat,M: complex,N: complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_complex @ ord_less_nat @ R2 )
=> ( ( ord_less_eq_complex @ M @ N )
=> ( ord_less_eq_nat @ ( R2 @ M ) @ ( R2 @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_1086_strict__mono__leD,axiom,
! [R2: complex > int,M: complex,N: complex] :
( ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_complex @ ord_less_int @ R2 )
=> ( ( ord_less_eq_complex @ M @ N )
=> ( ord_less_eq_int @ ( R2 @ M ) @ ( R2 @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_1087_strict__mono__leD,axiom,
! [R2: real > real,M: real,N: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ R2 )
=> ( ( ord_less_eq_real @ M @ N )
=> ( ord_less_eq_real @ ( R2 @ M ) @ ( R2 @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_1088_strict__mono__leD,axiom,
! [R2: real > nat,M: real,N: real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_real @ ord_less_nat @ R2 )
=> ( ( ord_less_eq_real @ M @ N )
=> ( ord_less_eq_nat @ ( R2 @ M ) @ ( R2 @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_1089_strict__mono__leD,axiom,
! [R2: real > int,M: real,N: real] :
( ( monotone_on_real_int @ top_top_set_real @ ord_less_real @ ord_less_int @ R2 )
=> ( ( ord_less_eq_real @ M @ N )
=> ( ord_less_eq_int @ ( R2 @ M ) @ ( R2 @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_1090_strict__mono__leD,axiom,
! [R2: nat > real,M: nat,N: nat] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_nat @ ord_less_real @ R2 )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_real @ ( R2 @ M ) @ ( R2 @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_1091_strict__mono__leD,axiom,
! [R2: nat > nat,M: nat,N: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R2 )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( R2 @ M ) @ ( R2 @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_1092_strict__mono__leD,axiom,
! [R2: nat > int,M: nat,N: nat] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_nat @ ord_less_int @ R2 )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_int @ ( R2 @ M ) @ ( R2 @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_1093_strict__mono__leD,axiom,
! [R2: int > real,M: int,N: int] :
( ( monotone_on_int_real @ top_top_set_int @ ord_less_int @ ord_less_real @ R2 )
=> ( ( ord_less_eq_int @ M @ N )
=> ( ord_less_eq_real @ ( R2 @ M ) @ ( R2 @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_1094_strict__mono__o,axiom,
! [R2: complex > real,S: complex > complex] :
( ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_complex @ ord_less_real @ R2 )
=> ( ( monoto2383616128001712325omplex @ top_top_set_complex @ ord_less_complex @ ord_less_complex @ S )
=> ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_complex @ ord_less_real @ ( comp_c2063761206571265261omplex @ R2 @ S ) ) ) ) ).
% strict_mono_o
thf(fact_1095_strict__mono__o,axiom,
! [R2: complex > real,S: real > complex] :
( ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_complex @ ord_less_real @ R2 )
=> ( ( monoto7309693138617929411omplex @ top_top_set_real @ ord_less_real @ ord_less_complex @ S )
=> ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ ( comp_c3333796836230738283l_real @ R2 @ S ) ) ) ) ).
% strict_mono_o
thf(fact_1096_strict__mono__o,axiom,
! [R2: complex > real,S: nat > complex] :
( ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_complex @ ord_less_real @ R2 )
=> ( ( monoto8010102104071190503omplex @ top_top_set_nat @ ord_less_nat @ ord_less_complex @ S )
=> ( monotone_on_nat_real @ top_top_set_nat @ ord_less_nat @ ord_less_real @ ( comp_c7990426058975542799al_nat @ R2 @ S ) ) ) ) ).
% strict_mono_o
thf(fact_1097_strict__mono__o,axiom,
! [R2: complex > real,S: int > complex] :
( ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_complex @ ord_less_real @ R2 )
=> ( ( monoto8985637569277367875omplex @ top_top_set_int @ ord_less_int @ ord_less_complex @ S )
=> ( monotone_on_int_real @ top_top_set_int @ ord_less_int @ ord_less_real @ ( comp_c7987935588466492523al_int @ R2 @ S ) ) ) ) ).
% strict_mono_o
thf(fact_1098_strict__mono__o,axiom,
! [R2: complex > nat,S: complex > complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_complex @ ord_less_nat @ R2 )
=> ( ( monoto2383616128001712325omplex @ top_top_set_complex @ ord_less_complex @ ord_less_complex @ S )
=> ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_complex @ ord_less_nat @ ( comp_c3648228598504721169omplex @ R2 @ S ) ) ) ) ).
% strict_mono_o
thf(fact_1099_strict__mono__o,axiom,
! [R2: complex > nat,S: real > complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_complex @ ord_less_nat @ R2 )
=> ( ( monoto7309693138617929411omplex @ top_top_set_real @ ord_less_real @ ord_less_complex @ S )
=> ( monotone_on_real_nat @ top_top_set_real @ ord_less_real @ ord_less_nat @ ( comp_c3423117485846644111t_real @ R2 @ S ) ) ) ) ).
% strict_mono_o
thf(fact_1100_strict__mono__o,axiom,
! [R2: complex > nat,S: nat > complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_complex @ ord_less_nat @ R2 )
=> ( ( monoto8010102104071190503omplex @ top_top_set_nat @ ord_less_nat @ ord_less_complex @ S )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ ( comp_complex_nat_nat @ R2 @ S ) ) ) ) ).
% strict_mono_o
thf(fact_1101_strict__mono__o,axiom,
! [R2: complex > nat,S: int > complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_complex @ ord_less_nat @ R2 )
=> ( ( monoto8985637569277367875omplex @ top_top_set_int @ ord_less_int @ ord_less_complex @ S )
=> ( monotone_on_int_nat @ top_top_set_int @ ord_less_int @ ord_less_nat @ ( comp_complex_nat_int @ R2 @ S ) ) ) ) ).
% strict_mono_o
thf(fact_1102_strict__mono__o,axiom,
! [R2: complex > int,S: complex > complex] :
( ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_complex @ ord_less_int @ R2 )
=> ( ( monoto2383616128001712325omplex @ top_top_set_complex @ ord_less_complex @ ord_less_complex @ S )
=> ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_complex @ ord_less_int @ ( comp_c4623764063710898541omplex @ R2 @ S ) ) ) ) ).
% strict_mono_o
thf(fact_1103_strict__mono__o,axiom,
! [R2: complex > int,S: real > complex] :
( ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_complex @ ord_less_int @ R2 )
=> ( ( monoto7309693138617929411omplex @ top_top_set_real @ ord_less_real @ ord_less_complex @ S )
=> ( monotone_on_real_int @ top_top_set_real @ ord_less_real @ ord_less_int @ ( comp_c5610039060539665899t_real @ R2 @ S ) ) ) ) ).
% strict_mono_o
thf(fact_1104_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y4: real] :
? [N2: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1105_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_1106_incseq__def,axiom,
! [X6: nat > real] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_real @ X6 )
= ( ! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_real @ ( X6 @ M2 ) @ ( X6 @ N3 ) ) ) ) ) ).
% incseq_def
thf(fact_1107_incseq__def,axiom,
! [X6: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ X6 )
= ( ! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ ( X6 @ M2 ) @ ( X6 @ N3 ) ) ) ) ) ).
% incseq_def
thf(fact_1108_incseq__def,axiom,
! [X6: nat > int] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_int @ X6 )
= ( ! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_int @ ( X6 @ M2 ) @ ( X6 @ N3 ) ) ) ) ) ).
% incseq_def
thf(fact_1109_incseqD,axiom,
! [F: nat > real,I: nat,J: nat] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% incseqD
thf(fact_1110_incseqD,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% incseqD
thf(fact_1111_incseqD,axiom,
! [F: nat > int,I: nat,J: nat] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% incseqD
thf(fact_1112_incseq__SucD,axiom,
! [A2: nat > real,I: nat] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_real @ A2 )
=> ( ord_less_eq_real @ ( A2 @ I ) @ ( A2 @ ( suc @ I ) ) ) ) ).
% incseq_SucD
thf(fact_1113_incseq__SucD,axiom,
! [A2: nat > nat,I: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ A2 )
=> ( ord_less_eq_nat @ ( A2 @ I ) @ ( A2 @ ( suc @ I ) ) ) ) ).
% incseq_SucD
thf(fact_1114_incseq__SucD,axiom,
! [A2: nat > int,I: nat] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_int @ A2 )
=> ( ord_less_eq_int @ ( A2 @ I ) @ ( A2 @ ( suc @ I ) ) ) ) ).
% incseq_SucD
thf(fact_1115_incseq__SucI,axiom,
! [X6: nat > real] :
( ! [N2: nat] : ( ord_less_eq_real @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
=> ( monotone_on_nat_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_real @ X6 ) ) ).
% incseq_SucI
thf(fact_1116_incseq__SucI,axiom,
! [X6: nat > nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ X6 ) ) ).
% incseq_SucI
thf(fact_1117_incseq__SucI,axiom,
! [X6: nat > int] :
( ! [N2: nat] : ( ord_less_eq_int @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
=> ( monotone_on_nat_int @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_int @ X6 ) ) ).
% incseq_SucI
thf(fact_1118_incseq__Suc__iff,axiom,
! [F: nat > real] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_real @ F )
= ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) ) ) ) ).
% incseq_Suc_iff
thf(fact_1119_incseq__Suc__iff,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
= ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) ) ) ) ).
% incseq_Suc_iff
thf(fact_1120_incseq__Suc__iff,axiom,
! [F: nat > int] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_int @ F )
= ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) ) ) ) ).
% incseq_Suc_iff
thf(fact_1121_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X: real] :
( ( ( times_times_real @ A @ X )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_1122_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: real] :
( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_1123_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_1124_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X: real,Y: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y ) )
= ( ( X = Y )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_1125_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_1126_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_1127_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X: real,B: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ B @ X ) )
= ( ( A = B )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_1128_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_1129_zdiv__mono2,axiom,
! [A: int,B6: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B6 )
=> ( ( ord_less_eq_int @ B6 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B6 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1130_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_1131_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_1132_zdiv__mono2__neg,axiom,
! [A: int,B6: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B6 )
=> ( ( ord_less_eq_int @ B6 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B6 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1133_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_1134_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_1135_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_1136_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_1137_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_1138_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_1139_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_1140_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_1141_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_1142_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_1143_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_1144_vector__space__over__itself_Oscale__scale,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X ) )
= ( times_times_real @ ( times_times_real @ A @ B ) @ X ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_1145_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X ) )
= ( times_times_real @ B @ ( times_times_real @ A @ X ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_1146_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: real,A: real,B: real] :
( ( X != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ B @ X ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_1147_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: real,X: real,Y: real] :
( ( A != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y ) )
=> ( X = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_1148_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_1149_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 )
=> ? [C3: real,D3: real] :
( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ C3 @ D3 ) )
& ( ord_less_eq_real @ C3 @ D3 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_comp__apply,axiom,
( comp_real_real_real
= ( ^ [F2: real > real,G2: real > real,X3: real] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% comp_apply
thf(fact_1157_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_1158_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_1159_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_1160_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_1161_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_1162_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_1163_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1164_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1165_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1166_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_1167_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_1168_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1169_lower__def,axiom,
( lower
= ( ^ [X3: real] : ( a_seg @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ n ) @ X3 ) @ a ) ) ) ) ) ) ).
% lower_def
thf(fact_1170_upper__def,axiom,
( upper
= ( ^ [X3: real] : ( a_seg @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ n ) @ X3 ) @ a ) ) ) ) ) ) ).
% upper_def
thf(fact_1171_k__def,axiom,
( k
= ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ ( divide_divide_real @ x @ a ) @ ( semiri5074537144036343181t_real @ n ) ) ) ) ) ).
% k_def
thf(fact_1172_n__def,axiom,
( n
= ( nat2 @ ( archim6058952711729229775r_real @ ( divide_divide_real @ a @ delta ) ) ) ) ).
% n_def
thf(fact_1173_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ Z3 @ zero_zero_int )
=> ( ( nat2 @ Z3 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1174_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1175_zless__nat__conj,axiom,
! [W2: int,Z3: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ( ord_less_int @ zero_zero_int @ Z3 )
& ( ord_less_int @ W2 @ Z3 ) ) ) ).
% zless_nat_conj
thf(fact_1176_zero__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% zero_less_nat_eq
thf(fact_1177_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_1178_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_1179_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_1180_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_1181_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_1182_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_1183_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_1184_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_1185_nat__mono__iff,axiom,
! [Z3: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ) ).
% nat_mono_iff
thf(fact_1186_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_1187_zless__nat__eq__int__zless,axiom,
! [M: nat,Z3: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z3 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z3 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1188_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_1189_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_1190_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_1191_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N3: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N3 ) )
=> ( P @ N3 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1192_nat__le__eq__zle,axiom,
! [W2: int,Z3: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z3 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ) ).
% nat_le_eq_zle
thf(fact_1193_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_1194_nat__less__eq__zless,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ) ).
% nat_less_eq_zless
thf(fact_1195_nat__mult__distrib,axiom,
! [Z3: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( nat2 @ ( times_times_int @ Z3 @ Z5 ) )
= ( times_times_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z5 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1196_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_1197_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_1198_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_1199_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_1200_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1201_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1202_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1203_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1204_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less
thf(fact_1205_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less
thf(fact_1206_seq__mono__lemma,axiom,
! [M: nat,D: nat > real,E: nat > real] :
( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_real @ ( D @ N2 ) @ ( E @ N2 ) ) )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_real @ ( E @ N2 ) @ ( E @ M ) ) )
=> ! [N6: nat] :
( ( ord_less_eq_nat @ M @ N6 )
=> ( ord_less_real @ ( D @ N6 ) @ ( E @ M ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1207_real__non__denum,axiom,
~ ? [F3: nat > real] :
( ( image_nat_real @ F3 @ top_top_set_nat )
= top_top_set_real ) ).
% real_non_denum
thf(fact_1208_complex__non__denum,axiom,
~ ? [F3: nat > complex] :
( ( image_nat_complex @ F3 @ top_top_set_nat )
= top_top_set_complex ) ).
% complex_non_denum
thf(fact_1209_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 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1210_one__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ one_one_int @ Z3 ) ) ).
% one_less_nat_eq
thf(fact_1211_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1212_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_1213_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_1214_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_1215_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_1216_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1217_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1218_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1219_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_1220_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1221_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1222_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_1223_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_1224_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q3: nat > $o] :
( ! [X2: nat > real] :
( ( P @ X2 )
=> ( P @ ( F @ X2 ) ) )
=> ( ! [X2: nat > real] :
( ( P @ X2 )
=> ! [I2: nat] :
( ( Q3 @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I2 ) )
& ( ord_less_eq_real @ ( X2 @ I2 ) @ one_one_real ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X5: nat > real,I4: nat] : ( ord_less_eq_nat @ ( L3 @ X5 @ I4 ) @ one_one_nat )
& ! [X5: nat > real,I4: nat] :
( ( ( P @ X5 )
& ( Q3 @ I4 )
& ( ( X5 @ I4 )
= zero_zero_real ) )
=> ( ( L3 @ X5 @ I4 )
= zero_zero_nat ) )
& ! [X5: nat > real,I4: nat] :
( ( ( P @ X5 )
& ( Q3 @ I4 )
& ( ( X5 @ I4 )
= one_one_real ) )
=> ( ( L3 @ X5 @ I4 )
= one_one_nat ) )
& ! [X5: nat > real,I4: nat] :
( ( ( P @ X5 )
& ( Q3 @ I4 )
& ( ( L3 @ X5 @ I4 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X5 @ I4 ) @ ( F @ X5 @ I4 ) ) )
& ! [X5: nat > real,I4: nat] :
( ( ( P @ X5 )
& ( Q3 @ I4 )
& ( ( L3 @ X5 @ I4 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X5 @ I4 ) @ ( X5 @ I4 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1225_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_1226_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1227_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1228_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_1229_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1230_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_1231_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_1232_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_1233_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_1234_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_1235_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_1236_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_1237_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_1238_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_1239_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_1240_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_1241_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_1242_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_1243_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_1244_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_1245_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1246_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_1247_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_1248_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_1249_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1250_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1251_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1252_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1253_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1254_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1255_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1256_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_1257_int__distrib_I2_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1258_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W2 )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(1)
thf(fact_1259_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1260_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1261_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1262_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1263_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1264_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1265_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1266_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
% Conjectures (2)
thf(conj_0,hypothesis,
( ( ord_less_eq_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ k ) ) ) @ y )
=> ( ( ord_less_real @ y @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ k ) ) ) ) )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------