TPTP Problem File: SLH0995^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Youngs_Inequality/0000_Youngs/prob_00201_008085__12940282_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1377 ( 437 unt; 110 typ; 0 def)
% Number of atoms : 4442 ( 953 equ; 0 cnn)
% Maximal formula atoms : 26 ( 3 avg)
% Number of connectives : 13140 ( 356 ~; 106 |; 277 &;10193 @)
% ( 0 <=>;2208 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 8 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 913 ( 913 >; 0 *; 0 +; 0 <<)
% Number of symbols : 106 ( 103 usr; 12 con; 0-4 aty)
% Number of variables : 4298 ( 200 ^;3955 !; 143 ?;4298 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:27:15.754
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
set_set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (103)
thf(sy_c_Abstract__Topology__2_Oretraction_001t__Int__Oint,type,
abstra7169501481011290569on_int: set_int > set_int > ( int > int ) > $o ).
thf(sy_c_Abstract__Topology__2_Oretraction_001t__Real__Oreal,type,
abstra2606333701016485833n_real: set_real > set_real > ( real > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Int__Oint_001t__Int__Oint,type,
monotone_on_int_int: set_int > ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Int__Oint_001t__Nat__Onat,type,
monotone_on_int_nat: set_int > ( int > int > $o ) > ( nat > nat > $o ) > ( int > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Int__Oint_001t__Real__Oreal,type,
monotone_on_int_real: set_int > ( int > int > $o ) > ( real > real > $o ) > ( int > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Int__Oint_001t__Set__Oset_It__Real__Oreal_J,type,
monoto4059737280683413495t_real: set_int > ( int > int > $o ) > ( set_real > set_real > $o ) > ( int > set_real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Int__Oint,type,
monotone_on_nat_int: set_nat > ( nat > nat > $o ) > ( int > int > $o ) > ( nat > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
monotone_on_nat_nat: set_nat > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Real__Oreal,type,
monotone_on_nat_real: set_nat > ( nat > nat > $o ) > ( real > real > $o ) > ( nat > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Int__Oint,type,
monotone_on_real_int: set_real > ( real > real > $o ) > ( int > int > $o ) > ( real > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Nat__Onat,type,
monotone_on_real_nat: set_real > ( real > real > $o ) > ( nat > nat > $o ) > ( real > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Real__Oreal,type,
monoto4017252874604999745l_real: set_real > ( real > real > $o ) > ( real > real > $o ) > ( real > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Set__Oset_It__Real__Oreal_J,type,
monoto3333417327835629687t_real: set_real > ( real > real > $o ) > ( set_real > set_real > $o ) > ( real > set_real ) > $o ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
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_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
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_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Int__Oint_J,type,
plus_plus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Real__Oreal_J,type,
plus_plus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
sup_sup_int: int > int > int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Real__Oreal,type,
sup_sup_real: real > real > real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Int__Oint_J,type,
sup_sup_set_int: set_int > set_int > set_int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Real__Oreal_J,type,
sup_sup_set_real: set_real > set_real > set_real ).
thf(sy_c_Linear__Algebra_Oinfnorm_001t__Real__Oreal,type,
linear_infnorm_real: real > real ).
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_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
ord_le7926960851185191020t_real: set_set_real > set_set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
ord_le3558479182127378552t_real: set_set_real > set_set_real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Real__Oreal,type,
image_int_real: ( int > real ) > set_int > set_real ).
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__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_OatLeastAtMost_001t__Set__Oset_It__Real__Oreal_J,type,
set_or7743017856606604397t_real: set_real > set_real > set_set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Int__Oint,type,
set_ord_atLeast_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
set_ord_atLeast_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Real__Oreal_J,type,
set_or6787452043721853143t_real: set_real > set_set_real ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
set_or6656581121297822940st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Real__Oreal,type,
set_or2392270231875598684t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_It__Real__Oreal_J,type,
set_or3605051941450457490t_real: set_real > set_real > set_set_real ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
set_or1633881224788618240n_real: real > real > set_real ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Int__Oint_001t__Int__Oint,type,
topolo2178910747331673048nt_int: set_int > ( int > int ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Int__Oint_001t__Nat__Onat,type,
topolo2181401217840723324nt_nat: set_int > ( int > nat ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Int__Oint_001t__Real__Oreal,type,
topolo9130188401337238104t_real: set_int > ( int > real ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Nat__Onat_001t__Int__Oint,type,
topolo1179557035430618492at_int: set_nat > ( nat > int ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Nat__Onat_001t__Nat__Onat,type,
topolo1182047505939668768at_nat: set_nat > ( nat > nat ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Int__Oint,type,
topolo2284712892409288920al_int: set_real > ( real > int ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Nat__Onat,type,
topolo2287203362918339196al_nat: set_real > ( real > nat ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
member_set_real: set_real > set_set_real > $o ).
thf(sy_v_a,type,
a: real ).
thf(sy_v_f,type,
f: real > real ).
thf(sy_v_g,type,
g: real > real ).
thf(sy_v_u____,type,
u: real ).
thf(sy_v_y____,type,
y: real ).
% Relevant facts (1263)
thf(fact_0_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_1_contf,axiom,
topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ a ) @ f ).
% contf
thf(fact_2_fim,axiom,
( ( image_real_real @ f @ ( set_ord_atLeast_real @ a ) )
= ( set_ord_atLeast_real @ ( f @ a ) ) ) ).
% fim
thf(fact_3_u_I2_J,axiom,
ord_less_eq_real @ a @ u ).
% u(2)
thf(fact_4_sm,axiom,
monoto4017252874604999745l_real @ ( set_ord_atLeast_real @ a ) @ ord_less_real @ ord_less_real @ f ).
% sm
thf(fact_5_g,axiom,
! [X: real] :
( ( ord_less_eq_real @ a @ X )
=> ( ( g @ ( f @ X ) )
= X ) ) ).
% g
thf(fact_6_calculation_I1_J,axiom,
topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ a @ u ) @ f ).
% calculation(1)
thf(fact_7__092_060open_062f_Aa_A_092_060le_062_Ay_092_060close_062,axiom,
ord_less_eq_real @ ( f @ a ) @ y ).
% \<open>f a \<le> y\<close>
thf(fact_8_calculation_I2_J,axiom,
monoto4017252874604999745l_real @ ( set_or1222579329274155063t_real @ a @ u ) @ ord_less_real @ ord_less_real @ f ).
% calculation(2)
thf(fact_9__092_060open_062_092_060And_062thesis_O_A_I_092_060lbrakk_062continuous__on_A_123a_O_Ou_125_Af_059_Astrict__mono__on_A_123a_O_Ou_125_Af_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ a @ u ) @ f )
=> ~ ( monoto4017252874604999745l_real @ ( set_or1222579329274155063t_real @ a @ u ) @ ord_less_real @ ord_less_real @ f ) ) ).
% \<open>\<And>thesis. (\<lbrakk>continuous_on {a..u} f; strict_mono_on {a..u} f\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_10_image__eqI,axiom,
! [B: int,F: nat > int,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_int @ B @ ( image_nat_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_11_image__eqI,axiom,
! [B: real,F: real > real,X: real,A: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A )
=> ( member_real @ B @ ( image_real_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_12_image__eqI,axiom,
! [B: int,F: real > int,X: real,A: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A )
=> ( member_int @ B @ ( image_real_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_13_image__eqI,axiom,
! [B: real,F: int > real,X: int,A: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A )
=> ( member_real @ B @ ( image_int_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_14_image__eqI,axiom,
! [B: int,F: int > int,X: int,A: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A )
=> ( member_int @ B @ ( image_int_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_15_continuous__image__closed__interval,axiom,
! [A2: real,B: real,F: real > real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B ) @ F )
=> ? [C: real,D: real] :
( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A2 @ B ) )
= ( set_or1222579329274155063t_real @ C @ D ) )
& ( ord_less_eq_real @ C @ D ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_16_strict__mono__image__endpoints,axiom,
! [A2: real,B: real,F: real > real] :
( ( monoto4017252874604999745l_real @ ( set_or1222579329274155063t_real @ A2 @ B ) @ ord_less_real @ ord_less_real @ F )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B ) @ F )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A2 @ B ) )
= ( set_or1222579329274155063t_real @ ( F @ A2 ) @ ( F @ B ) ) ) ) ) ) ).
% strict_mono_image_endpoints
thf(fact_17_strict__mono__image__endpoints,axiom,
! [A2: real,B: real,F: real > nat] :
( ( monotone_on_real_nat @ ( set_or1222579329274155063t_real @ A2 @ B ) @ ord_less_real @ ord_less_nat @ F )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A2 @ B ) @ F )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ( image_real_nat @ F @ ( set_or1222579329274155063t_real @ A2 @ B ) )
= ( set_or1269000886237332187st_nat @ ( F @ A2 ) @ ( F @ B ) ) ) ) ) ) ).
% strict_mono_image_endpoints
thf(fact_18_strict__mono__image__endpoints,axiom,
! [A2: real,B: real,F: real > int] :
( ( monotone_on_real_int @ ( set_or1222579329274155063t_real @ A2 @ B ) @ ord_less_real @ ord_less_int @ F )
=> ( ( topolo2284712892409288920al_int @ ( set_or1222579329274155063t_real @ A2 @ B ) @ F )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ( image_real_int @ F @ ( set_or1222579329274155063t_real @ A2 @ B ) )
= ( set_or1266510415728281911st_int @ ( F @ A2 ) @ ( F @ B ) ) ) ) ) ) ).
% strict_mono_image_endpoints
thf(fact_19_u_I1_J,axiom,
( ( plus_plus_real @ y @ one_one_real )
= ( f @ u ) ) ).
% u(1)
thf(fact_20_IVT_H,axiom,
! [F: real > nat,A2: real,Y: nat,B: real] :
( ( ord_less_eq_nat @ ( F @ A2 ) @ Y )
=> ( ( ord_less_eq_nat @ Y @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A2 @ B ) @ F )
=> ? [X2: real] :
( ( ord_less_eq_real @ A2 @ X2 )
& ( ord_less_eq_real @ X2 @ B )
& ( ( F @ X2 )
= Y ) ) ) ) ) ) ).
% IVT'
thf(fact_21_IVT_H,axiom,
! [F: real > int,A2: real,Y: int,B: real] :
( ( ord_less_eq_int @ ( F @ A2 ) @ Y )
=> ( ( ord_less_eq_int @ Y @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ( topolo2284712892409288920al_int @ ( set_or1222579329274155063t_real @ A2 @ B ) @ F )
=> ? [X2: real] :
( ( ord_less_eq_real @ A2 @ X2 )
& ( ord_less_eq_real @ X2 @ B )
& ( ( F @ X2 )
= Y ) ) ) ) ) ) ).
% IVT'
thf(fact_22_IVT_H,axiom,
! [F: real > real,A2: real,Y: real,B: real] :
( ( ord_less_eq_real @ ( F @ A2 ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B ) @ F )
=> ? [X2: real] :
( ( ord_less_eq_real @ A2 @ X2 )
& ( ord_less_eq_real @ X2 @ B )
& ( ( F @ X2 )
= Y ) ) ) ) ) ) ).
% IVT'
thf(fact_23_IVT2_H,axiom,
! [F: real > nat,B: real,Y: nat,A2: real] :
( ( ord_less_eq_nat @ ( F @ B ) @ Y )
=> ( ( ord_less_eq_nat @ Y @ ( F @ A2 ) )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A2 @ B ) @ F )
=> ? [X2: real] :
( ( ord_less_eq_real @ A2 @ X2 )
& ( ord_less_eq_real @ X2 @ B )
& ( ( F @ X2 )
= Y ) ) ) ) ) ) ).
% IVT2'
thf(fact_24_IVT2_H,axiom,
! [F: real > int,B: real,Y: int,A2: real] :
( ( ord_less_eq_int @ ( F @ B ) @ Y )
=> ( ( ord_less_eq_int @ Y @ ( F @ A2 ) )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ( topolo2284712892409288920al_int @ ( set_or1222579329274155063t_real @ A2 @ B ) @ F )
=> ? [X2: real] :
( ( ord_less_eq_real @ A2 @ X2 )
& ( ord_less_eq_real @ X2 @ B )
& ( ( F @ X2 )
= Y ) ) ) ) ) ) ).
% IVT2'
thf(fact_25_IVT2_H,axiom,
! [F: real > real,B: real,Y: real,A2: real] :
( ( ord_less_eq_real @ ( F @ B ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( F @ A2 ) )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B ) @ F )
=> ? [X2: real] :
( ( ord_less_eq_real @ A2 @ X2 )
& ( ord_less_eq_real @ X2 @ B )
& ( ( F @ X2 )
= Y ) ) ) ) ) ) ).
% IVT2'
thf(fact_26_invertible__fixpoint__property,axiom,
! [T2: set_nat,I: nat > int,S2: set_int,R: int > nat,G: nat > nat] :
( ( topolo1179557035430618492at_int @ T2 @ I )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ I @ T2 ) @ S2 )
=> ( ( topolo2181401217840723324nt_nat @ S2 @ R )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ R @ S2 ) @ T2 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F2: int > int] :
( ( topolo2178910747331673048nt_int @ S2 @ F2 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F2 @ S2 ) @ S2 )
=> ? [X3: int] :
( ( member_int @ X3 @ S2 )
& ( ( F2 @ X3 )
= X3 ) ) ) )
=> ( ( topolo1182047505939668768at_nat @ T2 @ G )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ T2 ) @ T2 )
=> ~ ! [Y2: nat] :
( ( member_nat @ Y2 @ T2 )
=> ( ( G @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_27_invertible__fixpoint__property,axiom,
! [T2: set_int,I: int > nat,S2: set_nat,R: nat > int,G: int > int] :
( ( topolo2181401217840723324nt_nat @ T2 @ I )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ I @ T2 ) @ S2 )
=> ( ( topolo1179557035430618492at_int @ S2 @ R )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ R @ S2 ) @ T2 )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F2: nat > nat] :
( ( topolo1182047505939668768at_nat @ S2 @ F2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ S2 ) @ S2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S2 )
& ( ( F2 @ X3 )
= X3 ) ) ) )
=> ( ( topolo2178910747331673048nt_int @ T2 @ G )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ G @ T2 ) @ T2 )
=> ~ ! [Y2: int] :
( ( member_int @ Y2 @ T2 )
=> ( ( G @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_28_invertible__fixpoint__property,axiom,
! [T2: set_int,I: int > real,S2: set_real,R: real > int,G: int > int] :
( ( topolo9130188401337238104t_real @ T2 @ I )
=> ( ( ord_less_eq_set_real @ ( image_int_real @ I @ T2 ) @ S2 )
=> ( ( topolo2284712892409288920al_int @ S2 @ R )
=> ( ( ord_less_eq_set_int @ ( image_real_int @ R @ S2 ) @ T2 )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F2: real > real] :
( ( topolo5044208981011980120l_real @ S2 @ F2 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ F2 @ S2 ) @ S2 )
=> ? [X3: real] :
( ( member_real @ X3 @ S2 )
& ( ( F2 @ X3 )
= X3 ) ) ) )
=> ( ( topolo2178910747331673048nt_int @ T2 @ G )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ G @ T2 ) @ T2 )
=> ~ ! [Y2: int] :
( ( member_int @ Y2 @ T2 )
=> ( ( G @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_29_invertible__fixpoint__property,axiom,
! [T2: set_real,I: real > real,S2: set_real,R: real > real,G: real > real] :
( ( topolo5044208981011980120l_real @ T2 @ I )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ I @ T2 ) @ S2 )
=> ( ( topolo5044208981011980120l_real @ S2 @ R )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ R @ S2 ) @ T2 )
=> ( ! [Y2: real] :
( ( member_real @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F2: real > real] :
( ( topolo5044208981011980120l_real @ S2 @ F2 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ F2 @ S2 ) @ S2 )
=> ? [X3: real] :
( ( member_real @ X3 @ S2 )
& ( ( F2 @ X3 )
= X3 ) ) ) )
=> ( ( topolo5044208981011980120l_real @ T2 @ G )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ G @ T2 ) @ T2 )
=> ~ ! [Y2: real] :
( ( member_real @ Y2 @ T2 )
=> ( ( G @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_30_continuous__on__eq,axiom,
! [S: set_real,F: real > real,G: real > real] :
( ( topolo5044208981011980120l_real @ S @ F )
=> ( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( topolo5044208981011980120l_real @ S @ G ) ) ) ).
% continuous_on_eq
thf(fact_31_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_32_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > int,D2: nat > int,Inf: set_int > int] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_nat_int @ C2 @ A ) )
= ( Inf @ ( image_nat_int @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_33_Inf_OINF__cong,axiom,
! [A: set_real,B2: set_real,C2: real > real,D2: real > real,Inf: set_real > real] :
( ( A = B2 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_real_real @ C2 @ A ) )
= ( Inf @ ( image_real_real @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_34_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > int,D2: nat > int,Sup: set_int > int] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_nat_int @ C2 @ A ) )
= ( Sup @ ( image_nat_int @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_35_Sup_OSUP__cong,axiom,
! [A: set_real,B2: set_real,C2: real > real,D2: real > real,Sup: set_real > real] :
( ( A = B2 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_real_real @ C2 @ A ) )
= ( Sup @ ( image_real_real @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_36_subsetI,axiom,
! [A: set_int,B2: set_int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_int @ X2 @ B2 ) )
=> ( ord_less_eq_set_int @ A @ B2 ) ) ).
% subsetI
thf(fact_37_subsetI,axiom,
! [A: set_real,B2: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_real @ X2 @ B2 ) )
=> ( ord_less_eq_set_real @ A @ B2 ) ) ).
% subsetI
thf(fact_38_psubsetI,axiom,
! [A: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_real @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_39_subset__antisym,axiom,
! [A: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A @ B2 )
=> ( ( ord_less_eq_set_real @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_40__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062u_O_A_092_060lbrakk_062y_A_L_A1_A_061_Af_Au_059_Aa_A_092_060le_062_Au_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [U: real] :
( ( ( plus_plus_real @ y @ one_one_real )
= ( f @ U ) )
=> ~ ( ord_less_eq_real @ a @ U ) ) ).
% \<open>\<And>thesis. (\<And>u. \<lbrakk>y + 1 = f u; a \<le> u\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_41_in__mono,axiom,
! [A: set_int,B2: set_int,X: int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( member_int @ X @ A )
=> ( member_int @ X @ B2 ) ) ) ).
% in_mono
thf(fact_42_in__mono,axiom,
! [A: set_real,B2: set_real,X: real] :
( ( ord_less_eq_set_real @ A @ B2 )
=> ( ( member_real @ X @ A )
=> ( member_real @ X @ B2 ) ) ) ).
% in_mono
thf(fact_43_subsetD,axiom,
! [A: set_int,B2: set_int,C3: int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( member_int @ C3 @ A )
=> ( member_int @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_44_subsetD,axiom,
! [A: set_real,B2: set_real,C3: real] :
( ( ord_less_eq_set_real @ A @ B2 )
=> ( ( member_real @ C3 @ A )
=> ( member_real @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_45_psubsetE,axiom,
! [A: set_real,B2: set_real] :
( ( ord_less_set_real @ A @ B2 )
=> ~ ( ( ord_less_eq_set_real @ A @ B2 )
=> ( ord_less_eq_set_real @ B2 @ A ) ) ) ).
% psubsetE
thf(fact_46_equalityE,axiom,
! [A: set_real,B2: set_real] :
( ( A = B2 )
=> ~ ( ( ord_less_eq_set_real @ A @ B2 )
=> ~ ( ord_less_eq_set_real @ B2 @ A ) ) ) ).
% equalityE
thf(fact_47_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A3: set_int,B3: set_int] :
! [X4: int] :
( ( member_int @ X4 @ A3 )
=> ( member_int @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_48_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
! [X4: real] :
( ( member_real @ X4 @ A3 )
=> ( member_real @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_49_equalityD1,axiom,
! [A: set_real,B2: set_real] :
( ( A = B2 )
=> ( ord_less_eq_set_real @ A @ B2 ) ) ).
% equalityD1
thf(fact_50_equalityD2,axiom,
! [A: set_real,B2: set_real] :
( ( A = B2 )
=> ( ord_less_eq_set_real @ B2 @ A ) ) ).
% equalityD2
thf(fact_51_psubset__eq,axiom,
( ord_less_set_real
= ( ^ [A3: set_real,B3: set_real] :
( ( ord_less_eq_set_real @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_52_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A3: set_int,B3: set_int] :
! [T3: int] :
( ( member_int @ T3 @ A3 )
=> ( member_int @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_53_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
! [T3: real] :
( ( member_real @ T3 @ A3 )
=> ( member_real @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_54_subset__refl,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ A ) ).
% subset_refl
thf(fact_55_Collect__mono,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X2: real] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% Collect_mono
thf(fact_56_subset__trans,axiom,
! [A: set_real,B2: set_real,C2: set_real] :
( ( ord_less_eq_set_real @ A @ B2 )
=> ( ( ord_less_eq_set_real @ B2 @ C2 )
=> ( ord_less_eq_set_real @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_57_set__eq__subset,axiom,
( ( ^ [Y3: set_real,Z: set_real] : ( Y3 = Z ) )
= ( ^ [A3: set_real,B3: set_real] :
( ( ord_less_eq_set_real @ A3 @ B3 )
& ( ord_less_eq_set_real @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_58_Collect__mono__iff,axiom,
! [P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
= ( ! [X4: real] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_59_psubset__imp__subset,axiom,
! [A: set_real,B2: set_real] :
( ( ord_less_set_real @ A @ B2 )
=> ( ord_less_eq_set_real @ A @ B2 ) ) ).
% psubset_imp_subset
thf(fact_60_psubset__subset__trans,axiom,
! [A: set_real,B2: set_real,C2: set_real] :
( ( ord_less_set_real @ A @ B2 )
=> ( ( ord_less_eq_set_real @ B2 @ C2 )
=> ( ord_less_set_real @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_61_subset__not__subset__eq,axiom,
( ord_less_set_real
= ( ^ [A3: set_real,B3: set_real] :
( ( ord_less_eq_set_real @ A3 @ B3 )
& ~ ( ord_less_eq_set_real @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_62_subset__psubset__trans,axiom,
! [A: set_real,B2: set_real,C2: set_real] :
( ( ord_less_eq_set_real @ A @ B2 )
=> ( ( ord_less_set_real @ B2 @ C2 )
=> ( ord_less_set_real @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_63_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
( ( ord_less_set_real @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A2: real,P: real > $o] :
( ( member_real @ A2 @ ( collect_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
! [A2: int,P: int > $o] :
( ( member_int @ A2 @ ( collect_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X4: int] : ( member_int @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_68_subset__image__iff,axiom,
! [B2: set_int,F: nat > int,A: set_nat] :
( ( ord_less_eq_set_int @ B2 @ ( image_nat_int @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B2
= ( image_nat_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_69_subset__image__iff,axiom,
! [B2: set_real,F: real > real,A: set_real] :
( ( ord_less_eq_set_real @ B2 @ ( image_real_real @ F @ A ) )
= ( ? [AA: set_real] :
( ( ord_less_eq_set_real @ AA @ A )
& ( B2
= ( image_real_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_70_image__subset__iff,axiom,
! [F: nat > int,A: set_nat,B2: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ B2 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_int @ ( F @ X4 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_71_image__subset__iff,axiom,
! [F: real > real,A: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ ( image_real_real @ F @ A ) @ B2 )
= ( ! [X4: real] :
( ( member_real @ X4 @ A )
=> ( member_real @ ( F @ X4 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_72_subset__imageE,axiom,
! [B2: set_int,F: nat > int,A: set_nat] :
( ( ord_less_eq_set_int @ B2 @ ( image_nat_int @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B2
!= ( image_nat_int @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_73_subset__imageE,axiom,
! [B2: set_real,F: real > real,A: set_real] :
( ( ord_less_eq_set_real @ B2 @ ( image_real_real @ F @ A ) )
=> ~ ! [C4: set_real] :
( ( ord_less_eq_set_real @ C4 @ A )
=> ( B2
!= ( image_real_real @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_74_image__subsetI,axiom,
! [A: set_nat,F: nat > int,B2: set_int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_int @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_75_image__subsetI,axiom,
! [A: set_real,F: real > int,B2: set_int] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_int @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_int @ ( image_real_int @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_76_image__subsetI,axiom,
! [A: set_int,F: int > int,B2: set_int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_int @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_int @ ( image_int_int @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_77_image__subsetI,axiom,
! [A: set_real,F: real > real,B2: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_real @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_78_image__subsetI,axiom,
! [A: set_int,F: int > real,B2: set_real] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_real @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_real @ ( image_int_real @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_79_image__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > int] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A ) @ ( image_nat_int @ F @ B2 ) ) ) ).
% image_mono
thf(fact_80_image__mono,axiom,
! [A: set_real,B2: set_real,F: real > real] :
( ( ord_less_eq_set_real @ A @ B2 )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A ) @ ( image_real_real @ F @ B2 ) ) ) ).
% image_mono
thf(fact_81_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: int,F: nat > int] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_nat_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_82_rev__image__eqI,axiom,
! [X: real,A: set_real,B: real,F: real > real] :
( ( member_real @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_real_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_83_rev__image__eqI,axiom,
! [X: real,A: set_real,B: int,F: real > int] :
( ( member_real @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_real_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_84_rev__image__eqI,axiom,
! [X: int,A: set_int,B: real,F: int > real] :
( ( member_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_int_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_85_rev__image__eqI,axiom,
! [X: int,A: set_int,B: int,F: int > int] :
( ( member_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_int_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_86_ball__imageD,axiom,
! [F: real > real,A: set_real,P: real > $o] :
( ! [X2: real] :
( ( member_real @ X2 @ ( image_real_real @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_87_ball__imageD,axiom,
! [F: nat > int,A: set_nat,P: int > $o] :
( ! [X2: int] :
( ( member_int @ X2 @ ( image_nat_int @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_88_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > int,G: nat > int] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_int @ F @ M )
= ( image_nat_int @ G @ N ) ) ) ) ).
% image_cong
thf(fact_89_image__cong,axiom,
! [M: set_real,N: set_real,F: real > real,G: real > real] :
( ( M = N )
=> ( ! [X2: real] :
( ( member_real @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_real_real @ F @ M )
= ( image_real_real @ G @ N ) ) ) ) ).
% image_cong
thf(fact_90_bex__imageD,axiom,
! [F: real > real,A: set_real,P: real > $o] :
( ? [X3: real] :
( ( member_real @ X3 @ ( image_real_real @ F @ A ) )
& ( P @ X3 ) )
=> ? [X2: real] :
( ( member_real @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_91_bex__imageD,axiom,
! [F: nat > int,A: set_nat,P: int > $o] :
( ? [X3: int] :
( ( member_int @ X3 @ ( image_nat_int @ F @ A ) )
& ( P @ X3 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_92_image__iff,axiom,
! [Z2: real,F: real > real,A: set_real] :
( ( member_real @ Z2 @ ( image_real_real @ F @ A ) )
= ( ? [X4: real] :
( ( member_real @ X4 @ A )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_93_image__iff,axiom,
! [Z2: int,F: nat > int,A: set_nat] :
( ( member_int @ Z2 @ ( image_nat_int @ F @ A ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_94_imageI,axiom,
! [X: nat,A: set_nat,F: nat > int] :
( ( member_nat @ X @ A )
=> ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ A ) ) ) ).
% imageI
thf(fact_95_imageI,axiom,
! [X: real,A: set_real,F: real > real] :
( ( member_real @ X @ A )
=> ( member_real @ ( F @ X ) @ ( image_real_real @ F @ A ) ) ) ).
% imageI
thf(fact_96_imageI,axiom,
! [X: real,A: set_real,F: real > int] :
( ( member_real @ X @ A )
=> ( member_int @ ( F @ X ) @ ( image_real_int @ F @ A ) ) ) ).
% imageI
thf(fact_97_imageI,axiom,
! [X: int,A: set_int,F: int > real] :
( ( member_int @ X @ A )
=> ( member_real @ ( F @ X ) @ ( image_int_real @ F @ A ) ) ) ).
% imageI
thf(fact_98_imageI,axiom,
! [X: int,A: set_int,F: int > int] :
( ( member_int @ X @ A )
=> ( member_int @ ( F @ X ) @ ( image_int_int @ F @ A ) ) ) ).
% imageI
thf(fact_99_Icc__subset__Ici__iff,axiom,
! [L: set_real,H: set_real,L2: set_real] :
( ( ord_le3558479182127378552t_real @ ( set_or7743017856606604397t_real @ L @ H ) @ ( set_or6787452043721853143t_real @ L2 ) )
= ( ~ ( ord_less_eq_set_real @ L @ H )
| ( ord_less_eq_set_real @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_100_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_101_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_102_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_103_image__add__atLeast,axiom,
! [K: int,I: int] :
( ( image_int_int @ ( plus_plus_int @ K ) @ ( set_ord_atLeast_int @ I ) )
= ( set_ord_atLeast_int @ ( plus_plus_int @ K @ I ) ) ) ).
% image_add_atLeast
thf(fact_104_image__add__atLeast,axiom,
! [K: real,I: real] :
( ( image_real_real @ ( plus_plus_real @ K ) @ ( set_ord_atLeast_real @ I ) )
= ( set_ord_atLeast_real @ ( plus_plus_real @ K @ I ) ) ) ).
% image_add_atLeast
thf(fact_105_image__add__atLeast,axiom,
! [K: nat,I: nat] :
( ( image_nat_nat @ ( plus_plus_nat @ K ) @ ( set_ord_atLeast_nat @ I ) )
= ( set_ord_atLeast_nat @ ( plus_plus_nat @ K @ I ) ) ) ).
% image_add_atLeast
thf(fact_106_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_107_atLeast__subset__iff,axiom,
! [X: set_real,Y: set_real] :
( ( ord_le3558479182127378552t_real @ ( set_or6787452043721853143t_real @ X ) @ ( set_or6787452043721853143t_real @ Y ) )
= ( ord_less_eq_set_real @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_108_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_109_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_110_image__add__atLeastAtMost,axiom,
! [K: real,I: real,J: real] :
( ( image_real_real @ ( plus_plus_real @ K ) @ ( set_or1222579329274155063t_real @ I @ J ) )
= ( set_or1222579329274155063t_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ K ) ) ) ).
% image_add_atLeastAtMost
thf(fact_111_image__add__atLeastAtMost,axiom,
! [K: nat,I: nat,J: nat] :
( ( image_nat_nat @ ( plus_plus_nat @ K ) @ ( set_or1269000886237332187st_nat @ I @ J ) )
= ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% image_add_atLeastAtMost
thf(fact_112_image__add__atLeastAtMost,axiom,
! [K: int,I: int,J: int] :
( ( image_int_int @ ( plus_plus_int @ K ) @ ( set_or1266510415728281911st_int @ I @ J ) )
= ( set_or1266510415728281911st_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).
% image_add_atLeastAtMost
thf(fact_113_atLeastatMost__subset__iff,axiom,
! [A2: set_real,B: set_real,C3: set_real,D3: set_real] :
( ( ord_le3558479182127378552t_real @ ( set_or7743017856606604397t_real @ A2 @ B ) @ ( set_or7743017856606604397t_real @ C3 @ D3 ) )
= ( ~ ( ord_less_eq_set_real @ A2 @ B )
| ( ( ord_less_eq_set_real @ C3 @ A2 )
& ( ord_less_eq_set_real @ B @ D3 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_114_atLeastatMost__subset__iff,axiom,
! [A2: real,B: real,C3: real,D3: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A2 @ B ) @ ( set_or1222579329274155063t_real @ C3 @ D3 ) )
= ( ~ ( ord_less_eq_real @ A2 @ B )
| ( ( ord_less_eq_real @ C3 @ A2 )
& ( ord_less_eq_real @ B @ D3 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_115_atLeastatMost__subset__iff,axiom,
! [A2: nat,B: nat,C3: nat,D3: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B ) @ ( set_or1269000886237332187st_nat @ C3 @ D3 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( ( ord_less_eq_nat @ C3 @ A2 )
& ( ord_less_eq_nat @ B @ D3 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_116_atLeastatMost__subset__iff,axiom,
! [A2: int,B: int,C3: int,D3: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A2 @ B ) @ ( set_or1266510415728281911st_int @ C3 @ D3 ) )
= ( ~ ( ord_less_eq_int @ A2 @ B )
| ( ( ord_less_eq_int @ C3 @ A2 )
& ( ord_less_eq_int @ B @ D3 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_117_subset__translation__eq,axiom,
! [A2: real,S: set_real,T: set_real] :
( ( ord_less_eq_set_real @ ( image_real_real @ ( plus_plus_real @ A2 ) @ S ) @ ( image_real_real @ ( plus_plus_real @ A2 ) @ T ) )
= ( ord_less_eq_set_real @ S @ T ) ) ).
% subset_translation_eq
thf(fact_118_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_119_atLeast__iff,axiom,
! [I: set_real,K: set_real] :
( ( member_set_real @ I @ ( set_or6787452043721853143t_real @ K ) )
= ( ord_less_eq_set_real @ K @ I ) ) ).
% atLeast_iff
thf(fact_120_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_121_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_122_atLeastAtMost__iff,axiom,
! [I: set_real,L: set_real,U2: set_real] :
( ( member_set_real @ I @ ( set_or7743017856606604397t_real @ L @ U2 ) )
= ( ( ord_less_eq_set_real @ L @ I )
& ( ord_less_eq_set_real @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_123_atLeastAtMost__iff,axiom,
! [I: real,L: real,U2: real] :
( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U2 ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_eq_real @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_124_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U2: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U2 ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_125_atLeastAtMost__iff,axiom,
! [I: int,L: int,U2: int] :
( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U2 ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_eq_int @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_126_Icc__eq__Icc,axiom,
! [L: set_real,H: set_real,L2: set_real,H2: set_real] :
( ( ( set_or7743017856606604397t_real @ L @ H )
= ( set_or7743017856606604397t_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_real @ L @ H )
& ~ ( ord_less_eq_set_real @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_127_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_128_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_129_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_130_add__less__cancel__left,axiom,
! [C3: real,A2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C3 @ A2 ) @ ( plus_plus_real @ C3 @ B ) )
= ( ord_less_real @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_131_add__less__cancel__left,axiom,
! [C3: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C3 @ A2 ) @ ( plus_plus_nat @ C3 @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_132_add__less__cancel__left,axiom,
! [C3: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C3 @ A2 ) @ ( plus_plus_int @ C3 @ B ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_133_add__less__cancel__right,axiom,
! [A2: real,C3: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ C3 ) @ ( plus_plus_real @ B @ C3 ) )
= ( ord_less_real @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_134_add__less__cancel__right,axiom,
! [A2: nat,C3: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_135_add__less__cancel__right,axiom,
! [A2: int,C3: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C3 ) @ ( plus_plus_int @ B @ C3 ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_136_add__right__cancel,axiom,
! [B: real,A2: real,C3: real] :
( ( ( plus_plus_real @ B @ A2 )
= ( plus_plus_real @ C3 @ A2 ) )
= ( B = C3 ) ) ).
% add_right_cancel
thf(fact_137_add__right__cancel,axiom,
! [B: nat,A2: nat,C3: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C3 @ A2 ) )
= ( B = C3 ) ) ).
% add_right_cancel
thf(fact_138_add__right__cancel,axiom,
! [B: int,A2: int,C3: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C3 @ A2 ) )
= ( B = C3 ) ) ).
% add_right_cancel
thf(fact_139_add__left__cancel,axiom,
! [A2: real,B: real,C3: real] :
( ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ A2 @ C3 ) )
= ( B = C3 ) ) ).
% add_left_cancel
thf(fact_140_add__left__cancel,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C3 ) )
= ( B = C3 ) ) ).
% add_left_cancel
thf(fact_141_add__left__cancel,axiom,
! [A2: int,B: int,C3: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C3 ) )
= ( B = C3 ) ) ).
% add_left_cancel
thf(fact_142_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_143_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_144_add__le__cancel__right,axiom,
! [A2: real,C3: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C3 ) @ ( plus_plus_real @ B @ C3 ) )
= ( ord_less_eq_real @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_145_add__le__cancel__right,axiom,
! [A2: nat,C3: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_146_add__le__cancel__right,axiom,
! [A2: int,C3: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C3 ) @ ( plus_plus_int @ B @ C3 ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_147_add__le__cancel__left,axiom,
! [C3: real,A2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C3 @ A2 ) @ ( plus_plus_real @ C3 @ B ) )
= ( ord_less_eq_real @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_148_add__le__cancel__left,axiom,
! [C3: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C3 @ A2 ) @ ( plus_plus_nat @ C3 @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_149_add__le__cancel__left,axiom,
! [C3: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C3 @ A2 ) @ ( plus_plus_int @ C3 @ B ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_150_psubsetD,axiom,
! [A: set_real,B2: set_real,C3: real] :
( ( ord_less_set_real @ A @ B2 )
=> ( ( member_real @ C3 @ A )
=> ( member_real @ C3 @ B2 ) ) ) ).
% psubsetD
thf(fact_151_psubsetD,axiom,
! [A: set_int,B2: set_int,C3: int] :
( ( ord_less_set_int @ A @ B2 )
=> ( ( member_int @ C3 @ A )
=> ( member_int @ C3 @ B2 ) ) ) ).
% psubsetD
thf(fact_152_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_153_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_154_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_155_add__right__imp__eq,axiom,
! [B: real,A2: real,C3: real] :
( ( ( plus_plus_real @ B @ A2 )
= ( plus_plus_real @ C3 @ A2 ) )
=> ( B = C3 ) ) ).
% add_right_imp_eq
thf(fact_156_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C3: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C3 @ A2 ) )
=> ( B = C3 ) ) ).
% add_right_imp_eq
thf(fact_157_add__right__imp__eq,axiom,
! [B: int,A2: int,C3: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C3 @ A2 ) )
=> ( B = C3 ) ) ).
% add_right_imp_eq
thf(fact_158_add__left__imp__eq,axiom,
! [A2: real,B: real,C3: real] :
( ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ A2 @ C3 ) )
=> ( B = C3 ) ) ).
% add_left_imp_eq
thf(fact_159_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C3 ) )
=> ( B = C3 ) ) ).
% add_left_imp_eq
thf(fact_160_add__left__imp__eq,axiom,
! [A2: int,B: int,C3: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C3 ) )
=> ( B = C3 ) ) ).
% add_left_imp_eq
thf(fact_161_add_Oleft__commute,axiom,
! [B: real,A2: real,C3: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A2 @ C3 ) )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C3 ) ) ) ).
% add.left_commute
thf(fact_162_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C3: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C3 ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C3 ) ) ) ).
% add.left_commute
thf(fact_163_add_Oleft__commute,axiom,
! [B: int,A2: int,C3: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A2 @ C3 ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C3 ) ) ) ).
% add.left_commute
thf(fact_164_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_165_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_166_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_167_add_Oright__cancel,axiom,
! [B: real,A2: real,C3: real] :
( ( ( plus_plus_real @ B @ A2 )
= ( plus_plus_real @ C3 @ A2 ) )
= ( B = C3 ) ) ).
% add.right_cancel
thf(fact_168_add_Oright__cancel,axiom,
! [B: int,A2: int,C3: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C3 @ A2 ) )
= ( B = C3 ) ) ).
% add.right_cancel
thf(fact_169_add_Oleft__cancel,axiom,
! [A2: real,B: real,C3: real] :
( ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ A2 @ C3 ) )
= ( B = C3 ) ) ).
% add.left_cancel
thf(fact_170_add_Oleft__cancel,axiom,
! [A2: int,B: int,C3: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C3 ) )
= ( B = C3 ) ) ).
% add.left_cancel
thf(fact_171_add_Oassoc,axiom,
! [A2: real,B: real,C3: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B ) @ C3 )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C3 ) ) ) ).
% add.assoc
thf(fact_172_add_Oassoc,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C3 )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C3 ) ) ) ).
% add.assoc
thf(fact_173_add_Oassoc,axiom,
! [A2: int,B: int,C3: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C3 )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C3 ) ) ) ).
% add.assoc
thf(fact_174_group__cancel_Oadd2,axiom,
! [B2: real,K: real,B: real,A2: real] :
( ( B2
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A2 @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_175_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_176_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A2: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_177_group__cancel_Oadd1,axiom,
! [A: real,K: real,A2: real,B: real] :
( ( A
= ( plus_plus_real @ K @ A2 ) )
=> ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_178_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_179_group__cancel_Oadd1,axiom,
! [A: int,K: int,A2: int,B: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_180_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_181_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_182_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_183_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: real,B: real,C3: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B ) @ C3 )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C3 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_184_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C3 )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C3 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_185_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: int,B: int,C3: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C3 )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C3 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_186_add__le__imp__le__right,axiom,
! [A2: real,C3: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C3 ) @ ( plus_plus_real @ B @ C3 ) )
=> ( ord_less_eq_real @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_187_add__le__imp__le__right,axiom,
! [A2: nat,C3: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_188_add__le__imp__le__right,axiom,
! [A2: int,C3: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C3 ) @ ( plus_plus_int @ B @ C3 ) )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_189_add__le__imp__le__left,axiom,
! [C3: real,A2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C3 @ A2 ) @ ( plus_plus_real @ C3 @ B ) )
=> ( ord_less_eq_real @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_190_add__le__imp__le__left,axiom,
! [C3: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C3 @ A2 ) @ ( plus_plus_nat @ C3 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_191_add__le__imp__le__left,axiom,
! [C3: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C3 @ A2 ) @ ( plus_plus_int @ C3 @ B ) )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_192_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C5: nat] :
( B4
= ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_193_add__right__mono,axiom,
! [A2: real,B: real,C3: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C3 ) @ ( plus_plus_real @ B @ C3 ) ) ) ).
% add_right_mono
thf(fact_194_add__right__mono,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C3 ) @ ( plus_plus_nat @ B @ C3 ) ) ) ).
% add_right_mono
thf(fact_195_add__right__mono,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C3 ) @ ( plus_plus_int @ B @ C3 ) ) ) ).
% add_right_mono
thf(fact_196_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C: nat] :
( B
!= ( plus_plus_nat @ A2 @ C ) ) ) ).
% less_eqE
thf(fact_197_add__left__mono,axiom,
! [A2: real,B: real,C3: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C3 @ A2 ) @ ( plus_plus_real @ C3 @ B ) ) ) ).
% add_left_mono
thf(fact_198_add__left__mono,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C3 @ A2 ) @ ( plus_plus_nat @ C3 @ B ) ) ) ).
% add_left_mono
thf(fact_199_add__left__mono,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C3 @ A2 ) @ ( plus_plus_int @ C3 @ B ) ) ) ).
% add_left_mono
thf(fact_200_add__mono,axiom,
! [A2: real,B: real,C3: real,D3: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ C3 @ D3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C3 ) @ ( plus_plus_real @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_201_add__mono,axiom,
! [A2: nat,B: nat,C3: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C3 @ D3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C3 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_202_add__mono,axiom,
! [A2: int,B: int,C3: int,D3: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C3 @ D3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C3 ) @ ( plus_plus_int @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_203_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_204_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_205_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_206_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_207_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_208_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_209_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_210_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_211_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_212_add__less__imp__less__right,axiom,
! [A2: real,C3: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ C3 ) @ ( plus_plus_real @ B @ C3 ) )
=> ( ord_less_real @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_213_add__less__imp__less__right,axiom,
! [A2: nat,C3: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_214_add__less__imp__less__right,axiom,
! [A2: int,C3: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C3 ) @ ( plus_plus_int @ B @ C3 ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_215_add__less__imp__less__left,axiom,
! [C3: real,A2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C3 @ A2 ) @ ( plus_plus_real @ C3 @ B ) )
=> ( ord_less_real @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_216_add__less__imp__less__left,axiom,
! [C3: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C3 @ A2 ) @ ( plus_plus_nat @ C3 @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_217_add__less__imp__less__left,axiom,
! [C3: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C3 @ A2 ) @ ( plus_plus_int @ C3 @ B ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_218_add__strict__right__mono,axiom,
! [A2: real,B: real,C3: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C3 ) @ ( plus_plus_real @ B @ C3 ) ) ) ).
% add_strict_right_mono
thf(fact_219_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C3 ) @ ( plus_plus_nat @ B @ C3 ) ) ) ).
% add_strict_right_mono
thf(fact_220_add__strict__right__mono,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C3 ) @ ( plus_plus_int @ B @ C3 ) ) ) ).
% add_strict_right_mono
thf(fact_221_add__strict__left__mono,axiom,
! [A2: real,B: real,C3: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( plus_plus_real @ C3 @ A2 ) @ ( plus_plus_real @ C3 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_222_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C3 @ A2 ) @ ( plus_plus_nat @ C3 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_223_add__strict__left__mono,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ C3 @ A2 ) @ ( plus_plus_int @ C3 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_224_add__strict__mono,axiom,
! [A2: real,B: real,C3: real,D3: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ C3 @ D3 )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C3 ) @ ( plus_plus_real @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_225_add__strict__mono,axiom,
! [A2: nat,B: nat,C3: nat,D3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C3 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C3 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_226_add__strict__mono,axiom,
! [A2: int,B: int,C3: int,D3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C3 @ D3 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C3 ) @ ( plus_plus_int @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_227_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_228_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_229_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_230_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_231_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_232_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_233_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_234_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_235_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_236_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_237_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_238_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_239_add__less__le__mono,axiom,
! [A2: real,B: real,C3: real,D3: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ C3 @ D3 )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C3 ) @ ( plus_plus_real @ B @ D3 ) ) ) ) ).
% add_less_le_mono
thf(fact_240_add__less__le__mono,axiom,
! [A2: nat,B: nat,C3: nat,D3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C3 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C3 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_less_le_mono
thf(fact_241_add__less__le__mono,axiom,
! [A2: int,B: int,C3: int,D3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C3 @ D3 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C3 ) @ ( plus_plus_int @ B @ D3 ) ) ) ) ).
% add_less_le_mono
thf(fact_242_add__le__less__mono,axiom,
! [A2: real,B: real,C3: real,D3: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real @ C3 @ D3 )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C3 ) @ ( plus_plus_real @ B @ D3 ) ) ) ) ).
% add_le_less_mono
thf(fact_243_add__le__less__mono,axiom,
! [A2: nat,B: nat,C3: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C3 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C3 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_le_less_mono
thf(fact_244_add__le__less__mono,axiom,
! [A2: int,B: int,C3: int,D3: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ C3 @ D3 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C3 ) @ ( plus_plus_int @ B @ D3 ) ) ) ) ).
% add_le_less_mono
thf(fact_245_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_246_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_247_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_248_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_249_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_250_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_251_atLeastatMost__psubset__iff,axiom,
! [A2: set_real,B: set_real,C3: set_real,D3: set_real] :
( ( ord_le7926960851185191020t_real @ ( set_or7743017856606604397t_real @ A2 @ B ) @ ( set_or7743017856606604397t_real @ C3 @ D3 ) )
= ( ( ~ ( ord_less_eq_set_real @ A2 @ B )
| ( ( ord_less_eq_set_real @ C3 @ A2 )
& ( ord_less_eq_set_real @ B @ D3 )
& ( ( ord_less_set_real @ C3 @ A2 )
| ( ord_less_set_real @ B @ D3 ) ) ) )
& ( ord_less_eq_set_real @ C3 @ D3 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_252_atLeastatMost__psubset__iff,axiom,
! [A2: real,B: real,C3: real,D3: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A2 @ B ) @ ( set_or1222579329274155063t_real @ C3 @ D3 ) )
= ( ( ~ ( ord_less_eq_real @ A2 @ B )
| ( ( ord_less_eq_real @ C3 @ A2 )
& ( ord_less_eq_real @ B @ D3 )
& ( ( ord_less_real @ C3 @ A2 )
| ( ord_less_real @ B @ D3 ) ) ) )
& ( ord_less_eq_real @ C3 @ D3 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_253_atLeastatMost__psubset__iff,axiom,
! [A2: nat,B: nat,C3: nat,D3: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B ) @ ( set_or1269000886237332187st_nat @ C3 @ D3 ) )
= ( ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( ( ord_less_eq_nat @ C3 @ A2 )
& ( ord_less_eq_nat @ B @ D3 )
& ( ( ord_less_nat @ C3 @ A2 )
| ( ord_less_nat @ B @ D3 ) ) ) )
& ( ord_less_eq_nat @ C3 @ D3 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_254_atLeastatMost__psubset__iff,axiom,
! [A2: int,B: int,C3: int,D3: int] :
( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A2 @ B ) @ ( set_or1266510415728281911st_int @ C3 @ D3 ) )
= ( ( ~ ( ord_less_eq_int @ A2 @ B )
| ( ( ord_less_eq_int @ C3 @ A2 )
& ( ord_less_eq_int @ B @ D3 )
& ( ( ord_less_int @ C3 @ A2 )
| ( ord_less_int @ B @ D3 ) ) ) )
& ( ord_less_eq_int @ C3 @ D3 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_255_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_256_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_257_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_258_mono__on__subset,axiom,
! [A: set_real,F: real > real,B2: set_real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_real @ B2 @ A )
=> ( monoto4017252874604999745l_real @ B2 @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ) ).
% mono_on_subset
thf(fact_259_mono__on__subset,axiom,
! [A: set_real,F: real > nat,B2: set_real] :
( ( monotone_on_real_nat @ A @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_real @ B2 @ A )
=> ( monotone_on_real_nat @ B2 @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_260_mono__on__subset,axiom,
! [A: set_real,F: real > int,B2: set_real] :
( ( monotone_on_real_int @ A @ ord_less_eq_real @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_set_real @ B2 @ A )
=> ( monotone_on_real_int @ B2 @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ) ).
% mono_on_subset
thf(fact_261_mono__on__subset,axiom,
! [A: set_nat,F: nat > real,B2: set_nat] :
( ( monotone_on_nat_real @ A @ ord_less_eq_nat @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( monotone_on_nat_real @ B2 @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ) ).
% mono_on_subset
thf(fact_262_mono__on__subset,axiom,
! [A: set_nat,F: nat > nat,B2: set_nat] :
( ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( monotone_on_nat_nat @ B2 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_263_mono__on__subset,axiom,
! [A: set_nat,F: nat > int,B2: set_nat] :
( ( monotone_on_nat_int @ A @ ord_less_eq_nat @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( monotone_on_nat_int @ B2 @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ) ).
% mono_on_subset
thf(fact_264_mono__on__subset,axiom,
! [A: set_int,F: int > real,B2: set_int] :
( ( monotone_on_int_real @ A @ ord_less_eq_int @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_int @ B2 @ A )
=> ( monotone_on_int_real @ B2 @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ) ).
% mono_on_subset
thf(fact_265_mono__on__subset,axiom,
! [A: set_int,F: int > nat,B2: set_int] :
( ( monotone_on_int_nat @ A @ ord_less_eq_int @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_int @ B2 @ A )
=> ( monotone_on_int_nat @ B2 @ ord_less_eq_int @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_266_mono__on__subset,axiom,
! [A: set_int,F: int > int,B2: set_int] :
( ( monotone_on_int_int @ A @ ord_less_eq_int @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_set_int @ B2 @ A )
=> ( monotone_on_int_int @ B2 @ ord_less_eq_int @ ord_less_eq_int @ F ) ) ) ).
% mono_on_subset
thf(fact_267_mono__on__subset,axiom,
! [A: set_real,F: real > set_real,B2: set_real] :
( ( monoto3333417327835629687t_real @ A @ ord_less_eq_real @ ord_less_eq_set_real @ F )
=> ( ( ord_less_eq_set_real @ B2 @ A )
=> ( monoto3333417327835629687t_real @ B2 @ ord_less_eq_real @ ord_less_eq_set_real @ F ) ) ) ).
% mono_on_subset
thf(fact_268_ord_Omono__on__subset,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > real,B2: set_real] :
( ( monoto4017252874604999745l_real @ A @ Less_eq @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_real @ B2 @ A )
=> ( monoto4017252874604999745l_real @ B2 @ Less_eq @ ord_less_eq_real @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_269_ord_Omono__on__subset,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat,B2: set_nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( monotone_on_nat_nat @ B2 @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_270_ord_Omono__on__subset,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > nat,B2: set_real] :
( ( monotone_on_real_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_real @ B2 @ A )
=> ( monotone_on_real_nat @ B2 @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_271_ord_Omono__on__subset,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > int,B2: set_real] :
( ( monotone_on_real_int @ A @ Less_eq @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_set_real @ B2 @ A )
=> ( monotone_on_real_int @ B2 @ Less_eq @ ord_less_eq_int @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_272_ord_Omono__on__subset,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > set_real,B2: set_real] :
( ( monoto3333417327835629687t_real @ A @ Less_eq @ ord_less_eq_set_real @ F )
=> ( ( ord_less_eq_set_real @ B2 @ A )
=> ( monoto3333417327835629687t_real @ B2 @ Less_eq @ ord_less_eq_set_real @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_273_strict__mono__on__imp__mono__on,axiom,
! [A: set_real,F: real > real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_real @ ord_less_real @ F )
=> ( monoto4017252874604999745l_real @ A @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_274_strict__mono__on__imp__mono__on,axiom,
! [A: set_real,F: real > nat] :
( ( monotone_on_real_nat @ A @ ord_less_real @ ord_less_nat @ F )
=> ( monotone_on_real_nat @ A @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_275_strict__mono__on__imp__mono__on,axiom,
! [A: set_real,F: real > int] :
( ( monotone_on_real_int @ A @ ord_less_real @ ord_less_int @ F )
=> ( monotone_on_real_int @ A @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_276_strict__mono__on__imp__mono__on,axiom,
! [A: set_nat,F: nat > real] :
( ( monotone_on_nat_real @ A @ ord_less_nat @ ord_less_real @ F )
=> ( monotone_on_nat_real @ A @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_277_strict__mono__on__imp__mono__on,axiom,
! [A: set_nat,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_278_strict__mono__on__imp__mono__on,axiom,
! [A: set_nat,F: nat > int] :
( ( monotone_on_nat_int @ A @ ord_less_nat @ ord_less_int @ F )
=> ( monotone_on_nat_int @ A @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_279_strict__mono__on__imp__mono__on,axiom,
! [A: set_int,F: int > real] :
( ( monotone_on_int_real @ A @ ord_less_int @ ord_less_real @ F )
=> ( monotone_on_int_real @ A @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_280_strict__mono__on__imp__mono__on,axiom,
! [A: set_int,F: int > nat] :
( ( monotone_on_int_nat @ A @ ord_less_int @ ord_less_nat @ F )
=> ( monotone_on_int_nat @ A @ ord_less_eq_int @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_281_strict__mono__on__imp__mono__on,axiom,
! [A: set_int,F: int > int] :
( ( monotone_on_int_int @ A @ ord_less_int @ ord_less_int @ F )
=> ( monotone_on_int_int @ A @ ord_less_eq_int @ ord_less_eq_int @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_282_strict__mono__on__imp__mono__on,axiom,
! [A: set_real,F: real > set_real] :
( ( monoto3333417327835629687t_real @ A @ ord_less_real @ ord_less_set_real @ F )
=> ( monoto3333417327835629687t_real @ A @ ord_less_eq_real @ ord_less_eq_set_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_283_strict__mono__on__leD,axiom,
! [A: set_real,F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_real @ ord_less_real @ F )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_284_strict__mono__on__leD,axiom,
! [A: set_real,F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ A @ ord_less_real @ ord_less_nat @ F )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_285_strict__mono__on__leD,axiom,
! [A: set_real,F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ A @ ord_less_real @ ord_less_int @ F )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_286_strict__mono__on__leD,axiom,
! [A: set_nat,F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ A @ ord_less_nat @ ord_less_real @ F )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_287_strict__mono__on__leD,axiom,
! [A: set_nat,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_288_strict__mono__on__leD,axiom,
! [A: set_nat,F: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ A @ ord_less_nat @ ord_less_int @ F )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_289_strict__mono__on__leD,axiom,
! [A: set_int,F: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ A @ ord_less_int @ ord_less_real @ F )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_290_strict__mono__on__leD,axiom,
! [A: set_int,F: int > nat,X: int,Y: int] :
( ( monotone_on_int_nat @ A @ ord_less_int @ ord_less_nat @ F )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_291_strict__mono__on__leD,axiom,
! [A: set_int,F: int > int,X: int,Y: int] :
( ( monotone_on_int_int @ A @ ord_less_int @ ord_less_int @ F )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_292_strict__mono__on__leD,axiom,
! [A: set_real,F: real > set_real,X: real,Y: real] :
( ( monoto3333417327835629687t_real @ A @ ord_less_real @ ord_less_set_real @ F )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_set_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_293_mono__on__greaterD,axiom,
! [A: set_real,G: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_eq_real @ ord_less_eq_real @ G )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( ( ord_less_real @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_real @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_294_mono__on__greaterD,axiom,
! [A: set_real,G: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ A @ ord_less_eq_real @ ord_less_eq_nat @ G )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_real @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_295_mono__on__greaterD,axiom,
! [A: set_real,G: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ A @ ord_less_eq_real @ ord_less_eq_int @ G )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( ( ord_less_int @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_real @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_296_mono__on__greaterD,axiom,
! [A: set_nat,G: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ A @ ord_less_eq_nat @ ord_less_eq_real @ G )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ord_less_real @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_nat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_297_mono__on__greaterD,axiom,
! [A: set_nat,G: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ G )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_nat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_298_mono__on__greaterD,axiom,
! [A: set_nat,G: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ A @ ord_less_eq_nat @ ord_less_eq_int @ G )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ord_less_int @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_nat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_299_mono__on__greaterD,axiom,
! [A: set_int,G: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ A @ ord_less_eq_int @ ord_less_eq_real @ G )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ( ord_less_real @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_int @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_300_mono__on__greaterD,axiom,
! [A: set_int,G: int > nat,X: int,Y: int] :
( ( monotone_on_int_nat @ A @ ord_less_eq_int @ ord_less_eq_nat @ G )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_int @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_301_mono__on__greaterD,axiom,
! [A: set_int,G: int > int,X: int,Y: int] :
( ( monotone_on_int_int @ A @ ord_less_eq_int @ ord_less_eq_int @ G )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ( ord_less_int @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_int @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_302_add__mono1,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_303_add__mono1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_304_add__mono1,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_305_less__add__one,axiom,
! [A2: real] : ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ one_one_real ) ) ).
% less_add_one
thf(fact_306_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_307_less__add__one,axiom,
! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).
% less_add_one
thf(fact_308_set__plus__mono2,axiom,
! [C2: set_real,D2: set_real,E: set_real,F3: set_real] :
( ( ord_less_eq_set_real @ C2 @ D2 )
=> ( ( ord_less_eq_set_real @ E @ F3 )
=> ( ord_less_eq_set_real @ ( plus_plus_set_real @ C2 @ E ) @ ( plus_plus_set_real @ D2 @ F3 ) ) ) ) ).
% set_plus_mono2
thf(fact_309_set__plus__intro,axiom,
! [A2: real,C2: set_real,B: real,D2: set_real] :
( ( member_real @ A2 @ C2 )
=> ( ( member_real @ B @ D2 )
=> ( member_real @ ( plus_plus_real @ A2 @ B ) @ ( plus_plus_set_real @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_310_set__plus__intro,axiom,
! [A2: nat,C2: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( plus_plus_nat @ A2 @ B ) @ ( plus_plus_set_nat @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_311_set__plus__intro,axiom,
! [A2: int,C2: set_int,B: int,D2: set_int] :
( ( member_int @ A2 @ C2 )
=> ( ( member_int @ B @ D2 )
=> ( member_int @ ( plus_plus_int @ A2 @ B ) @ ( plus_plus_set_int @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_312_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_313_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_314_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_315_dual__order_Orefl,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_316_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_317_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_318_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_319_order__refl,axiom,
! [X: set_real] : ( ord_less_eq_set_real @ X @ X ) ).
% order_refl
thf(fact_320_nle__le,axiom,
! [A2: real,B: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B ) )
= ( ( ord_less_eq_real @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_321_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_322_nle__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B ) )
= ( ( ord_less_eq_int @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_323_le__cases3,axiom,
! [X: real,Y: real,Z2: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_324_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_325_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_326_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_327_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_328_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_329_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_real,Z: set_real] : ( Y3 = Z ) )
= ( ^ [X4: set_real,Y4: set_real] :
( ( ord_less_eq_set_real @ X4 @ Y4 )
& ( ord_less_eq_set_real @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_330_ord__eq__le__trans,axiom,
! [A2: real,B: real,C3: real] :
( ( A2 = B )
=> ( ( ord_less_eq_real @ B @ C3 )
=> ( ord_less_eq_real @ A2 @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_331_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ord_less_eq_nat @ A2 @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_332_ord__eq__le__trans,axiom,
! [A2: int,B: int,C3: int] :
( ( A2 = B )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ord_less_eq_int @ A2 @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_333_ord__eq__le__trans,axiom,
! [A2: set_real,B: set_real,C3: set_real] :
( ( A2 = B )
=> ( ( ord_less_eq_set_real @ B @ C3 )
=> ( ord_less_eq_set_real @ A2 @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_334_ord__le__eq__trans,axiom,
! [A2: real,B: real,C3: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( B = C3 )
=> ( ord_less_eq_real @ A2 @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_335_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C3 )
=> ( ord_less_eq_nat @ A2 @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_336_ord__le__eq__trans,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( B = C3 )
=> ( ord_less_eq_int @ A2 @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_337_ord__le__eq__trans,axiom,
! [A2: set_real,B: set_real,C3: set_real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( B = C3 )
=> ( ord_less_eq_set_real @ A2 @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_338_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_339_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_340_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_341_order__antisym,axiom,
! [X: set_real,Y: set_real] :
( ( ord_less_eq_set_real @ X @ Y )
=> ( ( ord_less_eq_set_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_342_order_Otrans,axiom,
! [A2: real,B: real,C3: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ C3 )
=> ( ord_less_eq_real @ A2 @ C3 ) ) ) ).
% order.trans
thf(fact_343_order_Otrans,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ord_less_eq_nat @ A2 @ C3 ) ) ) ).
% order.trans
thf(fact_344_order_Otrans,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ord_less_eq_int @ A2 @ C3 ) ) ) ).
% order.trans
thf(fact_345_order_Otrans,axiom,
! [A2: set_real,B: set_real,C3: set_real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( ord_less_eq_set_real @ B @ C3 )
=> ( ord_less_eq_set_real @ A2 @ C3 ) ) ) ).
% order.trans
thf(fact_346_order__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_347_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_348_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_349_order__trans,axiom,
! [X: set_real,Y: set_real,Z2: set_real] :
( ( ord_less_eq_set_real @ X @ Y )
=> ( ( ord_less_eq_set_real @ Y @ Z2 )
=> ( ord_less_eq_set_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_350_linorder__wlog,axiom,
! [P: real > real > $o,A2: real,B: real] :
( ! [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: real,B5: real] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_351_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_352_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_353_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_354_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_355_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_356_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_real,Z: set_real] : ( Y3 = Z ) )
= ( ^ [A4: set_real,B4: set_real] :
( ( ord_less_eq_set_real @ B4 @ A4 )
& ( ord_less_eq_set_real @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_357_dual__order_Oantisym,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_358_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_359_dual__order_Oantisym,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_360_dual__order_Oantisym,axiom,
! [B: set_real,A2: set_real] :
( ( ord_less_eq_set_real @ B @ A2 )
=> ( ( ord_less_eq_set_real @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_361_dual__order_Otrans,axiom,
! [B: real,A2: real,C3: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C3 @ B )
=> ( ord_less_eq_real @ C3 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_362_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C3: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C3 @ B )
=> ( ord_less_eq_nat @ C3 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_363_dual__order_Otrans,axiom,
! [B: int,A2: int,C3: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C3 @ B )
=> ( ord_less_eq_int @ C3 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_364_dual__order_Otrans,axiom,
! [B: set_real,A2: set_real,C3: set_real] :
( ( ord_less_eq_set_real @ B @ A2 )
=> ( ( ord_less_eq_set_real @ C3 @ B )
=> ( ord_less_eq_set_real @ C3 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_365_antisym,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_366_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_367_antisym,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_368_antisym,axiom,
! [A2: set_real,B: set_real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( ord_less_eq_set_real @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_369_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_370_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_371_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_372_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_real,Z: set_real] : ( Y3 = Z ) )
= ( ^ [A4: set_real,B4: set_real] :
( ( ord_less_eq_set_real @ A4 @ B4 )
& ( ord_less_eq_set_real @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_373_order__subst1,axiom,
! [A2: real,F: real > real,B: real,C3: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_374_order__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C3: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_375_order__subst1,axiom,
! [A2: real,F: int > real,B: int,C3: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_376_order__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C3: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_377_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_378_order__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C3: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_379_order__subst1,axiom,
! [A2: int,F: real > int,B: real,C3: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_380_order__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C3: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_381_order__subst1,axiom,
! [A2: int,F: int > int,B: int,C3: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_382_order__subst1,axiom,
! [A2: real,F: set_real > real,B: set_real,C3: set_real] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_real @ B @ C3 )
=> ( ! [X2: set_real,Y2: set_real] :
( ( ord_less_eq_set_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_383_order__subst2,axiom,
! [A2: real,B: real,F: real > real,C3: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_384_order__subst2,axiom,
! [A2: real,B: real,F: real > nat,C3: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_385_order__subst2,axiom,
! [A2: real,B: real,F: real > int,C3: int] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_386_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C3: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_387_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_388_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C3: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_389_order__subst2,axiom,
! [A2: int,B: int,F: int > real,C3: real] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_390_order__subst2,axiom,
! [A2: int,B: int,F: int > nat,C3: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_391_order__subst2,axiom,
! [A2: int,B: int,F: int > int,C3: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_392_order__subst2,axiom,
! [A2: real,B: real,F: real > set_real,C3: set_real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_set_real @ ( F @ B ) @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_393_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_394_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_395_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_396_order__eq__refl,axiom,
! [X: set_real,Y: set_real] :
( ( X = Y )
=> ( ord_less_eq_set_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_397_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_398_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_399_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_400_ord__eq__le__subst,axiom,
! [A2: real,F: real > real,B: real,C3: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_401_ord__eq__le__subst,axiom,
! [A2: nat,F: real > nat,B: real,C3: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_402_ord__eq__le__subst,axiom,
! [A2: int,F: real > int,B: real,C3: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_403_ord__eq__le__subst,axiom,
! [A2: real,F: nat > real,B: nat,C3: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_404_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C3: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_405_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B: nat,C3: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_406_ord__eq__le__subst,axiom,
! [A2: real,F: int > real,B: int,C3: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_407_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B: int,C3: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_408_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B: int,C3: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_409_ord__eq__le__subst,axiom,
! [A2: set_real,F: real > set_real,B: real,C3: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_410_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > real,C3: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_411_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > nat,C3: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_412_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > int,C3: int] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_413_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > real,C3: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_414_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_415_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C3: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_416_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > real,C3: real] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_417_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C3: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_418_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C3: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_419_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > set_real,C3: set_real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_420_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_421_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_422_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_423_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_424_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_425_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_426_order__antisym__conv,axiom,
! [Y: set_real,X: set_real] :
( ( ord_less_eq_set_real @ Y @ X )
=> ( ( ord_less_eq_set_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_427_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_428_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_429_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_430_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_431_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_432_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y ) ) ) ).
% dense
thf(fact_433_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_434_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_435_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_436_order_Oasym,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ~ ( ord_less_real @ B @ A2 ) ) ).
% order.asym
thf(fact_437_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_438_order_Oasym,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order.asym
thf(fact_439_ord__eq__less__trans,axiom,
! [A2: real,B: real,C3: real] :
( ( A2 = B )
=> ( ( ord_less_real @ B @ C3 )
=> ( ord_less_real @ A2 @ C3 ) ) ) ).
% ord_eq_less_trans
thf(fact_440_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ord_less_nat @ A2 @ C3 ) ) ) ).
% ord_eq_less_trans
thf(fact_441_ord__eq__less__trans,axiom,
! [A2: int,B: int,C3: int] :
( ( A2 = B )
=> ( ( ord_less_int @ B @ C3 )
=> ( ord_less_int @ A2 @ C3 ) ) ) ).
% ord_eq_less_trans
thf(fact_442_ord__less__eq__trans,axiom,
! [A2: real,B: real,C3: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( B = C3 )
=> ( ord_less_real @ A2 @ C3 ) ) ) ).
% ord_less_eq_trans
thf(fact_443_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C3 )
=> ( ord_less_nat @ A2 @ C3 ) ) ) ).
% ord_less_eq_trans
thf(fact_444_ord__less__eq__trans,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( B = C3 )
=> ( ord_less_int @ A2 @ C3 ) ) ) ).
% ord_less_eq_trans
thf(fact_445_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_446_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_447_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_448_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_449_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_450_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_451_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_452_dual__order_Oasym,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ~ ( ord_less_real @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_453_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_454_dual__order_Oasym,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ~ ( ord_less_int @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_455_dual__order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_456_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_457_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_458_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_459_linorder__less__wlog,axiom,
! [P: real > real > $o,A2: real,B: real] :
( ! [A5: real,B5: real] :
( ( ord_less_real @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: real] : ( P @ A5 @ A5 )
=> ( ! [A5: real,B5: real] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_460_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_461_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A5: int,B5: int] :
( ( ord_less_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int] : ( P @ A5 @ A5 )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_462_order_Ostrict__trans,axiom,
! [A2: real,B: real,C3: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ B @ C3 )
=> ( ord_less_real @ A2 @ C3 ) ) ) ).
% order.strict_trans
thf(fact_463_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ord_less_nat @ A2 @ C3 ) ) ) ).
% order.strict_trans
thf(fact_464_order_Ostrict__trans,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C3 )
=> ( ord_less_int @ A2 @ C3 ) ) ) ).
% order.strict_trans
thf(fact_465_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_466_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_467_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_468_dual__order_Ostrict__trans,axiom,
! [B: real,A2: real,C3: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C3 @ B )
=> ( ord_less_real @ C3 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_469_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C3: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C3 @ B )
=> ( ord_less_nat @ C3 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_470_dual__order_Ostrict__trans,axiom,
! [B: int,A2: int,C3: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C3 @ B )
=> ( ord_less_int @ C3 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_471_order_Ostrict__implies__not__eq,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_472_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_473_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_474_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_475_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_476_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_477_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_478_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_479_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_480_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_481_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_482_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_483_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_484_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_485_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_486_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_487_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_488_order__less__asym_H,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ~ ( ord_less_real @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_489_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_490_order__less__asym_H,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_491_order__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_492_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_493_order__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_494_ord__eq__less__subst,axiom,
! [A2: real,F: real > real,B: real,C3: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_495_ord__eq__less__subst,axiom,
! [A2: nat,F: real > nat,B: real,C3: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_496_ord__eq__less__subst,axiom,
! [A2: int,F: real > int,B: real,C3: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_497_ord__eq__less__subst,axiom,
! [A2: real,F: nat > real,B: nat,C3: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_498_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C3: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_499_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B: nat,C3: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_500_ord__eq__less__subst,axiom,
! [A2: real,F: int > real,B: int,C3: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_501_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B: int,C3: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_502_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B: int,C3: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_503_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > real,C3: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_504_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > nat,C3: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_505_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > int,C3: int] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_506_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > real,C3: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_507_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_508_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C3: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_509_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > real,C3: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_510_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C3: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_511_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_512_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_513_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_514_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_515_order__less__subst1,axiom,
! [A2: real,F: real > real,B: real,C3: real] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_516_order__less__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C3: nat] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_517_order__less__subst1,axiom,
! [A2: real,F: int > real,B: int,C3: int] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_518_order__less__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C3: real] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_519_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C3: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_520_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C3: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_521_order__less__subst1,axiom,
! [A2: int,F: real > int,B: real,C3: real] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_522_order__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C3: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_523_order__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C3: int] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_524_order__less__subst2,axiom,
! [A2: real,B: real,F: real > real,C3: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_525_order__less__subst2,axiom,
! [A2: real,B: real,F: real > nat,C3: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_526_order__less__subst2,axiom,
! [A2: real,B: real,F: real > int,C3: int] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_527_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C3: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_528_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_529_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C3: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_530_order__less__subst2,axiom,
! [A2: int,B: int,F: int > real,C3: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_531_order__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C3: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_532_order__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_533_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_534_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_535_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_536_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_537_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_538_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_539_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_540_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_541_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_542_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_543_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_544_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_545_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_546_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_547_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_548_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_549_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_550_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_551_set__plus__elim,axiom,
! [X: real,A: set_real,B2: set_real] :
( ( member_real @ X @ ( plus_plus_set_real @ A @ B2 ) )
=> ~ ! [A5: real,B5: real] :
( ( X
= ( plus_plus_real @ A5 @ B5 ) )
=> ( ( member_real @ A5 @ A )
=> ~ ( member_real @ B5 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_552_set__plus__elim,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ X @ ( plus_plus_set_nat @ A @ B2 ) )
=> ~ ! [A5: nat,B5: nat] :
( ( X
= ( plus_plus_nat @ A5 @ B5 ) )
=> ( ( member_nat @ A5 @ A )
=> ~ ( member_nat @ B5 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_553_set__plus__elim,axiom,
! [X: int,A: set_int,B2: set_int] :
( ( member_int @ X @ ( plus_plus_set_int @ A @ B2 ) )
=> ~ ! [A5: int,B5: int] :
( ( X
= ( plus_plus_int @ A5 @ B5 ) )
=> ( ( member_int @ A5 @ A )
=> ~ ( member_int @ B5 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_554_monotone__onD,axiom,
! [A: set_real,Orda: real > real > $o,Ordb: real > real > $o,F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ A @ Orda @ Ordb @ F )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% monotone_onD
thf(fact_555_monotone__onD,axiom,
! [A: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A @ Orda @ Ordb @ F )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% monotone_onD
thf(fact_556_monotone__onI,axiom,
! [A: set_real,Orda: real > real > $o,Ordb: real > real > $o,F: real > real] :
( ! [X2: real,Y2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_real @ Y2 @ A )
=> ( ( Orda @ X2 @ Y2 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A @ Orda @ Ordb @ F ) ) ).
% monotone_onI
thf(fact_557_monotone__onI,axiom,
! [A: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat] :
( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( Orda @ X2 @ Y2 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ Orda @ Ordb @ F ) ) ).
% monotone_onI
thf(fact_558_monotone__on__def,axiom,
( monoto4017252874604999745l_real
= ( ^ [A3: set_real,Orda2: real > real > $o,Ordb2: real > real > $o,F4: real > real] :
! [X4: real] :
( ( member_real @ X4 @ A3 )
=> ! [Y4: real] :
( ( member_real @ Y4 @ A3 )
=> ( ( Orda2 @ X4 @ Y4 )
=> ( Ordb2 @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_559_monotone__on__def,axiom,
( monotone_on_nat_nat
= ( ^ [A3: set_nat,Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F4: nat > nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ A3 )
=> ( ( Orda2 @ X4 @ Y4 )
=> ( Ordb2 @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_560_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_561_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_562_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_563_leD,axiom,
! [Y: set_real,X: set_real] :
( ( ord_less_eq_set_real @ Y @ X )
=> ~ ( ord_less_set_real @ X @ Y ) ) ).
% leD
thf(fact_564_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_565_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_566_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_567_nless__le,axiom,
! [A2: real,B: real] :
( ( ~ ( ord_less_real @ A2 @ B ) )
= ( ~ ( ord_less_eq_real @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_568_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_569_nless__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_int @ A2 @ B ) )
= ( ~ ( ord_less_eq_int @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_570_nless__le,axiom,
! [A2: set_real,B: set_real] :
( ( ~ ( ord_less_set_real @ A2 @ B ) )
= ( ~ ( ord_less_eq_set_real @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_571_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_572_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_573_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_574_antisym__conv1,axiom,
! [X: set_real,Y: set_real] :
( ~ ( ord_less_set_real @ X @ Y )
=> ( ( ord_less_eq_set_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_575_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_576_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_577_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_578_antisym__conv2,axiom,
! [X: set_real,Y: set_real] :
( ( ord_less_eq_set_real @ X @ Y )
=> ( ( ~ ( ord_less_set_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_579_dense__ge,axiom,
! [Z2: real,Y: real] :
( ! [X2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ord_less_eq_real @ Y @ X2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_580_dense__le,axiom,
! [Y: real,Z2: real] :
( ! [X2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_le
thf(fact_581_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_582_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_583_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_584_less__le__not__le,axiom,
( ord_less_set_real
= ( ^ [X4: set_real,Y4: set_real] :
( ( ord_less_eq_set_real @ X4 @ Y4 )
& ~ ( ord_less_eq_set_real @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_585_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_586_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_587_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_588_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_589_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_590_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_591_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_real
= ( ^ [A4: set_real,B4: set_real] :
( ( ord_less_set_real @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_592_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_593_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_594_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_595_order_Ostrict__iff__order,axiom,
( ord_less_set_real
= ( ^ [A4: set_real,B4: set_real] :
( ( ord_less_eq_set_real @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_596_order_Ostrict__trans1,axiom,
! [A2: real,B: real,C3: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real @ B @ C3 )
=> ( ord_less_real @ A2 @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_597_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ord_less_nat @ A2 @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_598_order_Ostrict__trans1,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C3 )
=> ( ord_less_int @ A2 @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_599_order_Ostrict__trans1,axiom,
! [A2: set_real,B: set_real,C3: set_real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( ord_less_set_real @ B @ C3 )
=> ( ord_less_set_real @ A2 @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_600_order_Ostrict__trans2,axiom,
! [A2: real,B: real,C3: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ C3 )
=> ( ord_less_real @ A2 @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_601_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ord_less_nat @ A2 @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_602_order_Ostrict__trans2,axiom,
! [A2: int,B: int,C3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ord_less_int @ A2 @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_603_order_Ostrict__trans2,axiom,
! [A2: set_real,B: set_real,C3: set_real] :
( ( ord_less_set_real @ A2 @ B )
=> ( ( ord_less_eq_set_real @ B @ C3 )
=> ( ord_less_set_real @ A2 @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_604_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ~ ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_605_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_606_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_607_order_Ostrict__iff__not,axiom,
( ord_less_set_real
= ( ^ [A4: set_real,B4: set_real] :
( ( ord_less_eq_set_real @ A4 @ B4 )
& ~ ( ord_less_eq_set_real @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_608_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_609_dense__le__bounded,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_610_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_real @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_611_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_612_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_int @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_613_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_real
= ( ^ [B4: set_real,A4: set_real] :
( ( ord_less_set_real @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_614_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_615_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_616_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_617_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_real
= ( ^ [B4: set_real,A4: set_real] :
( ( ord_less_eq_set_real @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_618_dual__order_Ostrict__trans1,axiom,
! [B: real,A2: real,C3: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_real @ C3 @ B )
=> ( ord_less_real @ C3 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_619_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C3: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C3 @ B )
=> ( ord_less_nat @ C3 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_620_dual__order_Ostrict__trans1,axiom,
! [B: int,A2: int,C3: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_int @ C3 @ B )
=> ( ord_less_int @ C3 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_621_dual__order_Ostrict__trans1,axiom,
! [B: set_real,A2: set_real,C3: set_real] :
( ( ord_less_eq_set_real @ B @ A2 )
=> ( ( ord_less_set_real @ C3 @ B )
=> ( ord_less_set_real @ C3 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_622_dual__order_Ostrict__trans2,axiom,
! [B: real,A2: real,C3: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C3 @ B )
=> ( ord_less_real @ C3 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_623_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C3: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C3 @ B )
=> ( ord_less_nat @ C3 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_624_dual__order_Ostrict__trans2,axiom,
! [B: int,A2: int,C3: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C3 @ B )
=> ( ord_less_int @ C3 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_625_dual__order_Ostrict__trans2,axiom,
! [B: set_real,A2: set_real,C3: set_real] :
( ( ord_less_set_real @ B @ A2 )
=> ( ( ord_less_eq_set_real @ C3 @ B )
=> ( ord_less_set_real @ C3 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_626_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_627_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_628_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_629_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_real
= ( ^ [B4: set_real,A4: set_real] :
( ( ord_less_eq_set_real @ B4 @ A4 )
& ~ ( ord_less_eq_set_real @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_630_order_Ostrict__implies__order,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_eq_real @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_631_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_632_order_Ostrict__implies__order,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_633_order_Ostrict__implies__order,axiom,
! [A2: set_real,B: set_real] :
( ( ord_less_set_real @ A2 @ B )
=> ( ord_less_eq_set_real @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_634_dual__order_Ostrict__implies__order,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( ord_less_eq_real @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_635_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_636_dual__order_Ostrict__implies__order,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_637_dual__order_Ostrict__implies__order,axiom,
! [B: set_real,A2: set_real] :
( ( ord_less_set_real @ B @ A2 )
=> ( ord_less_eq_set_real @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_638_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_639_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_640_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_641_order__le__less,axiom,
( ord_less_eq_set_real
= ( ^ [X4: set_real,Y4: set_real] :
( ( ord_less_set_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_642_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_643_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_644_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_645_order__less__le,axiom,
( ord_less_set_real
= ( ^ [X4: set_real,Y4: set_real] :
( ( ord_less_eq_set_real @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_646_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_647_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_648_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_649_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_650_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_651_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_652_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_653_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_654_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_655_order__less__imp__le,axiom,
! [X: set_real,Y: set_real] :
( ( ord_less_set_real @ X @ Y )
=> ( ord_less_eq_set_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_656_order__le__neq__trans,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_real @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_657_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_658_order__le__neq__trans,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_659_order__le__neq__trans,axiom,
! [A2: set_real,B: set_real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_real @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_660_order__neq__le__trans,axiom,
! [A2: real,B: real] :
( ( A2 != B )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_real @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_661_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_662_order__neq__le__trans,axiom,
! [A2: int,B: int] :
( ( A2 != B )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_663_order__neq__le__trans,axiom,
! [A2: set_real,B: set_real] :
( ( A2 != B )
=> ( ( ord_less_eq_set_real @ A2 @ B )
=> ( ord_less_set_real @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_664_order__le__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_665_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_666_order__le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_667_order__le__less__trans,axiom,
! [X: set_real,Y: set_real,Z2: set_real] :
( ( ord_less_eq_set_real @ X @ Y )
=> ( ( ord_less_set_real @ Y @ Z2 )
=> ( ord_less_set_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_668_order__less__le__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_669_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_670_order__less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_671_order__less__le__trans,axiom,
! [X: set_real,Y: set_real,Z2: set_real] :
( ( ord_less_set_real @ X @ Y )
=> ( ( ord_less_eq_set_real @ Y @ Z2 )
=> ( ord_less_set_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_672_order__le__less__subst1,axiom,
! [A2: real,F: real > real,B: real,C3: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_673_order__le__less__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C3: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_674_order__le__less__subst1,axiom,
! [A2: real,F: int > real,B: int,C3: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_675_order__le__less__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C3: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_676_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_677_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C3: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_678_order__le__less__subst1,axiom,
! [A2: int,F: real > int,B: real,C3: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_679_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C3: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_680_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C3: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_681_order__le__less__subst1,axiom,
! [A2: set_real,F: real > set_real,B: real,C3: real] :
( ( ord_less_eq_set_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_set_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_682_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > real,C3: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_683_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > nat,C3: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_684_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > int,C3: int] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_685_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C3: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_686_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_687_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C3: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_688_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > real,C3: real] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_689_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C3: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_690_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C3: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_691_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > set_real,C3: set_real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_set_real @ ( F @ B ) @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_692_order__less__le__subst1,axiom,
! [A2: real,F: real > real,B: real,C3: real] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_693_order__less__le__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C3: real] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_694_order__less__le__subst1,axiom,
! [A2: int,F: real > int,B: real,C3: real] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_695_order__less__le__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C3: nat] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_696_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C3: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_697_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C3: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_698_order__less__le__subst1,axiom,
! [A2: real,F: int > real,B: int,C3: int] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_699_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C3: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_700_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B: int,C3: int] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_701_order__less__le__subst1,axiom,
! [A2: set_real,F: real > set_real,B: real,C3: real] :
( ( ord_less_set_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_real @ A2 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_702_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > real,C3: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_703_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C3: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_704_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > real,C3: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_705_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > nat,C3: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_706_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_707_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > nat,C3: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_708_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > int,C3: int] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_709_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C3: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C3 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_710_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > int,C3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C3 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_711_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > set_real,C3: set_real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_set_real @ ( F @ B ) @ C3 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_set_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_real @ ( F @ A2 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_712_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_713_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_714_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_715_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_716_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_717_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_718_order__le__imp__less__or__eq,axiom,
! [X: set_real,Y: set_real] :
( ( ord_less_eq_set_real @ X @ Y )
=> ( ( ord_less_set_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_719_ord_Omono__onD,axiom,
! [A: set_int,Less_eq: int > int > $o,F: int > real,R: int,S: int] :
( ( monotone_on_int_real @ A @ Less_eq @ ord_less_eq_real @ F )
=> ( ( member_int @ R @ A )
=> ( ( member_int @ S @ A )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_720_ord_Omono__onD,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > real,R: real,S: real] :
( ( monoto4017252874604999745l_real @ A @ Less_eq @ ord_less_eq_real @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S @ A )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_721_ord_Omono__onD,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > nat,R: real,S: real] :
( ( monotone_on_real_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S @ A )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_722_ord_Omono__onD,axiom,
! [A: set_int,Less_eq: int > int > $o,F: int > nat,R: int,S: int] :
( ( monotone_on_int_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_int @ R @ A )
=> ( ( member_int @ S @ A )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_723_ord_Omono__onD,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat,R: nat,S: nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S @ A )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_724_ord_Omono__onD,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > int,R: real,S: real] :
( ( monotone_on_real_int @ A @ Less_eq @ ord_less_eq_int @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S @ A )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_725_ord_Omono__onD,axiom,
! [A: set_int,Less_eq: int > int > $o,F: int > int,R: int,S: int] :
( ( monotone_on_int_int @ A @ Less_eq @ ord_less_eq_int @ F )
=> ( ( member_int @ R @ A )
=> ( ( member_int @ S @ A )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_726_ord_Omono__onD,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > set_real,R: real,S: real] :
( ( monoto3333417327835629687t_real @ A @ Less_eq @ ord_less_eq_set_real @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S @ A )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_set_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_727_ord_Omono__onD,axiom,
! [A: set_int,Less_eq: int > int > $o,F: int > set_real,R: int,S: int] :
( ( monoto4059737280683413495t_real @ A @ Less_eq @ ord_less_eq_set_real @ F )
=> ( ( member_int @ R @ A )
=> ( ( member_int @ S @ A )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_set_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_728_ord_Omono__onI,axiom,
! [A: set_int,Less_eq: int > int > $o,F: int > real] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A )
=> ( ( member_int @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_real @ A @ Less_eq @ ord_less_eq_real @ F ) ) ).
% ord.mono_onI
thf(fact_729_ord_Omono__onI,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > real] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A @ Less_eq @ ord_less_eq_real @ F ) ) ).
% ord.mono_onI
thf(fact_730_ord_Omono__onI,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > nat] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_731_ord_Omono__onI,axiom,
! [A: set_int,Less_eq: int > int > $o,F: int > nat] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A )
=> ( ( member_int @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_732_ord_Omono__onI,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_733_ord_Omono__onI,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > int] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_int @ A @ Less_eq @ ord_less_eq_int @ F ) ) ).
% ord.mono_onI
thf(fact_734_ord_Omono__onI,axiom,
! [A: set_int,Less_eq: int > int > $o,F: int > int] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A )
=> ( ( member_int @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_int @ A @ Less_eq @ ord_less_eq_int @ F ) ) ).
% ord.mono_onI
thf(fact_735_ord_Omono__onI,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > set_real] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_set_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto3333417327835629687t_real @ A @ Less_eq @ ord_less_eq_set_real @ F ) ) ).
% ord.mono_onI
thf(fact_736_ord_Omono__onI,axiom,
! [A: set_int,Less_eq: int > int > $o,F: int > set_real] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A )
=> ( ( member_int @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_set_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4059737280683413495t_real @ A @ Less_eq @ ord_less_eq_set_real @ F ) ) ).
% ord.mono_onI
thf(fact_737_ord_Omono__on__def,axiom,
! [A: set_int,Less_eq: int > int > $o,F: int > real] :
( ( monotone_on_int_real @ A @ Less_eq @ ord_less_eq_real @ F )
= ( ! [R3: int,S4: int] :
( ( ( member_int @ R3 @ A )
& ( member_int @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_738_ord_Omono__on__def,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > real] :
( ( monoto4017252874604999745l_real @ A @ Less_eq @ ord_less_eq_real @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A )
& ( member_real @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_739_ord_Omono__on__def,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > nat] :
( ( monotone_on_real_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A )
& ( member_real @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_740_ord_Omono__on__def,axiom,
! [A: set_int,Less_eq: int > int > $o,F: int > nat] :
( ( monotone_on_int_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: int,S4: int] :
( ( ( member_int @ R3 @ A )
& ( member_int @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_741_ord_Omono__on__def,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A )
& ( member_nat @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_742_ord_Omono__on__def,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > int] :
( ( monotone_on_real_int @ A @ Less_eq @ ord_less_eq_int @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A )
& ( member_real @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_int @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_743_ord_Omono__on__def,axiom,
! [A: set_int,Less_eq: int > int > $o,F: int > int] :
( ( monotone_on_int_int @ A @ Less_eq @ ord_less_eq_int @ F )
= ( ! [R3: int,S4: int] :
( ( ( member_int @ R3 @ A )
& ( member_int @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_int @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_744_ord_Omono__on__def,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > set_real] :
( ( monoto3333417327835629687t_real @ A @ Less_eq @ ord_less_eq_set_real @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A )
& ( member_real @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_set_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_745_ord_Omono__on__def,axiom,
! [A: set_int,Less_eq: int > int > $o,F: int > set_real] :
( ( monoto4059737280683413495t_real @ A @ Less_eq @ ord_less_eq_set_real @ F )
= ( ! [R3: int,S4: int] :
( ( ( member_int @ R3 @ A )
& ( member_int @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_set_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_746_mono__onD,axiom,
! [A: set_real,F: real > real,R: real,S: real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S @ A )
=> ( ( ord_less_eq_real @ R @ S )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_747_mono__onD,axiom,
! [A: set_real,F: real > nat,R: real,S: real] :
( ( monotone_on_real_nat @ A @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S @ A )
=> ( ( ord_less_eq_real @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_748_mono__onD,axiom,
! [A: set_real,F: real > int,R: real,S: real] :
( ( monotone_on_real_int @ A @ ord_less_eq_real @ ord_less_eq_int @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S @ A )
=> ( ( ord_less_eq_real @ R @ S )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_749_mono__onD,axiom,
! [A: set_nat,F: nat > real,R: nat,S: nat] :
( ( monotone_on_nat_real @ A @ ord_less_eq_nat @ ord_less_eq_real @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S @ A )
=> ( ( ord_less_eq_nat @ R @ S )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_750_mono__onD,axiom,
! [A: set_nat,F: nat > nat,R: nat,S: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S @ A )
=> ( ( ord_less_eq_nat @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_751_mono__onD,axiom,
! [A: set_nat,F: nat > int,R: nat,S: nat] :
( ( monotone_on_nat_int @ A @ ord_less_eq_nat @ ord_less_eq_int @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S @ A )
=> ( ( ord_less_eq_nat @ R @ S )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_752_mono__onD,axiom,
! [A: set_int,F: int > real,R: int,S: int] :
( ( monotone_on_int_real @ A @ ord_less_eq_int @ ord_less_eq_real @ F )
=> ( ( member_int @ R @ A )
=> ( ( member_int @ S @ A )
=> ( ( ord_less_eq_int @ R @ S )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_753_mono__onD,axiom,
! [A: set_int,F: int > nat,R: int,S: int] :
( ( monotone_on_int_nat @ A @ ord_less_eq_int @ ord_less_eq_nat @ F )
=> ( ( member_int @ R @ A )
=> ( ( member_int @ S @ A )
=> ( ( ord_less_eq_int @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_754_mono__onD,axiom,
! [A: set_int,F: int > int,R: int,S: int] :
( ( monotone_on_int_int @ A @ ord_less_eq_int @ ord_less_eq_int @ F )
=> ( ( member_int @ R @ A )
=> ( ( member_int @ S @ A )
=> ( ( ord_less_eq_int @ R @ S )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_755_mono__onD,axiom,
! [A: set_real,F: real > set_real,R: real,S: real] :
( ( monoto3333417327835629687t_real @ A @ ord_less_eq_real @ ord_less_eq_set_real @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S @ A )
=> ( ( ord_less_eq_real @ R @ S )
=> ( ord_less_eq_set_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_756_mono__onI,axiom,
! [A: set_real,F: real > real] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( ord_less_eq_real @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_757_mono__onI,axiom,
! [A: set_real,F: real > nat] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( ord_less_eq_real @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_758_mono__onI,axiom,
! [A: set_real,F: real > int] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( ord_less_eq_real @ R2 @ S3 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_int @ A @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_759_mono__onI,axiom,
! [A: set_nat,F: nat > real] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( ord_less_eq_nat @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_real @ A @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_760_mono__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( ord_less_eq_nat @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_761_mono__onI,axiom,
! [A: set_nat,F: nat > int] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( ord_less_eq_nat @ R2 @ S3 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_int @ A @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_762_mono__onI,axiom,
! [A: set_int,F: int > real] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A )
=> ( ( member_int @ S3 @ A )
=> ( ( ord_less_eq_int @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_real @ A @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_763_mono__onI,axiom,
! [A: set_int,F: int > nat] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A )
=> ( ( member_int @ S3 @ A )
=> ( ( ord_less_eq_int @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_nat @ A @ ord_less_eq_int @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_764_mono__onI,axiom,
! [A: set_int,F: int > int] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A )
=> ( ( member_int @ S3 @ A )
=> ( ( ord_less_eq_int @ R2 @ S3 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_int @ A @ ord_less_eq_int @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_765_mono__onI,axiom,
! [A: set_real,F: real > set_real] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( ord_less_eq_real @ R2 @ S3 )
=> ( ord_less_eq_set_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto3333417327835629687t_real @ A @ ord_less_eq_real @ ord_less_eq_set_real @ F ) ) ).
% mono_onI
thf(fact_766_ord_Ostrict__mono__onD,axiom,
! [A: set_int,Less: int > int > $o,F: int > real,R: int,S: int] :
( ( monotone_on_int_real @ A @ Less @ ord_less_real @ F )
=> ( ( member_int @ R @ A )
=> ( ( member_int @ S @ A )
=> ( ( Less @ R @ S )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_767_ord_Ostrict__mono__onD,axiom,
! [A: set_real,Less: real > real > $o,F: real > real,R: real,S: real] :
( ( monoto4017252874604999745l_real @ A @ Less @ ord_less_real @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S @ A )
=> ( ( Less @ R @ S )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_768_ord_Ostrict__mono__onD,axiom,
! [A: set_real,Less: real > real > $o,F: real > nat,R: real,S: real] :
( ( monotone_on_real_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S @ A )
=> ( ( Less @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_769_ord_Ostrict__mono__onD,axiom,
! [A: set_int,Less: int > int > $o,F: int > nat,R: int,S: int] :
( ( monotone_on_int_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_int @ R @ A )
=> ( ( member_int @ S @ A )
=> ( ( Less @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_770_ord_Ostrict__mono__onD,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > nat,R: nat,S: nat] :
( ( monotone_on_nat_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S @ A )
=> ( ( Less @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_771_ord_Ostrict__mono__onD,axiom,
! [A: set_real,Less: real > real > $o,F: real > int,R: real,S: real] :
( ( monotone_on_real_int @ A @ Less @ ord_less_int @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S @ A )
=> ( ( Less @ R @ S )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_772_ord_Ostrict__mono__onD,axiom,
! [A: set_int,Less: int > int > $o,F: int > int,R: int,S: int] :
( ( monotone_on_int_int @ A @ Less @ ord_less_int @ F )
=> ( ( member_int @ R @ A )
=> ( ( member_int @ S @ A )
=> ( ( Less @ R @ S )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_773_ord_Ostrict__mono__onI,axiom,
! [A: set_int,Less: int > int > $o,F: int > real] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A )
=> ( ( member_int @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_real @ A @ Less @ ord_less_real @ F ) ) ).
% ord.strict_mono_onI
thf(fact_774_ord_Ostrict__mono__onI,axiom,
! [A: set_real,Less: real > real > $o,F: real > real] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A @ Less @ ord_less_real @ F ) ) ).
% ord.strict_mono_onI
thf(fact_775_ord_Ostrict__mono__onI,axiom,
! [A: set_real,Less: real > real > $o,F: real > nat] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_776_ord_Ostrict__mono__onI,axiom,
! [A: set_int,Less: int > int > $o,F: int > nat] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A )
=> ( ( member_int @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_777_ord_Ostrict__mono__onI,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_778_ord_Ostrict__mono__onI,axiom,
! [A: set_real,Less: real > real > $o,F: real > int] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_int @ A @ Less @ ord_less_int @ F ) ) ).
% ord.strict_mono_onI
thf(fact_779_ord_Ostrict__mono__onI,axiom,
! [A: set_int,Less: int > int > $o,F: int > int] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A )
=> ( ( member_int @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_int @ A @ Less @ ord_less_int @ F ) ) ).
% ord.strict_mono_onI
thf(fact_780_ord_Ostrict__mono__on__def,axiom,
! [A: set_int,Less: int > int > $o,F: int > real] :
( ( monotone_on_int_real @ A @ Less @ ord_less_real @ F )
= ( ! [R3: int,S4: int] :
( ( ( member_int @ R3 @ A )
& ( member_int @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_781_ord_Ostrict__mono__on__def,axiom,
! [A: set_real,Less: real > real > $o,F: real > real] :
( ( monoto4017252874604999745l_real @ A @ Less @ ord_less_real @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A )
& ( member_real @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_782_ord_Ostrict__mono__on__def,axiom,
! [A: set_real,Less: real > real > $o,F: real > nat] :
( ( monotone_on_real_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A )
& ( member_real @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_783_ord_Ostrict__mono__on__def,axiom,
! [A: set_int,Less: int > int > $o,F: int > nat] :
( ( monotone_on_int_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R3: int,S4: int] :
( ( ( member_int @ R3 @ A )
& ( member_int @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_784_ord_Ostrict__mono__on__def,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A )
& ( member_nat @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_785_ord_Ostrict__mono__on__def,axiom,
! [A: set_real,Less: real > real > $o,F: real > int] :
( ( monotone_on_real_int @ A @ Less @ ord_less_int @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A )
& ( member_real @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_int @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_786_ord_Ostrict__mono__on__def,axiom,
! [A: set_int,Less: int > int > $o,F: int > int] :
( ( monotone_on_int_int @ A @ Less @ ord_less_int @ F )
= ( ! [R3: int,S4: int] :
( ( ( member_int @ R3 @ A )
& ( member_int @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_int @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_787_strict__mono__onD,axiom,
! [A: set_real,F: real > real,R: real,S: real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_real @ ord_less_real @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S @ A )
=> ( ( ord_less_real @ R @ S )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_788_strict__mono__onD,axiom,
! [A: set_real,F: real > nat,R: real,S: real] :
( ( monotone_on_real_nat @ A @ ord_less_real @ ord_less_nat @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S @ A )
=> ( ( ord_less_real @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_789_strict__mono__onD,axiom,
! [A: set_real,F: real > int,R: real,S: real] :
( ( monotone_on_real_int @ A @ ord_less_real @ ord_less_int @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S @ A )
=> ( ( ord_less_real @ R @ S )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_790_strict__mono__onD,axiom,
! [A: set_nat,F: nat > real,R: nat,S: nat] :
( ( monotone_on_nat_real @ A @ ord_less_nat @ ord_less_real @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S @ A )
=> ( ( ord_less_nat @ R @ S )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_791_strict__mono__onD,axiom,
! [A: set_nat,F: nat > nat,R: nat,S: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S @ A )
=> ( ( ord_less_nat @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_792_strict__mono__onD,axiom,
! [A: set_nat,F: nat > int,R: nat,S: nat] :
( ( monotone_on_nat_int @ A @ ord_less_nat @ ord_less_int @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S @ A )
=> ( ( ord_less_nat @ R @ S )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_793_strict__mono__onD,axiom,
! [A: set_int,F: int > real,R: int,S: int] :
( ( monotone_on_int_real @ A @ ord_less_int @ ord_less_real @ F )
=> ( ( member_int @ R @ A )
=> ( ( member_int @ S @ A )
=> ( ( ord_less_int @ R @ S )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_794_strict__mono__onD,axiom,
! [A: set_int,F: int > nat,R: int,S: int] :
( ( monotone_on_int_nat @ A @ ord_less_int @ ord_less_nat @ F )
=> ( ( member_int @ R @ A )
=> ( ( member_int @ S @ A )
=> ( ( ord_less_int @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_795_strict__mono__onD,axiom,
! [A: set_int,F: int > int,R: int,S: int] :
( ( monotone_on_int_int @ A @ ord_less_int @ ord_less_int @ F )
=> ( ( member_int @ R @ A )
=> ( ( member_int @ S @ A )
=> ( ( ord_less_int @ R @ S )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_796_strict__mono__onI,axiom,
! [A: set_real,F: real > real] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( ord_less_real @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A @ ord_less_real @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_797_strict__mono__onI,axiom,
! [A: set_real,F: real > nat] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( ord_less_real @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A @ ord_less_real @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_798_strict__mono__onI,axiom,
! [A: set_real,F: real > int] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( ord_less_real @ R2 @ S3 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_int @ A @ ord_less_real @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_799_strict__mono__onI,axiom,
! [A: set_nat,F: nat > real] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( ord_less_nat @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_real @ A @ ord_less_nat @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_800_strict__mono__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( ord_less_nat @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_801_strict__mono__onI,axiom,
! [A: set_nat,F: nat > int] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( ord_less_nat @ R2 @ S3 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_int @ A @ ord_less_nat @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_802_strict__mono__onI,axiom,
! [A: set_int,F: int > real] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A )
=> ( ( member_int @ S3 @ A )
=> ( ( ord_less_int @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_real @ A @ ord_less_int @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_803_strict__mono__onI,axiom,
! [A: set_int,F: int > nat] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A )
=> ( ( member_int @ S3 @ A )
=> ( ( ord_less_int @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_nat @ A @ ord_less_int @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_804_strict__mono__onI,axiom,
! [A: set_int,F: int > int] :
( ! [R2: int,S3: int] :
( ( member_int @ R2 @ A )
=> ( ( member_int @ S3 @ A )
=> ( ( ord_less_int @ R2 @ S3 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_int_int @ A @ ord_less_int @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_805_strict__mono__on__eqD,axiom,
! [A: set_real,F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_real @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_806_strict__mono__on__eqD,axiom,
! [A: set_real,F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ A @ ord_less_real @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_807_strict__mono__on__eqD,axiom,
! [A: set_real,F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ A @ ord_less_real @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_808_strict__mono__on__eqD,axiom,
! [A: set_nat,F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ A @ ord_less_nat @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_809_strict__mono__on__eqD,axiom,
! [A: set_nat,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_810_strict__mono__on__eqD,axiom,
! [A: set_nat,F: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ A @ ord_less_nat @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_811_strict__mono__on__eqD,axiom,
! [A: set_int,F: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ A @ ord_less_int @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_812_strict__mono__on__eqD,axiom,
! [A: set_int,F: int > nat,X: int,Y: int] :
( ( monotone_on_int_nat @ A @ ord_less_int @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_813_strict__mono__on__eqD,axiom,
! [A: set_int,F: int > int,X: int,Y: int] :
( ( monotone_on_int_int @ A @ ord_less_int @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_814_monotone__on__subset,axiom,
! [A: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,B2: set_nat] :
( ( monotone_on_nat_nat @ A @ Orda @ Ordb @ F )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( monotone_on_nat_nat @ B2 @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_815_monotone__on__subset,axiom,
! [A: set_real,Orda: real > real > $o,Ordb: real > real > $o,F: real > real,B2: set_real] :
( ( monoto4017252874604999745l_real @ A @ Orda @ Ordb @ F )
=> ( ( ord_less_eq_set_real @ B2 @ A )
=> ( monoto4017252874604999745l_real @ B2 @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_816_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_817_seq__mono__lemma,axiom,
! [M3: nat,D3: nat > real,E2: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_real @ ( D3 @ N3 ) @ ( E2 @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_real @ ( E2 @ N3 ) @ ( E2 @ M3 ) ) )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
=> ( ord_less_real @ ( D3 @ N4 ) @ ( E2 @ M3 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_818_all__subset__image,axiom,
! [F: nat > int,A: set_nat,P: set_int > $o] :
( ( ! [B3: set_int] :
( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( P @ ( image_nat_int @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_819_all__subset__image,axiom,
! [F: real > real,A: set_real,P: set_real > $o] :
( ( ! [B3: set_real] :
( ( ord_less_eq_set_real @ B3 @ ( image_real_real @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_real] :
( ( ord_less_eq_set_real @ B3 @ A )
=> ( P @ ( image_real_real @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_820_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_821_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_822_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_823_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_824_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_825_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_826_minf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ~ ( ord_less_eq_real @ T @ X3 ) ) ).
% minf(8)
thf(fact_827_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% minf(8)
thf(fact_828_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X3 ) ) ).
% minf(8)
thf(fact_829_minf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ord_less_eq_real @ X3 @ T ) ) ).
% minf(6)
thf(fact_830_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ T ) ) ).
% minf(6)
thf(fact_831_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ord_less_eq_int @ X3 @ T ) ) ).
% minf(6)
thf(fact_832_pinf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ord_less_eq_real @ T @ X3 ) ) ).
% pinf(8)
thf(fact_833_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_834_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ord_less_eq_int @ T @ X3 ) ) ).
% pinf(8)
thf(fact_835_pinf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ T ) ) ).
% pinf(6)
thf(fact_836_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T ) ) ).
% pinf(6)
thf(fact_837_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ T ) ) ).
% pinf(6)
thf(fact_838_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_839_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_840_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_841_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_842_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_843_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_844_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_845_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_846_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_847_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_848_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_849_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_850_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_851_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ~ ( ord_less_real @ X3 @ T ) ) ).
% pinf(5)
thf(fact_852_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ~ ( ord_less_nat @ X3 @ T ) ) ).
% pinf(5)
thf(fact_853_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ~ ( ord_less_int @ X3 @ T ) ) ).
% pinf(5)
thf(fact_854_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ord_less_real @ T @ X3 ) ) ).
% pinf(7)
thf(fact_855_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ord_less_nat @ T @ X3 ) ) ).
% pinf(7)
thf(fact_856_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ord_less_int @ T @ X3 ) ) ).
% pinf(7)
thf(fact_857_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_858_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_859_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_860_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_861_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_862_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_863_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_864_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_865_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_866_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_867_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_868_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_869_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ord_less_real @ X3 @ T ) ) ).
% minf(5)
thf(fact_870_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ord_less_nat @ X3 @ T ) ) ).
% minf(5)
thf(fact_871_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ord_less_int @ X3 @ T ) ) ).
% minf(5)
thf(fact_872_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ~ ( ord_less_real @ T @ X3 ) ) ).
% minf(7)
thf(fact_873_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ~ ( ord_less_nat @ T @ X3 ) ) ).
% minf(7)
thf(fact_874_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ~ ( ord_less_int @ T @ X3 ) ) ).
% minf(7)
thf(fact_875_is__num__normalize_I1_J,axiom,
! [A2: real,B: real,C3: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B ) @ C3 )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C3 ) ) ) ).
% is_num_normalize(1)
thf(fact_876_is__num__normalize_I1_J,axiom,
! [A2: int,B: int,C3: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C3 )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C3 ) ) ) ).
% is_num_normalize(1)
thf(fact_877_complete__real,axiom,
! [S2: set_real] :
( ? [X3: real] : ( member_real @ X3 @ S2 )
=> ( ? [Z4: real] :
! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ? [Y2: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Y2 ) )
& ! [Z4: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ( ord_less_eq_real @ Y2 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_878_nat__add__left__cancel__le,axiom,
! [K: nat,M3: nat,N5: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N5 ) )
= ( ord_less_eq_nat @ M3 @ N5 ) ) ).
% nat_add_left_cancel_le
thf(fact_879_complete__interval,axiom,
! [A2: real,B: real,P: real > $o] :
( ( ord_less_real @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C: real] :
( ( ord_less_eq_real @ A2 @ C )
& ( ord_less_eq_real @ C @ B )
& ! [X3: real] :
( ( ( ord_less_eq_real @ A2 @ X3 )
& ( ord_less_real @ X3 @ C ) )
=> ( P @ X3 ) )
& ! [D4: real] :
( ! [X2: real] :
( ( ( ord_less_eq_real @ A2 @ X2 )
& ( ord_less_real @ X2 @ D4 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_real @ D4 @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_880_complete__interval,axiom,
! [A2: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C: nat] :
( ( ord_less_eq_nat @ A2 @ C )
& ( ord_less_eq_nat @ C @ B )
& ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ C ) )
=> ( P @ X3 ) )
& ! [D4: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A2 @ X2 )
& ( ord_less_nat @ X2 @ D4 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D4 @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_881_complete__interval,axiom,
! [A2: int,B: int,P: int > $o] :
( ( ord_less_int @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C: int] :
( ( ord_less_eq_int @ A2 @ C )
& ( ord_less_eq_int @ C @ B )
& ! [X3: int] :
( ( ( ord_less_eq_int @ A2 @ X3 )
& ( ord_less_int @ X3 @ C ) )
=> ( P @ X3 ) )
& ! [D4: int] :
( ! [X2: int] :
( ( ( ord_less_eq_int @ A2 @ X2 )
& ( ord_less_int @ X2 @ D4 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_int @ D4 @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_882_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_883_verit__comp__simplify1_I3_J,axiom,
! [B6: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
= ( ord_less_real @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_884_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_885_verit__comp__simplify1_I3_J,axiom,
! [B6: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
= ( ord_less_int @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_886_retraction,axiom,
( abstra2606333701016485833n_real
= ( ^ [S5: set_real,T4: set_real,R3: real > real] :
( ( ord_less_eq_set_real @ T4 @ S5 )
& ( topolo5044208981011980120l_real @ S5 @ R3 )
& ( ( image_real_real @ R3 @ S5 )
= T4 )
& ! [X4: real] :
( ( member_real @ X4 @ T4 )
=> ( ( R3 @ X4 )
= X4 ) ) ) ) ) ).
% retraction
thf(fact_887_retractionE,axiom,
! [S2: set_int,T2: set_int,R: int > int] :
( ( abstra7169501481011290569on_int @ S2 @ T2 @ R )
=> ~ ( ( T2
= ( image_int_int @ R @ S2 ) )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ R @ S2 ) @ S2 )
=> ( ( topolo2178910747331673048nt_int @ S2 @ R )
=> ~ ! [X3: int] :
( ( member_int @ X3 @ S2 )
=> ( ( R @ ( R @ X3 ) )
= ( R @ X3 ) ) ) ) ) ) ) ).
% retractionE
thf(fact_888_retractionE,axiom,
! [S2: set_real,T2: set_real,R: real > real] :
( ( abstra2606333701016485833n_real @ S2 @ T2 @ R )
=> ~ ( ( T2
= ( image_real_real @ R @ S2 ) )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ R @ S2 ) @ S2 )
=> ( ( topolo5044208981011980120l_real @ S2 @ R )
=> ~ ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ( R @ ( R @ X3 ) )
= ( R @ X3 ) ) ) ) ) ) ) ).
% retractionE
thf(fact_889_retraction__def,axiom,
( abstra2606333701016485833n_real
= ( ^ [S5: set_real,T4: set_real,R3: real > real] :
( ( ord_less_eq_set_real @ T4 @ S5 )
& ( topolo5044208981011980120l_real @ S5 @ R3 )
& ( ord_less_eq_set_real @ ( image_real_real @ R3 @ S5 ) @ T4 )
& ! [X4: real] :
( ( member_real @ X4 @ T4 )
=> ( ( R3 @ X4 )
= X4 ) ) ) ) ) ).
% retraction_def
thf(fact_890_nat__add__left__cancel__less,axiom,
! [K: nat,M3: nat,N5: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N5 ) )
= ( ord_less_nat @ M3 @ N5 ) ) ).
% nat_add_left_cancel_less
thf(fact_891_retraction__idempotent,axiom,
! [S2: set_real,T2: set_real,R: real > real,X: real] :
( ( abstra2606333701016485833n_real @ S2 @ T2 @ R )
=> ( ( member_real @ X @ S2 )
=> ( ( R @ ( R @ X ) )
= ( R @ X ) ) ) ) ).
% retraction_idempotent
thf(fact_892_retraction__idempotent,axiom,
! [S2: set_int,T2: set_int,R: int > int,X: int] :
( ( abstra7169501481011290569on_int @ S2 @ T2 @ R )
=> ( ( member_int @ X @ S2 )
=> ( ( R @ ( R @ X ) )
= ( R @ X ) ) ) ) ).
% retraction_idempotent
thf(fact_893_less__add__eq__less,axiom,
! [K: nat,L: nat,M3: nat,N5: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M3 @ L )
= ( plus_plus_nat @ K @ N5 ) )
=> ( ord_less_nat @ M3 @ N5 ) ) ) ).
% less_add_eq_less
thf(fact_894_trans__less__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_less_add2
thf(fact_895_trans__less__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_less_add1
thf(fact_896_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_897_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_898_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_899_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_900_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_901_retraction__subset,axiom,
! [S2: set_real,T2: set_real,R: real > real,S6: set_real] :
( ( abstra2606333701016485833n_real @ S2 @ T2 @ R )
=> ( ( ord_less_eq_set_real @ T2 @ S6 )
=> ( ( ord_less_eq_set_real @ S6 @ S2 )
=> ( abstra2606333701016485833n_real @ S6 @ T2 @ R ) ) ) ) ).
% retraction_subset
thf(fact_902_verit__comp__simplify1_I2_J,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_903_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_904_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_905_verit__comp__simplify1_I2_J,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_906_verit__la__disequality,axiom,
! [A2: real,B: real] :
( ( A2 = B )
| ~ ( ord_less_eq_real @ A2 @ B )
| ~ ( ord_less_eq_real @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_907_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_908_verit__la__disequality,axiom,
! [A2: int,B: int] :
( ( A2 = B )
| ~ ( ord_less_eq_int @ A2 @ B )
| ~ ( ord_less_eq_int @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_909_verit__comp__simplify1_I1_J,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_910_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_911_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_912_ex__gt__or__lt,axiom,
! [A2: real] :
? [B5: real] :
( ( ord_less_real @ A2 @ B5 )
| ( ord_less_real @ B5 @ A2 ) ) ).
% ex_gt_or_lt
thf(fact_913_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_914_less__imp__le__nat,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_nat @ M3 @ N5 )
=> ( ord_less_eq_nat @ M3 @ N5 ) ) ).
% less_imp_le_nat
thf(fact_915_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_916_less__or__eq__imp__le,axiom,
! [M3: nat,N5: nat] :
( ( ( ord_less_nat @ M3 @ N5 )
| ( M3 = N5 ) )
=> ( ord_less_eq_nat @ M3 @ N5 ) ) ).
% less_or_eq_imp_le
thf(fact_917_le__neq__implies__less,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
=> ( ( M3 != N5 )
=> ( ord_less_nat @ M3 @ N5 ) ) ) ).
% le_neq_implies_less
thf(fact_918_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_919_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_920_nat__le__linear,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
| ( ord_less_eq_nat @ N5 @ M3 ) ) ).
% nat_le_linear
thf(fact_921_le__antisym,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
=> ( ( ord_less_eq_nat @ N5 @ M3 )
=> ( M3 = N5 ) ) ) ).
% le_antisym
thf(fact_922_eq__imp__le,axiom,
! [M3: nat,N5: nat] :
( ( M3 = N5 )
=> ( ord_less_eq_nat @ M3 @ N5 ) ) ).
% eq_imp_le
thf(fact_923_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_924_le__refl,axiom,
! [N5: nat] : ( ord_less_eq_nat @ N5 @ N5 ) ).
% le_refl
thf(fact_925_mono__nat__linear__lb,axiom,
! [F: nat > nat,M3: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K ) @ ( F @ ( plus_plus_nat @ M3 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_926_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_927_trans__le__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_le_add2
thf(fact_928_trans__le__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_le_add1
thf(fact_929_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_930_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_931_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_932_add__leD2,axiom,
! [M3: nat,K: nat,N5: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N5 )
=> ( ord_less_eq_nat @ K @ N5 ) ) ).
% add_leD2
thf(fact_933_add__leD1,axiom,
! [M3: nat,K: nat,N5: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N5 )
=> ( ord_less_eq_nat @ M3 @ N5 ) ) ).
% add_leD1
thf(fact_934_le__add2,axiom,
! [N5: nat,M3: nat] : ( ord_less_eq_nat @ N5 @ ( plus_plus_nat @ M3 @ N5 ) ) ).
% le_add2
thf(fact_935_le__add1,axiom,
! [N5: nat,M3: nat] : ( ord_less_eq_nat @ N5 @ ( plus_plus_nat @ N5 @ M3 ) ) ).
% le_add1
thf(fact_936_add__leE,axiom,
! [M3: nat,K: nat,N5: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N5 )
=> ~ ( ( ord_less_eq_nat @ M3 @ N5 )
=> ~ ( ord_less_eq_nat @ K @ N5 ) ) ) ).
% add_leE
thf(fact_937_idempotent__imp__retraction,axiom,
! [S2: set_int,F: int > int] :
( ( topolo2178910747331673048nt_int @ S2 @ F )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F @ S2 ) @ S2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S2 )
=> ( ( F @ ( F @ X2 ) )
= ( F @ X2 ) ) )
=> ( abstra7169501481011290569on_int @ S2 @ ( image_int_int @ F @ S2 ) @ F ) ) ) ) ).
% idempotent_imp_retraction
thf(fact_938_idempotent__imp__retraction,axiom,
! [S2: set_real,F: real > real] :
( ( topolo5044208981011980120l_real @ S2 @ F )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ F @ S2 ) @ S2 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ( F @ ( F @ X2 ) )
= ( F @ X2 ) ) )
=> ( abstra2606333701016485833n_real @ S2 @ ( image_real_real @ F @ S2 ) @ F ) ) ) ) ).
% idempotent_imp_retraction
thf(fact_939_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A2: real,B: real,C3: real,D3: real] :
( ( ord_less_eq_set_real @ ( set_or1633881224788618240n_real @ A2 @ B ) @ ( set_or1222579329274155063t_real @ C3 @ D3 ) )
= ( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ C3 @ A2 )
& ( ord_less_eq_real @ B @ D3 ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_940_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N2: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_941_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X4: real] : ( plus_plus_real @ ( plus_plus_real @ X4 @ X4 ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_942_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X4: int] : ( plus_plus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_943_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A2: real,B: real,C3: real,D3: real] :
( ( ord_less_eq_set_real @ ( set_or2392270231875598684t_real @ A2 @ B ) @ ( set_or1222579329274155063t_real @ C3 @ D3 ) )
= ( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ C3 @ A2 )
& ( ord_less_eq_real @ B @ D3 ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_944_infnorm__triangle,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( linear_infnorm_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( linear_infnorm_real @ X ) @ ( linear_infnorm_real @ Y ) ) ) ).
% infnorm_triangle
thf(fact_945_of__nat__eq__iff,axiom,
! [M3: nat,N5: nat] :
( ( ( semiri5074537144036343181t_real @ M3 )
= ( semiri5074537144036343181t_real @ N5 ) )
= ( M3 = N5 ) ) ).
% of_nat_eq_iff
thf(fact_946_of__nat__eq__iff,axiom,
! [M3: nat,N5: nat] :
( ( ( semiri1314217659103216013at_int @ M3 )
= ( semiri1314217659103216013at_int @ N5 ) )
= ( M3 = N5 ) ) ).
% of_nat_eq_iff
thf(fact_947_greaterThanLessThan__iff,axiom,
! [I: nat,L: nat,U2: nat] :
( ( member_nat @ I @ ( set_or5834768355832116004an_nat @ L @ U2 ) )
= ( ( ord_less_nat @ L @ I )
& ( ord_less_nat @ I @ U2 ) ) ) ).
% greaterThanLessThan_iff
thf(fact_948_greaterThanLessThan__iff,axiom,
! [I: real,L: real,U2: real] :
( ( member_real @ I @ ( set_or1633881224788618240n_real @ L @ U2 ) )
= ( ( ord_less_real @ L @ I )
& ( ord_less_real @ I @ U2 ) ) ) ).
% greaterThanLessThan_iff
thf(fact_949_greaterThanLessThan__iff,axiom,
! [I: int,L: int,U2: int] :
( ( member_int @ I @ ( set_or5832277885323065728an_int @ L @ U2 ) )
= ( ( ord_less_int @ L @ I )
& ( ord_less_int @ I @ U2 ) ) ) ).
% greaterThanLessThan_iff
thf(fact_950_of__nat__le__iff,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N5 ) )
= ( ord_less_eq_nat @ M3 @ N5 ) ) ).
% of_nat_le_iff
thf(fact_951_of__nat__le__iff,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N5 ) )
= ( ord_less_eq_nat @ M3 @ N5 ) ) ).
% of_nat_le_iff
thf(fact_952_of__nat__le__iff,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N5 ) )
= ( ord_less_eq_nat @ M3 @ N5 ) ) ).
% of_nat_le_iff
thf(fact_953_of__nat__less__iff,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N5 ) )
= ( ord_less_nat @ M3 @ N5 ) ) ).
% of_nat_less_iff
thf(fact_954_of__nat__less__iff,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N5 ) )
= ( ord_less_nat @ M3 @ N5 ) ) ).
% of_nat_less_iff
thf(fact_955_of__nat__less__iff,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N5 ) )
= ( ord_less_nat @ M3 @ N5 ) ) ).
% of_nat_less_iff
thf(fact_956_of__nat__add,axiom,
! [M3: nat,N5: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M3 @ N5 ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N5 ) ) ) ).
% of_nat_add
thf(fact_957_of__nat__add,axiom,
! [M3: nat,N5: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M3 @ N5 ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N5 ) ) ) ).
% of_nat_add
thf(fact_958_of__nat__add,axiom,
! [M3: nat,N5: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M3 @ N5 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N5 ) ) ) ).
% of_nat_add
thf(fact_959_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_960_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_961_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_962_of__nat__1__eq__iff,axiom,
! [N5: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N5 ) )
= ( N5 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_963_of__nat__1__eq__iff,axiom,
! [N5: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N5 ) )
= ( N5 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_964_of__nat__1__eq__iff,axiom,
! [N5: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N5 ) )
= ( N5 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_965_of__nat__eq__1__iff,axiom,
! [N5: nat] :
( ( ( semiri1316708129612266289at_nat @ N5 )
= one_one_nat )
= ( N5 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_966_of__nat__eq__1__iff,axiom,
! [N5: nat] :
( ( ( semiri5074537144036343181t_real @ N5 )
= one_one_real )
= ( N5 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_967_of__nat__eq__1__iff,axiom,
! [N5: nat] :
( ( ( semiri1314217659103216013at_int @ N5 )
= one_one_int )
= ( N5 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_968_greaterThanAtMost__iff,axiom,
! [I: real,L: real,U2: real] :
( ( member_real @ I @ ( set_or2392270231875598684t_real @ L @ U2 ) )
= ( ( ord_less_real @ L @ I )
& ( ord_less_eq_real @ I @ U2 ) ) ) ).
% greaterThanAtMost_iff
thf(fact_969_greaterThanAtMost__iff,axiom,
! [I: nat,L: nat,U2: nat] :
( ( member_nat @ I @ ( set_or6659071591806873216st_nat @ L @ U2 ) )
= ( ( ord_less_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U2 ) ) ) ).
% greaterThanAtMost_iff
thf(fact_970_greaterThanAtMost__iff,axiom,
! [I: set_real,L: set_real,U2: set_real] :
( ( member_set_real @ I @ ( set_or3605051941450457490t_real @ L @ U2 ) )
= ( ( ord_less_set_real @ L @ I )
& ( ord_less_eq_set_real @ I @ U2 ) ) ) ).
% greaterThanAtMost_iff
thf(fact_971_greaterThanAtMost__iff,axiom,
! [I: int,L: int,U2: int] :
( ( member_int @ I @ ( set_or6656581121297822940st_int @ L @ U2 ) )
= ( ( ord_less_int @ L @ I )
& ( ord_less_eq_int @ I @ U2 ) ) ) ).
% greaterThanAtMost_iff
thf(fact_972_image__add__greaterThanAtMost,axiom,
! [C3: real,A2: real,B: real] :
( ( image_real_real @ ( plus_plus_real @ C3 ) @ ( set_or2392270231875598684t_real @ A2 @ B ) )
= ( set_or2392270231875598684t_real @ ( plus_plus_real @ C3 @ A2 ) @ ( plus_plus_real @ C3 @ B ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_973_image__add__greaterThanAtMost,axiom,
! [C3: nat,A2: nat,B: nat] :
( ( image_nat_nat @ ( plus_plus_nat @ C3 ) @ ( set_or6659071591806873216st_nat @ A2 @ B ) )
= ( set_or6659071591806873216st_nat @ ( plus_plus_nat @ C3 @ A2 ) @ ( plus_plus_nat @ C3 @ B ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_974_image__add__greaterThanAtMost,axiom,
! [C3: int,A2: int,B: int] :
( ( image_int_int @ ( plus_plus_int @ C3 ) @ ( set_or6656581121297822940st_int @ A2 @ B ) )
= ( set_or6656581121297822940st_int @ ( plus_plus_int @ C3 @ A2 ) @ ( plus_plus_int @ C3 @ B ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_975_nat__neq__iff,axiom,
! [M3: nat,N5: nat] :
( ( M3 != N5 )
= ( ( ord_less_nat @ M3 @ N5 )
| ( ord_less_nat @ N5 @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_976_less__not__refl,axiom,
! [N5: nat] :
~ ( ord_less_nat @ N5 @ N5 ) ).
% less_not_refl
thf(fact_977_less__not__refl2,axiom,
! [N5: nat,M3: nat] :
( ( ord_less_nat @ N5 @ M3 )
=> ( M3 != N5 ) ) ).
% less_not_refl2
thf(fact_978_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_979_less__irrefl__nat,axiom,
! [N5: nat] :
~ ( ord_less_nat @ N5 @ N5 ) ).
% less_irrefl_nat
thf(fact_980_nat__less__induct,axiom,
! [P: nat > $o,N5: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N5 ) ) ).
% nat_less_induct
thf(fact_981_infinite__descent,axiom,
! [P: nat > $o,N5: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) )
=> ( P @ N5 ) ) ).
% infinite_descent
thf(fact_982_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_983_less__imp__of__nat__less,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_nat @ M3 @ N5 )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N5 ) ) ) ).
% less_imp_of_nat_less
thf(fact_984_less__imp__of__nat__less,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_nat @ M3 @ N5 )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N5 ) ) ) ).
% less_imp_of_nat_less
thf(fact_985_less__imp__of__nat__less,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_nat @ M3 @ N5 )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N5 ) ) ) ).
% less_imp_of_nat_less
thf(fact_986_of__nat__less__imp__less,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N5 ) )
=> ( ord_less_nat @ M3 @ N5 ) ) ).
% of_nat_less_imp_less
thf(fact_987_of__nat__less__imp__less,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N5 ) )
=> ( ord_less_nat @ M3 @ N5 ) ) ).
% of_nat_less_imp_less
thf(fact_988_of__nat__less__imp__less,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N5 ) )
=> ( ord_less_nat @ M3 @ N5 ) ) ).
% of_nat_less_imp_less
thf(fact_989_Ioc__inj,axiom,
! [A2: real,B: real,C3: real,D3: real] :
( ( ( set_or2392270231875598684t_real @ A2 @ B )
= ( set_or2392270231875598684t_real @ C3 @ D3 ) )
= ( ( ( ord_less_eq_real @ B @ A2 )
& ( ord_less_eq_real @ D3 @ C3 ) )
| ( ( A2 = C3 )
& ( B = D3 ) ) ) ) ).
% Ioc_inj
thf(fact_990_Ioc__inj,axiom,
! [A2: nat,B: nat,C3: nat,D3: nat] :
( ( ( set_or6659071591806873216st_nat @ A2 @ B )
= ( set_or6659071591806873216st_nat @ C3 @ D3 ) )
= ( ( ( ord_less_eq_nat @ B @ A2 )
& ( ord_less_eq_nat @ D3 @ C3 ) )
| ( ( A2 = C3 )
& ( B = D3 ) ) ) ) ).
% Ioc_inj
thf(fact_991_Ioc__inj,axiom,
! [A2: int,B: int,C3: int,D3: int] :
( ( ( set_or6656581121297822940st_int @ A2 @ B )
= ( set_or6656581121297822940st_int @ C3 @ D3 ) )
= ( ( ( ord_less_eq_int @ B @ A2 )
& ( ord_less_eq_int @ D3 @ C3 ) )
| ( ( A2 = C3 )
& ( B = D3 ) ) ) ) ).
% Ioc_inj
thf(fact_992_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A2: real,B: real,C3: real,D3: real] :
( ( ord_less_eq_set_real @ ( set_or1633881224788618240n_real @ A2 @ B ) @ ( set_or2392270231875598684t_real @ C3 @ D3 ) )
= ( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ C3 @ A2 )
& ( ord_less_eq_real @ B @ D3 ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_993_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_994_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_995_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_996_Ioc__subset__iff,axiom,
! [A2: nat,B: nat,C3: nat,D3: nat] :
( ( ord_less_eq_set_nat @ ( set_or6659071591806873216st_nat @ A2 @ B ) @ ( set_or6659071591806873216st_nat @ C3 @ D3 ) )
= ( ( ord_less_eq_nat @ B @ A2 )
| ( ( ord_less_eq_nat @ C3 @ A2 )
& ( ord_less_eq_nat @ B @ D3 ) ) ) ) ).
% Ioc_subset_iff
thf(fact_997_Ioc__subset__iff,axiom,
! [A2: real,B: real,C3: real,D3: real] :
( ( ord_less_eq_set_real @ ( set_or2392270231875598684t_real @ A2 @ B ) @ ( set_or2392270231875598684t_real @ C3 @ D3 ) )
= ( ( ord_less_eq_real @ B @ A2 )
| ( ( ord_less_eq_real @ C3 @ A2 )
& ( ord_less_eq_real @ B @ D3 ) ) ) ) ).
% Ioc_subset_iff
thf(fact_998_Ioc__subset__iff,axiom,
! [A2: int,B: int,C3: int,D3: int] :
( ( ord_less_eq_set_int @ ( set_or6656581121297822940st_int @ A2 @ B ) @ ( set_or6656581121297822940st_int @ C3 @ D3 ) )
= ( ( ord_less_eq_int @ B @ A2 )
| ( ( ord_less_eq_int @ C3 @ A2 )
& ( ord_less_eq_int @ B @ D3 ) ) ) ) ).
% Ioc_subset_iff
thf(fact_999_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A2: real,B: real,C3: real,D3: real] :
( ( ord_less_eq_set_real @ ( set_or1633881224788618240n_real @ A2 @ B ) @ ( set_or1633881224788618240n_real @ C3 @ D3 ) )
= ( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ C3 @ A2 )
& ( ord_less_eq_real @ B @ D3 ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_1000_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N2: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% nat_less_real_le
thf(fact_1001_real__of__nat__ge__one__iff,axiom,
! [N5: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N5 ) )
= ( ord_less_eq_nat @ one_one_nat @ N5 ) ) ).
% real_of_nat_ge_one_iff
thf(fact_1002_one__less__of__natD,axiom,
! [N5: nat] :
( ( ord_less_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ N5 ) )
=> ( ord_less_nat @ one_one_nat @ N5 ) ) ).
% one_less_of_natD
thf(fact_1003_one__less__of__natD,axiom,
! [N5: nat] :
( ( ord_less_real @ one_one_real @ ( semiri5074537144036343181t_real @ N5 ) )
=> ( ord_less_nat @ one_one_nat @ N5 ) ) ).
% one_less_of_natD
thf(fact_1004_one__less__of__natD,axiom,
! [N5: nat] :
( ( ord_less_int @ one_one_int @ ( semiri1314217659103216013at_int @ N5 ) )
=> ( ord_less_nat @ one_one_nat @ N5 ) ) ).
% one_less_of_natD
thf(fact_1005_continuous__ge__on__Ioo,axiom,
! [C3: real,D3: real,G: real > real,A2: real,X: real] :
( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ C3 @ D3 ) @ G )
=> ( ! [X2: real] :
( ( member_real @ X2 @ ( set_or1633881224788618240n_real @ C3 @ D3 ) )
=> ( ord_less_eq_real @ A2 @ ( G @ X2 ) ) )
=> ( ( ord_less_real @ C3 @ D3 )
=> ( ( member_real @ X @ ( set_or1222579329274155063t_real @ C3 @ D3 ) )
=> ( ord_less_eq_real @ A2 @ ( G @ X ) ) ) ) ) ) ).
% continuous_ge_on_Ioo
thf(fact_1006_nat__descend__induct,axiom,
! [N5: nat,P: nat > $o,M3: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N5 @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N5 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K3 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K3 ) ) )
=> ( P @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_1007_reals__Archimedean2,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% reals_Archimedean2
thf(fact_1008_real__arch__simple,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_1009_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1010_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1011_int__plus,axiom,
! [N5: nat,M3: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N5 @ M3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N5 ) @ ( semiri1314217659103216013at_int @ M3 ) ) ) ).
% int_plus
thf(fact_1012_int__ops_I5_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1013_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1014_greaterThanLessThan__eq__iff,axiom,
! [R: real,S: real,T: real,U2: real] :
( ( ( set_or1633881224788618240n_real @ R @ S )
= ( set_or1633881224788618240n_real @ T @ U2 ) )
= ( ( ( ord_less_eq_real @ S @ R )
& ( ord_less_eq_real @ U2 @ T ) )
| ( ( R = T )
& ( S = U2 ) ) ) ) ).
% greaterThanLessThan_eq_iff
thf(fact_1015_ivl__disj__un__two_I5_J,axiom,
! [L: nat,M3: nat,U2: nat] :
( ( ord_less_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U2 )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ M3 ) @ ( set_or1269000886237332187st_nat @ M3 @ U2 ) )
= ( set_or6659071591806873216st_nat @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_1016_ivl__disj__un__two_I5_J,axiom,
! [L: real,M3: real,U2: real] :
( ( ord_less_real @ L @ M3 )
=> ( ( ord_less_eq_real @ M3 @ U2 )
=> ( ( sup_sup_set_real @ ( set_or1633881224788618240n_real @ L @ M3 ) @ ( set_or1222579329274155063t_real @ M3 @ U2 ) )
= ( set_or2392270231875598684t_real @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_1017_ivl__disj__un__two_I5_J,axiom,
! [L: int,M3: int,U2: int] :
( ( ord_less_int @ L @ M3 )
=> ( ( ord_less_eq_int @ M3 @ U2 )
=> ( ( sup_sup_set_int @ ( set_or5832277885323065728an_int @ L @ M3 ) @ ( set_or1266510415728281911st_int @ M3 @ U2 ) )
= ( set_or6656581121297822940st_int @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_1018_ivl__disj__un__two_I2_J,axiom,
! [L: nat,M3: nat,U2: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_nat @ M3 @ U2 )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M3 ) @ ( set_or5834768355832116004an_nat @ M3 @ U2 ) )
= ( set_or5834768355832116004an_nat @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_1019_ivl__disj__un__two_I2_J,axiom,
! [L: real,M3: real,U2: real] :
( ( ord_less_eq_real @ L @ M3 )
=> ( ( ord_less_real @ M3 @ U2 )
=> ( ( sup_sup_set_real @ ( set_or2392270231875598684t_real @ L @ M3 ) @ ( set_or1633881224788618240n_real @ M3 @ U2 ) )
= ( set_or1633881224788618240n_real @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_1020_ivl__disj__un__two_I2_J,axiom,
! [L: int,M3: int,U2: int] :
( ( ord_less_eq_int @ L @ M3 )
=> ( ( ord_less_int @ M3 @ U2 )
=> ( ( sup_sup_set_int @ ( set_or6656581121297822940st_int @ L @ M3 ) @ ( set_or5832277885323065728an_int @ M3 @ U2 ) )
= ( set_or5832277885323065728an_int @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_1021_UnCI,axiom,
! [C3: real,B2: set_real,A: set_real] :
( ( ~ ( member_real @ C3 @ B2 )
=> ( member_real @ C3 @ A ) )
=> ( member_real @ C3 @ ( sup_sup_set_real @ A @ B2 ) ) ) ).
% UnCI
thf(fact_1022_UnCI,axiom,
! [C3: int,B2: set_int,A: set_int] :
( ( ~ ( member_int @ C3 @ B2 )
=> ( member_int @ C3 @ A ) )
=> ( member_int @ C3 @ ( sup_sup_set_int @ A @ B2 ) ) ) ).
% UnCI
thf(fact_1023_UnCI,axiom,
! [C3: nat,B2: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C3 @ B2 )
=> ( member_nat @ C3 @ A ) )
=> ( member_nat @ C3 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% UnCI
thf(fact_1024_Un__iff,axiom,
! [C3: real,A: set_real,B2: set_real] :
( ( member_real @ C3 @ ( sup_sup_set_real @ A @ B2 ) )
= ( ( member_real @ C3 @ A )
| ( member_real @ C3 @ B2 ) ) ) ).
% Un_iff
thf(fact_1025_Un__iff,axiom,
! [C3: int,A: set_int,B2: set_int] :
( ( member_int @ C3 @ ( sup_sup_set_int @ A @ B2 ) )
= ( ( member_int @ C3 @ A )
| ( member_int @ C3 @ B2 ) ) ) ).
% Un_iff
thf(fact_1026_Un__iff,axiom,
! [C3: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C3 @ ( sup_sup_set_nat @ A @ B2 ) )
= ( ( member_nat @ C3 @ A )
| ( member_nat @ C3 @ B2 ) ) ) ).
% Un_iff
thf(fact_1027_Un__subset__iff,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C2 )
= ( ( ord_less_eq_set_nat @ A @ C2 )
& ( ord_less_eq_set_nat @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1028_Un__subset__iff,axiom,
! [A: set_real,B2: set_real,C2: set_real] :
( ( ord_less_eq_set_real @ ( sup_sup_set_real @ A @ B2 ) @ C2 )
= ( ( ord_less_eq_set_real @ A @ C2 )
& ( ord_less_eq_set_real @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1029_nat__int__comparison_I1_J,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1030_int__if,axiom,
! [P: $o,A2: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_1031_image__int__atLeastAtMost,axiom,
! [A2: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A2 @ B ) )
= ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastAtMost
thf(fact_1032_verit__la__generic,axiom,
! [A2: int,X: int] :
( ( ord_less_eq_int @ A2 @ X )
| ( A2 = X )
| ( ord_less_eq_int @ X @ A2 ) ) ).
% verit_la_generic
thf(fact_1033_UnE,axiom,
! [C3: real,A: set_real,B2: set_real] :
( ( member_real @ C3 @ ( sup_sup_set_real @ A @ B2 ) )
=> ( ~ ( member_real @ C3 @ A )
=> ( member_real @ C3 @ B2 ) ) ) ).
% UnE
thf(fact_1034_UnE,axiom,
! [C3: int,A: set_int,B2: set_int] :
( ( member_int @ C3 @ ( sup_sup_set_int @ A @ B2 ) )
=> ( ~ ( member_int @ C3 @ A )
=> ( member_int @ C3 @ B2 ) ) ) ).
% UnE
thf(fact_1035_UnE,axiom,
! [C3: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C3 @ ( sup_sup_set_nat @ A @ B2 ) )
=> ( ~ ( member_nat @ C3 @ A )
=> ( member_nat @ C3 @ B2 ) ) ) ).
% UnE
thf(fact_1036_UnI1,axiom,
! [C3: real,A: set_real,B2: set_real] :
( ( member_real @ C3 @ A )
=> ( member_real @ C3 @ ( sup_sup_set_real @ A @ B2 ) ) ) ).
% UnI1
thf(fact_1037_UnI1,axiom,
! [C3: int,A: set_int,B2: set_int] :
( ( member_int @ C3 @ A )
=> ( member_int @ C3 @ ( sup_sup_set_int @ A @ B2 ) ) ) ).
% UnI1
thf(fact_1038_UnI1,axiom,
! [C3: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C3 @ A )
=> ( member_nat @ C3 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% UnI1
thf(fact_1039_UnI2,axiom,
! [C3: real,B2: set_real,A: set_real] :
( ( member_real @ C3 @ B2 )
=> ( member_real @ C3 @ ( sup_sup_set_real @ A @ B2 ) ) ) ).
% UnI2
thf(fact_1040_UnI2,axiom,
! [C3: int,B2: set_int,A: set_int] :
( ( member_int @ C3 @ B2 )
=> ( member_int @ C3 @ ( sup_sup_set_int @ A @ B2 ) ) ) ).
% UnI2
thf(fact_1041_UnI2,axiom,
! [C3: nat,B2: set_nat,A: set_nat] :
( ( member_nat @ C3 @ B2 )
=> ( member_nat @ C3 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% UnI2
thf(fact_1042_bex__Un,axiom,
! [A: set_nat,B2: set_nat,P: nat > $o] :
( ( ? [X4: nat] :
( ( member_nat @ X4 @ ( sup_sup_set_nat @ A @ B2 ) )
& ( P @ X4 ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) )
| ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_1043_ball__Un,axiom,
! [A: set_nat,B2: set_nat,P: nat > $o] :
( ( ! [X4: nat] :
( ( member_nat @ X4 @ ( sup_sup_set_nat @ A @ B2 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ X4 ) )
& ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_1044_Un__assoc,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C2 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_1045_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_1046_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_1047_Un__left__absorb,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ A @ B2 ) ) ).
% Un_left_absorb
thf(fact_1048_Un__left__commute,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_1049_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U2: int] :
( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U2 )
= ( set_or6656581121297822940st_int @ L @ U2 ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_1050_image__Un,axiom,
! [F: real > real,A: set_real,B2: set_real] :
( ( image_real_real @ F @ ( sup_sup_set_real @ A @ B2 ) )
= ( sup_sup_set_real @ ( image_real_real @ F @ A ) @ ( image_real_real @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1051_image__Un,axiom,
! [F: nat > int,A: set_nat,B2: set_nat] :
( ( image_nat_int @ F @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_set_int @ ( image_nat_int @ F @ A ) @ ( image_nat_int @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1052_image__Un,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1053_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_1054_subset__Un__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
( ( sup_sup_set_real @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_1055_subset__UnE,axiom,
! [C2: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B2 ) )
=> ~ ! [A7: set_nat] :
( ( ord_less_eq_set_nat @ A7 @ A )
=> ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ B2 )
=> ( C2
!= ( sup_sup_set_nat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_1056_subset__UnE,axiom,
! [C2: set_real,A: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ C2 @ ( sup_sup_set_real @ A @ B2 ) )
=> ~ ! [A7: set_real] :
( ( ord_less_eq_set_real @ A7 @ A )
=> ! [B7: set_real] :
( ( ord_less_eq_set_real @ B7 @ B2 )
=> ( C2
!= ( sup_sup_set_real @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_1057_Un__absorb2,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( sup_sup_set_nat @ A @ B2 )
= A ) ) ).
% Un_absorb2
thf(fact_1058_Un__absorb2,axiom,
! [B2: set_real,A: set_real] :
( ( ord_less_eq_set_real @ B2 @ A )
=> ( ( sup_sup_set_real @ A @ B2 )
= A ) ) ).
% Un_absorb2
thf(fact_1059_Un__absorb1,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( sup_sup_set_nat @ A @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_1060_Un__absorb1,axiom,
! [A: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A @ B2 )
=> ( ( sup_sup_set_real @ A @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_1061_Un__upper2,axiom,
! [B2: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% Un_upper2
thf(fact_1062_Un__upper2,axiom,
! [B2: set_real,A: set_real] : ( ord_less_eq_set_real @ B2 @ ( sup_sup_set_real @ A @ B2 ) ) ).
% Un_upper2
thf(fact_1063_Un__upper1,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% Un_upper1
thf(fact_1064_Un__upper1,axiom,
! [A: set_real,B2: set_real] : ( ord_less_eq_set_real @ A @ ( sup_sup_set_real @ A @ B2 ) ) ).
% Un_upper1
thf(fact_1065_Un__least,axiom,
! [A: set_nat,C2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_1066_Un__least,axiom,
! [A: set_real,C2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B2 @ C2 )
=> ( ord_less_eq_set_real @ ( sup_sup_set_real @ A @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_1067_Un__mono,axiom,
! [A: set_nat,C2: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1068_Un__mono,axiom,
! [A: set_real,C2: set_real,B2: set_real,D2: set_real] :
( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B2 @ D2 )
=> ( ord_less_eq_set_real @ ( sup_sup_set_real @ A @ B2 ) @ ( sup_sup_set_real @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1069_set__plus__Un,axiom,
! [C2: set_nat,A: set_nat,B2: set_nat] :
( ( plus_plus_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ ( plus_plus_set_nat @ C2 @ A ) @ ( plus_plus_set_nat @ C2 @ B2 ) ) ) ).
% set_plus_Un
thf(fact_1070_Un__set__plus,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( plus_plus_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C2 )
= ( sup_sup_set_nat @ ( plus_plus_set_nat @ A @ C2 ) @ ( plus_plus_set_nat @ B2 @ C2 ) ) ) ).
% Un_set_plus
thf(fact_1071_ivl__disj__un__two__touch_I4_J,axiom,
! [L: real,M3: real,U2: real] :
( ( ord_less_eq_real @ L @ M3 )
=> ( ( ord_less_eq_real @ M3 @ U2 )
=> ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M3 ) @ ( set_or1222579329274155063t_real @ M3 @ U2 ) )
= ( set_or1222579329274155063t_real @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_1072_ivl__disj__un__two__touch_I4_J,axiom,
! [L: nat,M3: nat,U2: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U2 )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M3 ) @ ( set_or1269000886237332187st_nat @ M3 @ U2 ) )
= ( set_or1269000886237332187st_nat @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_1073_ivl__disj__un__two__touch_I4_J,axiom,
! [L: int,M3: int,U2: int] :
( ( ord_less_eq_int @ L @ M3 )
=> ( ( ord_less_eq_int @ M3 @ U2 )
=> ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M3 ) @ ( set_or1266510415728281911st_int @ M3 @ U2 ) )
= ( set_or1266510415728281911st_int @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_1074_ivl__disj__un__two_I6_J,axiom,
! [L: real,M3: real,U2: real] :
( ( ord_less_eq_real @ L @ M3 )
=> ( ( ord_less_eq_real @ M3 @ U2 )
=> ( ( sup_sup_set_real @ ( set_or2392270231875598684t_real @ L @ M3 ) @ ( set_or2392270231875598684t_real @ M3 @ U2 ) )
= ( set_or2392270231875598684t_real @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_1075_ivl__disj__un__two_I6_J,axiom,
! [L: nat,M3: nat,U2: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U2 )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M3 ) @ ( set_or6659071591806873216st_nat @ M3 @ U2 ) )
= ( set_or6659071591806873216st_nat @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_1076_ivl__disj__un__two_I6_J,axiom,
! [L: int,M3: int,U2: int] :
( ( ord_less_eq_int @ L @ M3 )
=> ( ( ord_less_eq_int @ M3 @ U2 )
=> ( ( sup_sup_set_int @ ( set_or6656581121297822940st_int @ L @ M3 ) @ ( set_or6656581121297822940st_int @ M3 @ U2 ) )
= ( set_or6656581121297822940st_int @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_1077_ivl__disj__un__two_I8_J,axiom,
! [L: real,M3: real,U2: real] :
( ( ord_less_eq_real @ L @ M3 )
=> ( ( ord_less_eq_real @ M3 @ U2 )
=> ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M3 ) @ ( set_or2392270231875598684t_real @ M3 @ U2 ) )
= ( set_or1222579329274155063t_real @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_1078_ivl__disj__un__two_I8_J,axiom,
! [L: nat,M3: nat,U2: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U2 )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M3 ) @ ( set_or6659071591806873216st_nat @ M3 @ U2 ) )
= ( set_or1269000886237332187st_nat @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_1079_ivl__disj__un__two_I8_J,axiom,
! [L: int,M3: int,U2: int] :
( ( ord_less_eq_int @ L @ M3 )
=> ( ( ord_less_eq_int @ M3 @ U2 )
=> ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M3 ) @ ( set_or6656581121297822940st_int @ M3 @ U2 ) )
= ( set_or1266510415728281911st_int @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_1080_ivl__disj__un__two__touch_I3_J,axiom,
! [L: real,M3: real,U2: real] :
( ( ord_less_real @ L @ M3 )
=> ( ( ord_less_eq_real @ M3 @ U2 )
=> ( ( sup_sup_set_real @ ( set_or2392270231875598684t_real @ L @ M3 ) @ ( set_or1222579329274155063t_real @ M3 @ U2 ) )
= ( set_or2392270231875598684t_real @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_1081_ivl__disj__un__two__touch_I3_J,axiom,
! [L: nat,M3: nat,U2: nat] :
( ( ord_less_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U2 )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M3 ) @ ( set_or1269000886237332187st_nat @ M3 @ U2 ) )
= ( set_or6659071591806873216st_nat @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_1082_ivl__disj__un__two__touch_I3_J,axiom,
! [L: int,M3: int,U2: int] :
( ( ord_less_int @ L @ M3 )
=> ( ( ord_less_eq_int @ M3 @ U2 )
=> ( ( sup_sup_set_int @ ( set_or6656581121297822940st_int @ L @ M3 ) @ ( set_or1266510415728281911st_int @ M3 @ U2 ) )
= ( set_or6656581121297822940st_int @ L @ U2 ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_1083_zle__add1__eq__le,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1084_le__sup__iff,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X @ Z2 )
& ( ord_less_eq_set_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_1085_le__sup__iff,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_real @ X @ Z2 )
& ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_1086_le__sup__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_1087_le__sup__iff,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_int @ X @ Z2 )
& ( ord_less_eq_int @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_1088_le__sup__iff,axiom,
! [X: set_real,Y: set_real,Z2: set_real] :
( ( ord_less_eq_set_real @ ( sup_sup_set_real @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_set_real @ X @ Z2 )
& ( ord_less_eq_set_real @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_1089_sup_Obounded__iff,axiom,
! [B: set_nat,C3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C3 ) @ A2 )
= ( ( ord_less_eq_set_nat @ B @ A2 )
& ( ord_less_eq_set_nat @ C3 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1090_sup_Obounded__iff,axiom,
! [B: real,C3: real,A2: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B @ C3 ) @ A2 )
= ( ( ord_less_eq_real @ B @ A2 )
& ( ord_less_eq_real @ C3 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1091_sup_Obounded__iff,axiom,
! [B: nat,C3: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C3 ) @ A2 )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( ord_less_eq_nat @ C3 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1092_sup_Obounded__iff,axiom,
! [B: int,C3: int,A2: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B @ C3 ) @ A2 )
= ( ( ord_less_eq_int @ B @ A2 )
& ( ord_less_eq_int @ C3 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1093_sup_Obounded__iff,axiom,
! [B: set_real,C3: set_real,A2: set_real] :
( ( ord_less_eq_set_real @ ( sup_sup_set_real @ B @ C3 ) @ A2 )
= ( ( ord_less_eq_set_real @ B @ A2 )
& ( ord_less_eq_set_real @ C3 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1094_zless__imp__add1__zle,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1095_add1__zle__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1096_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1097_inf__sup__ord_I4_J,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1098_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1099_inf__sup__ord_I4_J,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1100_inf__sup__ord_I4_J,axiom,
! [Y: set_real,X: set_real] : ( ord_less_eq_set_real @ Y @ ( sup_sup_set_real @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1101_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1102_inf__sup__ord_I3_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sup_sup_real @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1103_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1104_inf__sup__ord_I3_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1105_inf__sup__ord_I3_J,axiom,
! [X: set_real,Y: set_real] : ( ord_less_eq_set_real @ X @ ( sup_sup_set_real @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1106_le__supE,axiom,
! [A2: set_nat,B: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X )
=> ~ ( ord_less_eq_set_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_1107_le__supE,axiom,
! [A2: real,B: real,X: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_real @ A2 @ X )
=> ~ ( ord_less_eq_real @ B @ X ) ) ) ).
% le_supE
thf(fact_1108_le__supE,axiom,
! [A2: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_1109_le__supE,axiom,
! [A2: int,B: int,X: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_int @ A2 @ X )
=> ~ ( ord_less_eq_int @ B @ X ) ) ) ).
% le_supE
thf(fact_1110_le__supE,axiom,
! [A2: set_real,B: set_real,X: set_real] :
( ( ord_less_eq_set_real @ ( sup_sup_set_real @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_set_real @ A2 @ X )
=> ~ ( ord_less_eq_set_real @ B @ X ) ) ) ).
% le_supE
thf(fact_1111_le__supI,axiom,
! [A2: set_nat,X: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X )
=> ( ( ord_less_eq_set_nat @ B @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1112_le__supI,axiom,
! [A2: real,X: real,B: real] :
( ( ord_less_eq_real @ A2 @ X )
=> ( ( ord_less_eq_real @ B @ X )
=> ( ord_less_eq_real @ ( sup_sup_real @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1113_le__supI,axiom,
! [A2: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1114_le__supI,axiom,
! [A2: int,X: int,B: int] :
( ( ord_less_eq_int @ A2 @ X )
=> ( ( ord_less_eq_int @ B @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1115_le__supI,axiom,
! [A2: set_real,X: set_real,B: set_real] :
( ( ord_less_eq_set_real @ A2 @ X )
=> ( ( ord_less_eq_set_real @ B @ X )
=> ( ord_less_eq_set_real @ ( sup_sup_set_real @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1116_sup__ge1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_1117_sup__ge1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sup_sup_real @ X @ Y ) ) ).
% sup_ge1
thf(fact_1118_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_1119_sup__ge1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge1
thf(fact_1120_sup__ge1,axiom,
! [X: set_real,Y: set_real] : ( ord_less_eq_set_real @ X @ ( sup_sup_set_real @ X @ Y ) ) ).
% sup_ge1
thf(fact_1121_sup__ge2,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_1122_sup__ge2,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X @ Y ) ) ).
% sup_ge2
thf(fact_1123_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_1124_sup__ge2,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge2
thf(fact_1125_sup__ge2,axiom,
! [Y: set_real,X: set_real] : ( ord_less_eq_set_real @ Y @ ( sup_sup_set_real @ X @ Y ) ) ).
% sup_ge2
thf(fact_1126_le__supI1,axiom,
! [X: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_1127_le__supI1,axiom,
! [X: real,A2: real,B: real] :
( ( ord_less_eq_real @ X @ A2 )
=> ( ord_less_eq_real @ X @ ( sup_sup_real @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_1128_le__supI1,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_1129_le__supI1,axiom,
! [X: int,A2: int,B: int] :
( ( ord_less_eq_int @ X @ A2 )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_1130_le__supI1,axiom,
! [X: set_real,A2: set_real,B: set_real] :
( ( ord_less_eq_set_real @ X @ A2 )
=> ( ord_less_eq_set_real @ X @ ( sup_sup_set_real @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_1131_le__supI2,axiom,
! [X: real,B: real,A2: real] :
( ( ord_less_eq_real @ X @ B )
=> ( ord_less_eq_real @ X @ ( sup_sup_real @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_1132_le__supI2,axiom,
! [X: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_1133_le__supI2,axiom,
! [X: int,B: int,A2: int] :
( ( ord_less_eq_int @ X @ B )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_1134_le__supI2,axiom,
! [X: set_real,B: set_real,A2: set_real] :
( ( ord_less_eq_set_real @ X @ B )
=> ( ord_less_eq_set_real @ X @ ( sup_sup_set_real @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_1135_zle__int,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N5 ) )
= ( ord_less_eq_nat @ M3 @ N5 ) ) ).
% zle_int
thf(fact_1136_zadd__int__left,axiom,
! [M3: nat,N5: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N5 ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M3 @ N5 ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_1137_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1138_zless__add1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z2 )
| ( W2 = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1139_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1140_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W3: int,Z5: int] :
? [N2: nat] :
( Z5
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1141_image__int__atLeastLessThan,axiom,
! [A2: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A2 @ B ) )
= ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastLessThan
thf(fact_1142_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U2: int] :
( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U2 )
= ( set_or5832277885323065728an_int @ L @ U2 ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_1143_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_1144_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U2: int] :
( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U2 @ one_one_int ) )
= ( set_or1266510415728281911st_int @ L @ U2 ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_1145_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1146_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P4: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1147_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P4: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1148_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1149_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1150_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1151_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N5: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ord_less_eq_nat @ N5 @ ( F @ N5 ) ) ) ).
% strict_mono_imp_increasing
thf(fact_1152_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1153_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1154_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1155_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N2: int,M2: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M2 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_1156_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N2: int,M2: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M2 ) ) ) ) ).
% int_less_real_le
thf(fact_1157_le0,axiom,
! [N5: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N5 ) ).
% le0
thf(fact_1158_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1159_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1160_neq0__conv,axiom,
! [N5: nat] :
( ( N5 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N5 ) ) ).
% neq0_conv
thf(fact_1161_less__nat__zero__code,axiom,
! [N5: nat] :
~ ( ord_less_nat @ N5 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1162_atLeast__0,axiom,
( ( set_ord_atLeast_nat @ zero_zero_nat )
= top_top_set_nat ) ).
% atLeast_0
thf(fact_1163_add__is__0,axiom,
! [M3: nat,N5: nat] :
( ( ( plus_plus_nat @ M3 @ N5 )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N5 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1164_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_1165_add__gr__0,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N5 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N5 ) ) ) ).
% add_gr_0
thf(fact_1166_less__one,axiom,
! [N5: nat] :
( ( ord_less_nat @ N5 @ one_one_nat )
= ( N5 = zero_zero_nat ) ) ).
% less_one
thf(fact_1167_le__0__eq,axiom,
! [N5: nat] :
( ( ord_less_eq_nat @ N5 @ zero_zero_nat )
= ( N5 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1168_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1169_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1170_less__eq__nat_Osimps_I1_J,axiom,
! [N5: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N5 ) ).
% less_eq_nat.simps(1)
thf(fact_1171_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1172_gr0I,axiom,
! [N5: nat] :
( ( N5 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N5 ) ) ).
% gr0I
thf(fact_1173_not__gr0,axiom,
! [N5: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N5 ) )
= ( N5 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1174_not__less0,axiom,
! [N5: nat] :
~ ( ord_less_nat @ N5 @ zero_zero_nat ) ).
% not_less0
thf(fact_1175_less__zeroE,axiom,
! [N5: nat] :
~ ( ord_less_nat @ N5 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1176_gr__implies__not0,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_nat @ M3 @ N5 )
=> ( N5 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1177_infinite__descent0,axiom,
! [P: nat > $o,N5: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N5 ) ) ) ).
% infinite_descent0
thf(fact_1178_plus__nat_Oadd__0,axiom,
! [N5: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N5 )
= N5 ) ).
% plus_nat.add_0
thf(fact_1179_add__eq__self__zero,axiom,
! [M3: nat,N5: nat] :
( ( ( plus_plus_nat @ M3 @ N5 )
= M3 )
=> ( N5 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1180_ex__least__nat__le,axiom,
! [P: nat > $o,N5: nat] :
( ( P @ N5 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N5 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1181_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1182_ex__nat__less,axiom,
! [N5: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N5 )
& ( P @ M2 ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N5 ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_1183_all__nat__less,axiom,
! [N5: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N5 )
=> ( P @ M2 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N5 ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_1184_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1185_all__nat__less__eq,axiom,
! [N5: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N5 )
=> ( P @ M2 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N5 ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less_eq
thf(fact_1186_ex__nat__less__eq,axiom,
! [N5: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_nat @ M2 @ N5 )
& ( P @ M2 ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N5 ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1187_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1188_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1189_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X2: nat > real] :
( ( P @ X2 )
=> ( P @ ( F @ X2 ) ) )
=> ( ! [X2: nat > real] :
( ( P @ X2 )
=> ! [I2: nat] :
( ( Q @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I2 ) )
& ( ord_less_eq_real @ ( X2 @ I2 ) @ one_one_real ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X3: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L3 @ X3 @ I3 ) @ one_one_nat )
& ! [X3: nat > real,I3: nat] :
( ( ( P @ X3 )
& ( Q @ I3 )
& ( ( X3 @ I3 )
= zero_zero_real ) )
=> ( ( L3 @ X3 @ I3 )
= zero_zero_nat ) )
& ! [X3: nat > real,I3: nat] :
( ( ( P @ X3 )
& ( Q @ I3 )
& ( ( X3 @ I3 )
= one_one_real ) )
=> ( ( L3 @ X3 @ I3 )
= one_one_nat ) )
& ! [X3: nat > real,I3: nat] :
( ( ( P @ X3 )
& ( Q @ I3 )
& ( ( L3 @ X3 @ I3 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X3 @ I3 ) @ ( F @ X3 @ I3 ) ) )
& ! [X3: nat > real,I3: nat] :
( ( ( P @ X3 )
& ( Q @ I3 )
& ( ( L3 @ X3 @ I3 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X3 @ I3 ) @ ( X3 @ I3 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1190_kuhn__lemma,axiom,
! [P5: nat,N5: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P5 )
=> ( ! [X2: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N5 )
=> ( ord_less_eq_nat @ ( X2 @ I3 ) @ P5 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N5 )
=> ( ( ( Label @ X2 @ I2 )
= zero_zero_nat )
| ( ( Label @ X2 @ I2 )
= one_one_nat ) ) ) )
=> ( ! [X2: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N5 )
=> ( ord_less_eq_nat @ ( X2 @ I3 ) @ P5 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N5 )
=> ( ( ( X2 @ I2 )
= zero_zero_nat )
=> ( ( Label @ X2 @ I2 )
= zero_zero_nat ) ) ) )
=> ( ! [X2: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N5 )
=> ( ord_less_eq_nat @ ( X2 @ I3 ) @ P5 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N5 )
=> ( ( ( X2 @ I2 )
= P5 )
=> ( ( Label @ X2 @ I2 )
= one_one_nat ) ) ) )
=> ~ ! [Q3: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N5 )
=> ( ord_less_nat @ ( Q3 @ I3 ) @ P5 ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ N5 )
=> ? [R2: nat > nat] :
( ! [J3: nat] :
( ( ord_less_nat @ J3 @ N5 )
=> ( ( ord_less_eq_nat @ ( Q3 @ J3 ) @ ( R2 @ J3 ) )
& ( ord_less_eq_nat @ ( R2 @ J3 ) @ ( plus_plus_nat @ ( Q3 @ J3 ) @ one_one_nat ) ) ) )
& ? [S3: nat > nat] :
( ! [J3: nat] :
( ( ord_less_nat @ J3 @ N5 )
=> ( ( ord_less_eq_nat @ ( Q3 @ J3 ) @ ( S3 @ J3 ) )
& ( ord_less_eq_nat @ ( S3 @ J3 ) @ ( plus_plus_nat @ ( Q3 @ J3 ) @ one_one_nat ) ) ) )
& ( ( Label @ R2 @ I3 )
!= ( Label @ S3 @ I3 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1191_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
= ( P @ B5 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1192_aset_I8_J,axiom,
! [D2: int,A: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ T @ X3 )
=> ( ord_less_eq_int @ T @ ( plus_plus_int @ X3 @ D2 ) ) ) ) ) ).
% aset(8)
thf(fact_1193_zle__diff1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1194_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1195_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1196_bset_I1_J,axiom,
! [D2: int,B2: set_int,P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B2 )
=> ( X2
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D2 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B2 )
=> ( X2
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X2 )
=> ( Q @ ( minus_minus_int @ X2 @ D2 ) ) ) )
=> ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
=> ( ( P @ ( minus_minus_int @ X3 @ D2 ) )
& ( Q @ ( minus_minus_int @ X3 @ D2 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_1197_bset_I2_J,axiom,
! [D2: int,B2: set_int,P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B2 )
=> ( X2
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D2 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B2 )
=> ( X2
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X2 )
=> ( Q @ ( minus_minus_int @ X2 @ D2 ) ) ) )
=> ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
=> ( ( P @ ( minus_minus_int @ X3 @ D2 ) )
| ( Q @ ( minus_minus_int @ X3 @ D2 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_1198_aset_I1_J,axiom,
! [D2: int,A: set_int,P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A )
=> ( X2
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D2 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A )
=> ( X2
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X2 )
=> ( Q @ ( plus_plus_int @ X2 @ D2 ) ) ) )
=> ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
=> ( ( P @ ( plus_plus_int @ X3 @ D2 ) )
& ( Q @ ( plus_plus_int @ X3 @ D2 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_1199_aset_I2_J,axiom,
! [D2: int,A: set_int,P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A )
=> ( X2
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D2 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A )
=> ( X2
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X2 )
=> ( Q @ ( plus_plus_int @ X2 @ D2 ) ) ) )
=> ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
=> ( ( P @ ( plus_plus_int @ X3 @ D2 ) )
| ( Q @ ( plus_plus_int @ X3 @ D2 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_1200_bset_I3_J,axiom,
! [D2: int,T: int,B2: set_int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
=> ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( X3 = T )
=> ( ( minus_minus_int @ X3 @ D2 )
= T ) ) ) ) ) ).
% bset(3)
thf(fact_1201_bset_I4_J,axiom,
! [D2: int,T: int,B2: set_int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ( member_int @ T @ B2 )
=> ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( X3 != T )
=> ( ( minus_minus_int @ X3 @ D2 )
!= T ) ) ) ) ) ).
% bset(4)
thf(fact_1202_bset_I5_J,axiom,
! [D2: int,B2: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ X3 @ T )
=> ( ord_less_int @ ( minus_minus_int @ X3 @ D2 ) @ T ) ) ) ) ).
% bset(5)
thf(fact_1203_bset_I7_J,axiom,
! [D2: int,T: int,B2: set_int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ( member_int @ T @ B2 )
=> ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ T @ X3 )
=> ( ord_less_int @ T @ ( minus_minus_int @ X3 @ D2 ) ) ) ) ) ) ).
% bset(7)
thf(fact_1204_aset_I3_J,axiom,
! [D2: int,T: int,A: set_int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A )
=> ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( X3 = T )
=> ( ( plus_plus_int @ X3 @ D2 )
= T ) ) ) ) ) ).
% aset(3)
thf(fact_1205_aset_I4_J,axiom,
! [D2: int,T: int,A: set_int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ( member_int @ T @ A )
=> ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( X3 != T )
=> ( ( plus_plus_int @ X3 @ D2 )
!= T ) ) ) ) ) ).
% aset(4)
thf(fact_1206_aset_I5_J,axiom,
! [D2: int,T: int,A: set_int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ( member_int @ T @ A )
=> ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ X3 @ T )
=> ( ord_less_int @ ( plus_plus_int @ X3 @ D2 ) @ T ) ) ) ) ) ).
% aset(5)
thf(fact_1207_aset_I7_J,axiom,
! [D2: int,A: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ T @ X3 )
=> ( ord_less_int @ T @ ( plus_plus_int @ X3 @ D2 ) ) ) ) ) ).
% aset(7)
thf(fact_1208_bset_I6_J,axiom,
! [D2: int,B2: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ X3 @ T )
=> ( ord_less_eq_int @ ( minus_minus_int @ X3 @ D2 ) @ T ) ) ) ) ).
% bset(6)
thf(fact_1209_bset_I8_J,axiom,
! [D2: int,T: int,B2: set_int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
=> ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ T @ X3 )
=> ( ord_less_eq_int @ T @ ( minus_minus_int @ X3 @ D2 ) ) ) ) ) ) ).
% bset(8)
thf(fact_1210_aset_I6_J,axiom,
! [D2: int,T: int,A: set_int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A )
=> ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ X3 @ T )
=> ( ord_less_eq_int @ ( plus_plus_int @ X3 @ D2 ) @ T ) ) ) ) ) ).
% aset(6)
thf(fact_1211_nat0__intermed__int__val,axiom,
! [N5: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N5 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N5 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N5 )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1212_cpmi,axiom,
! [D2: int,P: int > $o,P4: int > $o,B2: set_int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B2 )
=> ( X2
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D2 ) ) ) )
=> ( ! [X2: int,K3: int] :
( ( P4 @ X2 )
= ( P4 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X4: int] :
( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
& ( P4 @ X4 ) )
| ? [X4: int] :
( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
& ? [Y4: int] :
( ( member_int @ Y4 @ B2 )
& ( P @ ( plus_plus_int @ Y4 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_1213_diff__0__eq__0,axiom,
! [N5: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N5 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1214_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1215_diff__diff__cancel,axiom,
! [I: nat,N5: nat] :
( ( ord_less_eq_nat @ I @ N5 )
=> ( ( minus_minus_nat @ N5 @ ( minus_minus_nat @ N5 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1216_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1217_diff__is__0__eq_H,axiom,
! [M3: nat,N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
=> ( ( minus_minus_nat @ M3 @ N5 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1218_diff__is__0__eq,axiom,
! [M3: nat,N5: nat] :
( ( ( minus_minus_nat @ M3 @ N5 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N5 ) ) ).
% diff_is_0_eq
thf(fact_1219_zero__less__diff,axiom,
! [N5: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N5 @ M3 ) )
= ( ord_less_nat @ M3 @ N5 ) ) ).
% zero_less_diff
thf(fact_1220_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_1221_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_1222_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_1223_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1224_eq__diff__iff,axiom,
! [K: nat,M3: nat,N5: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N5 )
=> ( ( ( minus_minus_nat @ M3 @ K )
= ( minus_minus_nat @ N5 @ K ) )
= ( M3 = N5 ) ) ) ) ).
% eq_diff_iff
thf(fact_1225_le__diff__iff,axiom,
! [K: nat,M3: nat,N5: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N5 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N5 @ K ) )
= ( ord_less_eq_nat @ M3 @ N5 ) ) ) ) ).
% le_diff_iff
thf(fact_1226_Nat_Odiff__diff__eq,axiom,
! [K: nat,M3: nat,N5: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N5 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N5 @ K ) )
= ( minus_minus_nat @ M3 @ N5 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1227_diff__le__mono,axiom,
! [M3: nat,N5: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N5 @ L ) ) ) ).
% diff_le_mono
thf(fact_1228_diff__le__self,axiom,
! [M3: nat,N5: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N5 ) @ M3 ) ).
% diff_le_self
thf(fact_1229_le__diff__iff_H,axiom,
! [A2: nat,C3: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C3 )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C3 @ A2 ) @ ( minus_minus_nat @ C3 @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1230_diff__le__mono2,axiom,
! [M3: nat,N5: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N5 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_1231_less__imp__diff__less,axiom,
! [J: nat,K: nat,N5: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N5 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1232_diff__less__mono2,axiom,
! [M3: nat,N5: nat,L: nat] :
( ( ord_less_nat @ M3 @ N5 )
=> ( ( ord_less_nat @ M3 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N5 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ) ).
% diff_less_mono2
thf(fact_1233_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1234_diff__add__inverse2,axiom,
! [M3: nat,N5: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N5 ) @ N5 )
= M3 ) ).
% diff_add_inverse2
thf(fact_1235_diff__add__inverse,axiom,
! [N5: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N5 @ M3 ) @ N5 )
= M3 ) ).
% diff_add_inverse
thf(fact_1236_diff__cancel2,axiom,
! [M3: nat,K: nat,N5: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K ) @ ( plus_plus_nat @ N5 @ K ) )
= ( minus_minus_nat @ M3 @ N5 ) ) ).
% diff_cancel2
thf(fact_1237_Nat_Odiff__cancel,axiom,
! [K: nat,M3: nat,N5: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N5 ) )
= ( minus_minus_nat @ M3 @ N5 ) ) ).
% Nat.diff_cancel
thf(fact_1238_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_1239_diffs0__imp__equal,axiom,
! [M3: nat,N5: nat] :
( ( ( minus_minus_nat @ M3 @ N5 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N5 @ M3 )
= zero_zero_nat )
=> ( M3 = N5 ) ) ) ).
% diffs0_imp_equal
thf(fact_1240_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_1241_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_1242_diff__less,axiom,
! [N5: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_nat @ ( minus_minus_nat @ M3 @ N5 ) @ M3 ) ) ) ).
% diff_less
thf(fact_1243_diff__less__mono,axiom,
! [A2: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C3 @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C3 ) @ ( minus_minus_nat @ B @ C3 ) ) ) ) ).
% diff_less_mono
thf(fact_1244_less__diff__iff,axiom,
! [K: nat,M3: nat,N5: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N5 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N5 @ K ) )
= ( ord_less_nat @ M3 @ N5 ) ) ) ) ).
% less_diff_iff
thf(fact_1245_diff__add__0,axiom,
! [N5: nat,M3: nat] :
( ( minus_minus_nat @ N5 @ ( plus_plus_nat @ N5 @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1246_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1247_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1248_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1249_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1250_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1251_add__diff__inverse__nat,axiom,
! [M3: nat,N5: nat] :
( ~ ( ord_less_nat @ M3 @ N5 )
=> ( ( plus_plus_nat @ N5 @ ( minus_minus_nat @ M3 @ N5 ) )
= M3 ) ) ).
% add_diff_inverse_nat
thf(fact_1252_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1253_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_1254_incr__lemma,axiom,
! [D3: int,Z2: int,X: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ord_less_int @ Z2 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D3 ) ) ) ) ).
% incr_lemma
thf(fact_1255_decr__lemma,axiom,
! [D3: int,X: int,Z2: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D3 ) ) @ Z2 ) ) ).
% decr_lemma
thf(fact_1256_Bolzano,axiom,
! [A2: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [A5: real,B5: real,C: real] :
( ( P @ A5 @ B5 )
=> ( ( P @ B5 @ C )
=> ( ( ord_less_eq_real @ A5 @ B5 )
=> ( ( ord_less_eq_real @ B5 @ C )
=> ( P @ A5 @ C ) ) ) ) )
=> ( ! [X2: real] :
( ( ord_less_eq_real @ A2 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [A5: real,B5: real] :
( ( ( ord_less_eq_real @ A5 @ X2 )
& ( ord_less_eq_real @ X2 @ B5 )
& ( ord_less_real @ ( minus_minus_real @ B5 @ A5 ) @ D4 ) )
=> ( P @ A5 @ B5 ) ) ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Bolzano
thf(fact_1257_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D5: nat] :
( ( A2
= ( plus_plus_nat @ B @ D5 ) )
& ~ ( P @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1258_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ( ( ord_less_nat @ A2 @ B )
=> ( P @ zero_zero_nat ) )
& ! [D5: nat] :
( ( A2
= ( plus_plus_nat @ B @ D5 ) )
=> ( P @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1259_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1260_pos__zmult__eq__1__iff,axiom,
! [M3: int,N5: int] :
( ( ord_less_int @ zero_zero_int @ M3 )
=> ( ( ( times_times_int @ M3 @ N5 )
= one_one_int )
= ( ( M3 = one_one_int )
& ( N5 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1261_plusinfinity,axiom,
! [D3: int,P4: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ! [X2: int,K3: int] :
( ( P4 @ X2 )
= ( P4 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1262_minusinfinity,axiom,
! [D3: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ! [X2: int,K3: int] :
( ( P1 @ X2 )
= ( P1 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P1 @ X2 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
topolo5044208981011980120l_real @ ( image_real_real @ f @ ( set_or1222579329274155063t_real @ a @ u ) ) @ g ).
%------------------------------------------------------------------------------