TPTP Problem File: SLH0362^1.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_00428_017413__13066356_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1388 ( 504 unt; 114 typ; 0 def)
% Number of atoms : 4170 (1086 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 12206 ( 320 ~; 100 |; 203 &;9612 @)
% ( 0 <=>;1971 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 1052 (1052 >; 0 *; 0 +; 0 <<)
% Number of symbols : 111 ( 108 usr; 14 con; 0-4 aty)
% Number of variables : 3849 ( 164 ^;3595 !; 90 ?;3849 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:30:37.375
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (108)
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
archim6058952711729229775r_real: real > int ).
thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
comp_int_int_int: ( int > int ) > ( int > int ) > int > int ).
thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Nat__Onat,type,
comp_int_int_nat: ( int > int ) > ( nat > int ) > nat > int ).
thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
comp_int_nat_nat: ( int > nat ) > ( nat > int ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Real__Oreal_001t__Int__Oint,type,
comp_int_real_int: ( int > real ) > ( int > int ) > int > real ).
thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Real__Oreal_001t__Nat__Onat,type,
comp_int_real_nat: ( int > real ) > ( nat > int ) > nat > real ).
thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Real__Oreal_001t__Real__Oreal,type,
comp_int_real_real: ( int > real ) > ( real > int ) > real > real ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Int__Oint_001t__Int__Oint,type,
comp_nat_int_int: ( nat > int ) > ( int > nat ) > int > int ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
comp_nat_int_nat: ( nat > int ) > ( nat > nat ) > nat > int ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Int__Oint_001t__Int__Oint,type,
comp_real_int_int: ( real > int ) > ( int > real ) > int > int ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Int__Oint_001t__Nat__Onat,type,
comp_real_int_nat: ( real > int ) > ( nat > real ) > nat > int ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Int__Oint_001t__Real__Oreal,type,
comp_real_int_real: ( real > int ) > ( real > real ) > real > int ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
comp_real_real_real: ( real > real ) > ( real > real ) > real > real ).
thf(sy_c_Fun_Omonotone__on_001t__Int__Oint_001t__Int__Oint,type,
monotone_on_int_int: set_int > ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Int__Oint_001t__Nat__Onat,type,
monotone_on_int_nat: set_int > ( int > int > $o ) > ( nat > nat > $o ) > ( int > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Int__Oint_001t__Real__Oreal,type,
monotone_on_int_real: set_int > ( int > int > $o ) > ( real > real > $o ) > ( int > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Int__Oint,type,
monotone_on_nat_int: set_nat > ( nat > nat > $o ) > ( int > int > $o ) > ( nat > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
monotone_on_nat_nat: set_nat > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Real__Oreal,type,
monotone_on_nat_real: set_nat > ( nat > nat > $o ) > ( real > real > $o ) > ( nat > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Int__Oint,type,
monotone_on_real_int: set_real > ( real > real > $o ) > ( int > int > $o ) > ( real > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Nat__Onat,type,
monotone_on_real_nat: set_real > ( real > real > $o ) > ( nat > nat > $o ) > ( real > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Real__Oreal,type,
monoto4017252874604999745l_real: set_real > ( real > real > $o ) > ( real > real > $o ) > ( real > real ) > $o ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
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_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_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_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Henstock__Kurzweil__Integration_Ointegrable__on_001t__Real__Oreal_001t__Real__Oreal,type,
hensto5963834015518849588l_real: ( real > real ) > set_real > $o ).
thf(sy_c_Henstock__Kurzweil__Integration_Ointegral_001t__Real__Oreal_001t__Real__Oreal,type,
hensto2714581292692559302l_real: set_real > ( real > real ) > real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Int__Oint,type,
order_Greatest_int: ( int > $o ) > int ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal,type,
order_Greatest_real: ( real > $o ) > real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
top_top_set_int: set_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
real_V7735802525324610683m_real: real > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__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__Nat__Onat_001t__Real__Oreal,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Int__Oint,type,
image_real_int: ( real > int ) > set_real > set_int ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Nat__Onat,type,
image_real_nat: ( real > nat ) > set_real > set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Int__Oint,type,
set_ord_atLeast_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
set_ord_atLeast_real: real > set_real ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Int__Oint_001t__Int__Oint,type,
topolo2178910747331673048nt_int: set_int > ( int > int ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Int__Oint_001t__Nat__Onat,type,
topolo2181401217840723324nt_nat: set_int > ( int > nat ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__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_Topological__Spaces_Ouniformly__continuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
topolo8845477368217174713l_real: set_real > ( real > real ) > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v__092_060delta_062____,type,
delta: real ).
thf(sy_v__092_060epsilon_062____,type,
epsilon: real ).
thf(sy_v_a,type,
a: real ).
thf(sy_v_a__seg____,type,
a_seg: real > real ).
thf(sy_v_b,type,
b: real ).
thf(sy_v_del____,type,
del: real > real ).
thf(sy_v_f,type,
f: real > real ).
thf(sy_v_f1____,type,
f1: real > real ).
thf(sy_v_g,type,
g: real > real ).
thf(sy_v_lower____,type,
lower: real > real ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_x____,type,
x: real ).
% Relevant facts (1270)
thf(fact_0_f_I1_J,axiom,
( ( f @ zero_zero_real )
= zero_zero_real ) ).
% f(1)
thf(fact_1__092_060open_062lower_Ax_A_092_060le_062_Ax_092_060close_062,axiom,
ord_less_eq_real @ ( lower @ x ) @ x ).
% \<open>lower x \<le> x\<close>
thf(fact_2__092_060open_0620_A_092_060le_062_Alower_Ax_092_060close_062,axiom,
ord_less_eq_real @ zero_zero_real @ ( lower @ x ) ).
% \<open>0 \<le> lower x\<close>
thf(fact_3_that_I1_J,axiom,
ord_less_eq_real @ zero_zero_real @ x ).
% that(1)
thf(fact_4_that_I2_J,axiom,
ord_less_eq_real @ x @ a ).
% that(2)
thf(fact_5_f1__def,axiom,
( f1
= ( comp_real_real_real @ f @ lower ) ) ).
% f1_def
thf(fact_6_f__iff_I2_J,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( f @ X ) @ ( f @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% f_iff(2)
thf(fact_7_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_8_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_9_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_10_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_11_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_12_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_13_a,axiom,
ord_less_eq_real @ zero_zero_real @ a ).
% a
thf(fact_14__092_060open_0620_A_092_060le_062_Ab_092_060close_062,axiom,
ord_less_eq_real @ zero_zero_real @ b ).
% \<open>0 \<le> b\<close>
thf(fact_15_complete__real,axiom,
! [S: set_real] :
( ? [X2: real] : ( member_real @ X2 @ S )
=> ( ? [Z: real] :
! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ? [Y2: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Y2 ) )
& ! [Z: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ( ord_less_eq_real @ Y2 @ Z ) ) ) ) ) ).
% complete_real
thf(fact_16_f_I2_J,axiom,
( ( f @ a )
= b ) ).
% f(2)
thf(fact_17_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_18_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_19_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_20_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_21_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_22_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_23_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_24_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_25_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_26_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_27_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_28_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z3: int] : ( Y3 = Z3 ) )
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_29_False,axiom,
a != zero_zero_real ).
% False
thf(fact_30_g,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ a )
=> ( ( g @ ( f @ X ) )
= X ) ) ) ).
% g
thf(fact_31__092_060open_062_092_060delta_062_A_092_060le_062_Aa_092_060close_062,axiom,
ord_less_eq_real @ delta @ a ).
% \<open>\<delta> \<le> a\<close>
thf(fact_32_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_33_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_34_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_35_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_36_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_37_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_38_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_39_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_40_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_41_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_42_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_43_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_44_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_45_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_46_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_47_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_48_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_49_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_50_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_51_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_52_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_53_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_54_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_55_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_56_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_57_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_58_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_59_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_60_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_61_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_62_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_63_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_64_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_65_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_66_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_67_order__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_68_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_69_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_70_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_71_order__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_72_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_73_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_74_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_75_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_76_order__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_77_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_78_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_79_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_80_order__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_81_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_82_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_83_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_84_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_85_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z3: int] : ( Y3 = Z3 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_86_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_87_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_88_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_89_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_90_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_91_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_92_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_93_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_94_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_95_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_96_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_97_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z3: int] : ( Y3 = Z3 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_98_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: real,B3: real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_99_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_100_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_101_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_102_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_103_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_104_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_105_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_106_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_107_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_108_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_109_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_110_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_111_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_112_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_113_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_114_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_115_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_116_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_117_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_118_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_119_Collect__mem__eq,axiom,
! [A4: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_120_Collect__mem__eq,axiom,
! [A4: set_int] :
( ( collect_int
@ ^ [X4: int] : ( member_int @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_121__092_060open_0620_A_060_Aa_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ a ).
% \<open>0 < a\<close>
thf(fact_122_comp__apply,axiom,
( comp_real_real_real
= ( ^ [F2: real > real,G: real > real,X4: real] : ( F2 @ ( G @ X4 ) ) ) ) ).
% comp_apply
thf(fact_123_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_124_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_125_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_126_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_127_f__iff_I1_J,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( f @ X ) @ ( f @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% f_iff(1)
thf(fact_128_fim,axiom,
( ( image_real_real @ f @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
= ( set_or1222579329274155063t_real @ zero_zero_real @ b ) ) ).
% fim
thf(fact_129__092_060open_0620_A_060_A_092_060epsilon_062_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ epsilon ).
% \<open>0 < \<epsilon>\<close>
thf(fact_130_Greatest__equality,axiom,
! [P: real > $o,X: real] :
( ( P @ X )
=> ( ! [Y2: real] :
( ( P @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) )
=> ( ( order_Greatest_real @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_131_Greatest__equality,axiom,
! [P: int > $o,X: int] :
( ( P @ X )
=> ( ! [Y2: int] :
( ( P @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) )
=> ( ( order_Greatest_int @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_132_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_133_GreatestI2__order,axiom,
! [P: real > $o,X: real,Q: real > $o] :
( ( P @ X )
=> ( ! [Y2: real] :
( ( P @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) )
=> ( ! [X3: real] :
( ( P @ X3 )
=> ( ! [Y5: real] :
( ( P @ Y5 )
=> ( ord_less_eq_real @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_134_GreatestI2__order,axiom,
! [P: int > $o,X: int,Q: int > $o] :
( ( P @ X )
=> ( ! [Y2: int] :
( ( P @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( ! [Y5: int] :
( ( P @ Y5 )
=> ( ord_less_eq_int @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_135_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_136__092_060open_062uniformly__continuous__on_A_1230_O_Oa_125_Af_092_060close_062,axiom,
topolo8845477368217174713l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) @ f ).
% \<open>uniformly_continuous_on {0..a} f\<close>
thf(fact_137__092_060open_0620_A_060_A_092_060delta_062_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ delta ).
% \<open>0 < \<delta>\<close>
thf(fact_138_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_139_del__gt0,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_real @ zero_zero_real @ ( del @ E ) ) ) ).
% del_gt0
thf(fact_140_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_141_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_142_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_143_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_144_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_145_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_146_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_147_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_148_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z4: real] :
( ( ord_less_real @ X @ Z4 )
& ( ord_less_real @ Z4 @ Y ) ) ) ).
% dense
thf(fact_149_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_150_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_151_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_152_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_153_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_154_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_155_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_156_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_157_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_158_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_159_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_160_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_161_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_162_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_163_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_164_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_165_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_166_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_167_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_168_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_169_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_170_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_171_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_172_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_173_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_174_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_175_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: real] : ( P @ A3 @ A3 )
=> ( ! [A3: real,B3: real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_176_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_177_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_178_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_179_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_180_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_181_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_182_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_183_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_184_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_185_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_186_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_187_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_188_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_189_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_190_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_191_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_192_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_193_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_194_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_195_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_196_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_197_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_198_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_199_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_200_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_201_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_202_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_203_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_204_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_205_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_206_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_207_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_208_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_209_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_210_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_211_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_212_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_213_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_214_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_215_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_216_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_217_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_218_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_219_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_220_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_221_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_222_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_223_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_224_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_225_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_226_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_227_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_228_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_229_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_230_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_231_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_232_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_233_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_234_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_235_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_236_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_237_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_238_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_239_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_240_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_241_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_242_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_243_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_244_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_245_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_246_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_247_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_248_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_249_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_250_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_251_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_252_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_253_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_254_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_255_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_256_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_257_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_258_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_259_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_260_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_261_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_262_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_263_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_264_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_265_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_266_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_267_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_268_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_269_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_270_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_271_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_272_image__eq__imp__comp,axiom,
! [F: nat > real,A4: set_nat,G2: real > real,B4: set_real,H: real > int] :
( ( ( image_nat_real @ F @ A4 )
= ( image_real_real @ G2 @ B4 ) )
=> ( ( image_nat_int @ ( comp_real_int_nat @ H @ F ) @ A4 )
= ( image_real_int @ ( comp_real_int_real @ H @ G2 ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_273_image__eq__imp__comp,axiom,
! [F: int > real,A4: set_int,G2: real > real,B4: set_real,H: real > int] :
( ( ( image_int_real @ F @ A4 )
= ( image_real_real @ G2 @ B4 ) )
=> ( ( image_int_int @ ( comp_real_int_int @ H @ F ) @ A4 )
= ( image_real_int @ ( comp_real_int_real @ H @ G2 ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_274_image__eq__imp__comp,axiom,
! [F: real > int,A4: set_real,G2: nat > int,B4: set_nat,H: int > real] :
( ( ( image_real_int @ F @ A4 )
= ( image_nat_int @ G2 @ B4 ) )
=> ( ( image_real_real @ ( comp_int_real_real @ H @ F ) @ A4 )
= ( image_nat_real @ ( comp_int_real_nat @ H @ G2 ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_275_image__eq__imp__comp,axiom,
! [F: real > int,A4: set_real,G2: int > int,B4: set_int,H: int > real] :
( ( ( image_real_int @ F @ A4 )
= ( image_int_int @ G2 @ B4 ) )
=> ( ( image_real_real @ ( comp_int_real_real @ H @ F ) @ A4 )
= ( image_int_real @ ( comp_int_real_int @ H @ G2 ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_276_image__eq__imp__comp,axiom,
! [F: real > real,A4: set_real,G2: nat > real,B4: set_nat,H: real > int] :
( ( ( image_real_real @ F @ A4 )
= ( image_nat_real @ G2 @ B4 ) )
=> ( ( image_real_int @ ( comp_real_int_real @ H @ F ) @ A4 )
= ( image_nat_int @ ( comp_real_int_nat @ H @ G2 ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_277_image__eq__imp__comp,axiom,
! [F: real > real,A4: set_real,G2: int > real,B4: set_int,H: real > int] :
( ( ( image_real_real @ F @ A4 )
= ( image_int_real @ G2 @ B4 ) )
=> ( ( image_real_int @ ( comp_real_int_real @ H @ F ) @ A4 )
= ( image_int_int @ ( comp_real_int_int @ H @ G2 ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_278_image__eq__imp__comp,axiom,
! [F: real > real,A4: set_real,G2: real > real,B4: set_real,H: real > real] :
( ( ( image_real_real @ F @ A4 )
= ( image_real_real @ G2 @ B4 ) )
=> ( ( image_real_real @ ( comp_real_real_real @ H @ F ) @ A4 )
= ( image_real_real @ ( comp_real_real_real @ H @ G2 ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_279_image__eq__imp__comp,axiom,
! [F: nat > int,A4: set_nat,G2: real > int,B4: set_real,H: int > real] :
( ( ( image_nat_int @ F @ A4 )
= ( image_real_int @ G2 @ B4 ) )
=> ( ( image_nat_real @ ( comp_int_real_nat @ H @ F ) @ A4 )
= ( image_real_real @ ( comp_int_real_real @ H @ G2 ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_280_image__eq__imp__comp,axiom,
! [F: nat > int,A4: set_nat,G2: nat > int,B4: set_nat,H: int > int] :
( ( ( image_nat_int @ F @ A4 )
= ( image_nat_int @ G2 @ B4 ) )
=> ( ( image_nat_int @ ( comp_int_int_nat @ H @ F ) @ A4 )
= ( image_nat_int @ ( comp_int_int_nat @ H @ G2 ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_281_image__eq__imp__comp,axiom,
! [F: nat > int,A4: set_nat,G2: int > int,B4: set_int,H: int > int] :
( ( ( image_nat_int @ F @ A4 )
= ( image_int_int @ G2 @ B4 ) )
=> ( ( image_nat_int @ ( comp_int_int_nat @ H @ F ) @ A4 )
= ( image_int_int @ ( comp_int_int_int @ H @ G2 ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_282_image__comp,axiom,
! [F: real > real,G2: real > real,R: set_real] :
( ( image_real_real @ F @ ( image_real_real @ G2 @ R ) )
= ( image_real_real @ ( comp_real_real_real @ F @ G2 ) @ R ) ) ).
% image_comp
thf(fact_283_image__comp,axiom,
! [F: nat > int,G2: nat > nat,R: set_nat] :
( ( image_nat_int @ F @ ( image_nat_nat @ G2 @ R ) )
= ( image_nat_int @ ( comp_nat_int_nat @ F @ G2 ) @ R ) ) ).
% image_comp
thf(fact_284_image__comp,axiom,
! [F: nat > int,G2: int > nat,R: set_int] :
( ( image_nat_int @ F @ ( image_int_nat @ G2 @ R ) )
= ( image_int_int @ ( comp_nat_int_int @ F @ G2 ) @ R ) ) ).
% image_comp
thf(fact_285_image__comp,axiom,
! [F: int > int,G2: nat > int,R: set_nat] :
( ( image_int_int @ F @ ( image_nat_int @ G2 @ R ) )
= ( image_nat_int @ ( comp_int_int_nat @ F @ G2 ) @ R ) ) ).
% image_comp
thf(fact_286_image__comp,axiom,
! [F: int > int,G2: int > int,R: set_int] :
( ( image_int_int @ F @ ( image_int_int @ G2 @ R ) )
= ( image_int_int @ ( comp_int_int_int @ F @ G2 ) @ R ) ) ).
% image_comp
thf(fact_287_verit__comp__simplify1_I3_J,axiom,
! [B5: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B5 @ A5 ) )
= ( ord_less_real @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_288_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_289_verit__comp__simplify1_I3_J,axiom,
! [B5: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
= ( ord_less_int @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_290_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_291_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_292_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_293_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_294_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_295_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_296_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_297_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_298_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_299_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_300_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_301_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_302_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_303_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_304_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_305_dense__ge,axiom,
! [Z2: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_306_dense__le,axiom,
! [Y: real,Z2: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_le
thf(fact_307_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_308_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_309_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_310_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_311_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_312_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_313_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_314_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_315_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_316_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_317_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_318_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_319_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_320_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_321_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_322_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_323_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_324_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_325_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_326_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_327_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_328_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_329_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_330_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_331_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_332_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_333_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_334_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_335_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_336_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_337_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_338_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_339_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_340_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_341_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_342_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ~ ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_343_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_344_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_345_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_346_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_347_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_348_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_349_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_350_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_351_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_352_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_353_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_354_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_355_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_356_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_357_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_358_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_359_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_360_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_361_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_362_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_363_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_364_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_365_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_366_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_367_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_368_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_369_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_370_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_371_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_372_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_373_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_374_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_375_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_376_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_377_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_378_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_379_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_380_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_381_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_382_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_383_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_384_order__le__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_385_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_386_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_387_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_388_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_389_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_390_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_391_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_392_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_393_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_394_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_395_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_396_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_397_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_398_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_399_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_400_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_401_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_402_order__less__le__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_403_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_404_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_405_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_406_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_407_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_408_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_409_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_410_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_411_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_412_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_413_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_414_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_415_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_416_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_417_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_418_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_419_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_420_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_421_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_422_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_423_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_424_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_425_comp__eq__dest__lhs,axiom,
! [A: real > real,B: real > real,C: real > real,V: real] :
( ( ( comp_real_real_real @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_426_comp__eq__elim,axiom,
! [A: real > real,B: real > real,C: real > real,D: real > real] :
( ( ( comp_real_real_real @ A @ B )
= ( comp_real_real_real @ C @ D ) )
=> ! [V2: real] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_427_comp__eq__dest,axiom,
! [A: real > real,B: real > real,C: real > real,D: real > real,V: real] :
( ( ( comp_real_real_real @ A @ B )
= ( comp_real_real_real @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_428_comp__assoc,axiom,
! [F: real > real,G2: real > real,H: real > real] :
( ( comp_real_real_real @ ( comp_real_real_real @ F @ G2 ) @ H )
= ( comp_real_real_real @ F @ ( comp_real_real_real @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_429_comp__def,axiom,
( comp_real_real_real
= ( ^ [F2: real > real,G: real > real,X4: real] : ( F2 @ ( G @ X4 ) ) ) ) ).
% comp_def
thf(fact_430_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_431_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_432_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_433_atLeastAtMost__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_eq_real @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_434_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_435_atLeastAtMost__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_eq_int @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_436_atLeastatMost__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_437_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_438_atLeastatMost__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_439__092_060open_062continuous__on_A_1230_O_Ob_125_Ag_092_060close_062,axiom,
topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) @ g ).
% \<open>continuous_on {0..b} g\<close>
thf(fact_440_intgb__g,axiom,
hensto5963834015518849588l_real @ g @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) ).
% intgb_g
thf(fact_441_sm__0a,axiom,
monoto4017252874604999745l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) @ ord_less_real @ ord_less_real @ f ).
% sm_0a
thf(fact_442_cont__0a,axiom,
topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) @ f ).
% cont_0a
thf(fact_443_intgb__f,axiom,
hensto5963834015518849588l_real @ f @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ).
% intgb_f
thf(fact_444_atLeastatMost__psubset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D )
& ( ( ord_less_real @ C @ A )
| ( ord_less_real @ B @ D ) ) ) )
& ( ord_less_eq_real @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_445_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_446_atLeastatMost__psubset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
= ( ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D )
& ( ( ord_less_int @ C @ A )
| ( ord_less_int @ B @ D ) ) ) )
& ( ord_less_eq_int @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_447_image__eqI,axiom,
! [B: int,F: nat > int,X: nat,A4: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A4 )
=> ( member_int @ B @ ( image_nat_int @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_448_image__eqI,axiom,
! [B: real,F: real > real,X: real,A4: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A4 )
=> ( member_real @ B @ ( image_real_real @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_449_image__eqI,axiom,
! [B: int,F: real > int,X: real,A4: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A4 )
=> ( member_int @ B @ ( image_real_int @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_450_image__eqI,axiom,
! [B: real,F: int > real,X: int,A4: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A4 )
=> ( member_real @ B @ ( image_int_real @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_451_image__eqI,axiom,
! [B: int,F: int > int,X: int,A4: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A4 )
=> ( member_int @ B @ ( image_int_int @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_452_subsetI,axiom,
! [A4: set_real,B4: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( member_real @ X3 @ B4 ) )
=> ( ord_less_eq_set_real @ A4 @ B4 ) ) ).
% subsetI
thf(fact_453_subsetI,axiom,
! [A4: set_int,B4: set_int] :
( ! [X3: int] :
( ( member_int @ X3 @ A4 )
=> ( member_int @ X3 @ B4 ) )
=> ( ord_less_eq_set_int @ A4 @ B4 ) ) ).
% subsetI
thf(fact_454_cont,axiom,
topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ f ).
% cont
thf(fact_455_sm,axiom,
monoto4017252874604999745l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ ord_less_real @ ord_less_real @ f ).
% sm
thf(fact_456_in__mono,axiom,
! [A4: set_real,B4: set_real,X: real] :
( ( ord_less_eq_set_real @ A4 @ B4 )
=> ( ( member_real @ X @ A4 )
=> ( member_real @ X @ B4 ) ) ) ).
% in_mono
thf(fact_457_in__mono,axiom,
! [A4: set_int,B4: set_int,X: int] :
( ( ord_less_eq_set_int @ A4 @ B4 )
=> ( ( member_int @ X @ A4 )
=> ( member_int @ X @ B4 ) ) ) ).
% in_mono
thf(fact_458_subsetD,axiom,
! [A4: set_real,B4: set_real,C: real] :
( ( ord_less_eq_set_real @ A4 @ B4 )
=> ( ( member_real @ C @ A4 )
=> ( member_real @ C @ B4 ) ) ) ).
% subsetD
thf(fact_459_subsetD,axiom,
! [A4: set_int,B4: set_int,C: int] :
( ( ord_less_eq_set_int @ A4 @ B4 )
=> ( ( member_int @ C @ A4 )
=> ( member_int @ C @ B4 ) ) ) ).
% subsetD
thf(fact_460_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [X4: real] :
( ( member_real @ X4 @ A6 )
=> ( member_real @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_461_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [X4: int] :
( ( member_int @ X4 @ A6 )
=> ( member_int @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_462_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [T: real] :
( ( member_real @ T @ A6 )
=> ( member_real @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_463_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [T: int] :
( ( member_int @ T @ A6 )
=> ( member_int @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_464_monotone__on__subset,axiom,
! [A4: set_real,Orda: real > real > $o,Ordb: real > real > $o,F: real > real,B4: set_real] :
( ( monoto4017252874604999745l_real @ A4 @ Orda @ Ordb @ F )
=> ( ( ord_less_eq_set_real @ B4 @ A4 )
=> ( monoto4017252874604999745l_real @ B4 @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_465_monotone__on__subset,axiom,
! [A4: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,B4: set_nat] :
( ( monotone_on_nat_nat @ A4 @ Orda @ Ordb @ F )
=> ( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( monotone_on_nat_nat @ B4 @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_466_monotone__on__def,axiom,
( monoto4017252874604999745l_real
= ( ^ [A6: set_real,Orda2: real > real > $o,Ordb2: real > real > $o,F2: real > real] :
! [X4: real] :
( ( member_real @ X4 @ A6 )
=> ! [Y4: real] :
( ( member_real @ Y4 @ A6 )
=> ( ( Orda2 @ X4 @ Y4 )
=> ( Ordb2 @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_467_monotone__on__def,axiom,
( monotone_on_nat_nat
= ( ^ [A6: set_nat,Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F2: nat > nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ A6 )
=> ( ( Orda2 @ X4 @ Y4 )
=> ( Ordb2 @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_468_monotone__onI,axiom,
! [A4: set_real,Orda: real > real > $o,Ordb: real > real > $o,F: real > real] :
( ! [X3: real,Y2: real] :
( ( member_real @ X3 @ A4 )
=> ( ( member_real @ Y2 @ A4 )
=> ( ( Orda @ X3 @ Y2 )
=> ( Ordb @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A4 @ Orda @ Ordb @ F ) ) ).
% monotone_onI
thf(fact_469_monotone__onI,axiom,
! [A4: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat] :
( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( member_nat @ Y2 @ A4 )
=> ( ( Orda @ X3 @ Y2 )
=> ( Ordb @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) )
=> ( monotone_on_nat_nat @ A4 @ Orda @ Ordb @ F ) ) ).
% monotone_onI
thf(fact_470_monotone__onD,axiom,
! [A4: set_real,Orda: real > real > $o,Ordb: real > real > $o,F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ A4 @ Orda @ Ordb @ F )
=> ( ( member_real @ X @ A4 )
=> ( ( member_real @ Y @ A4 )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% monotone_onD
thf(fact_471_monotone__onD,axiom,
! [A4: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A4 @ Orda @ Ordb @ F )
=> ( ( member_nat @ X @ A4 )
=> ( ( member_nat @ Y @ A4 )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% monotone_onD
thf(fact_472_psubsetD,axiom,
! [A4: set_real,B4: set_real,C: real] :
( ( ord_less_set_real @ A4 @ B4 )
=> ( ( member_real @ C @ A4 )
=> ( member_real @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_473_psubsetD,axiom,
! [A4: set_int,B4: set_int,C: int] :
( ( ord_less_set_int @ A4 @ B4 )
=> ( ( member_int @ C @ A4 )
=> ( member_int @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_474_invertible__fixpoint__property,axiom,
! [T2: set_nat,I: nat > int,S: set_int,R: int > nat,G2: nat > nat] :
( ( topolo1179557035430618492at_int @ T2 @ I )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ I @ T2 ) @ S )
=> ( ( topolo2181401217840723324nt_nat @ S @ R )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ R @ S ) @ T2 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F3: int > int] :
( ( topolo2178910747331673048nt_int @ S @ F3 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F3 @ S ) @ S )
=> ? [X2: int] :
( ( member_int @ X2 @ S )
& ( ( F3 @ X2 )
= X2 ) ) ) )
=> ( ( topolo1182047505939668768at_nat @ T2 @ G2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G2 @ T2 ) @ T2 )
=> ~ ! [Y2: nat] :
( ( member_nat @ Y2 @ T2 )
=> ( ( G2 @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_475_invertible__fixpoint__property,axiom,
! [T2: set_int,I: int > nat,S: set_nat,R: nat > int,G2: int > int] :
( ( topolo2181401217840723324nt_nat @ T2 @ I )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ I @ T2 ) @ S )
=> ( ( topolo1179557035430618492at_int @ S @ R )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ R @ S ) @ T2 )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F3: nat > nat] :
( ( topolo1182047505939668768at_nat @ S @ F3 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ S ) @ S )
=> ? [X2: nat] :
( ( member_nat @ X2 @ S )
& ( ( F3 @ X2 )
= X2 ) ) ) )
=> ( ( topolo2178910747331673048nt_int @ T2 @ G2 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ G2 @ T2 ) @ T2 )
=> ~ ! [Y2: int] :
( ( member_int @ Y2 @ T2 )
=> ( ( G2 @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_476_invertible__fixpoint__property,axiom,
! [T2: set_int,I: int > int,S: set_int,R: int > int,G2: int > int] :
( ( topolo2178910747331673048nt_int @ T2 @ I )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ I @ T2 ) @ S )
=> ( ( topolo2178910747331673048nt_int @ S @ R )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ R @ S ) @ T2 )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F3: int > int] :
( ( topolo2178910747331673048nt_int @ S @ F3 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F3 @ S ) @ S )
=> ? [X2: int] :
( ( member_int @ X2 @ S )
& ( ( F3 @ X2 )
= X2 ) ) ) )
=> ( ( topolo2178910747331673048nt_int @ T2 @ G2 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ G2 @ T2 ) @ T2 )
=> ~ ! [Y2: int] :
( ( member_int @ Y2 @ T2 )
=> ( ( G2 @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_477_invertible__fixpoint__property,axiom,
! [T2: set_real,I: real > int,S: set_int,R: int > real,G2: real > real] :
( ( topolo2284712892409288920al_int @ T2 @ I )
=> ( ( ord_less_eq_set_int @ ( image_real_int @ I @ T2 ) @ S )
=> ( ( topolo9130188401337238104t_real @ S @ R )
=> ( ( ord_less_eq_set_real @ ( image_int_real @ R @ S ) @ T2 )
=> ( ! [Y2: real] :
( ( member_real @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F3: int > int] :
( ( topolo2178910747331673048nt_int @ S @ F3 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F3 @ S ) @ S )
=> ? [X2: int] :
( ( member_int @ X2 @ S )
& ( ( F3 @ X2 )
= X2 ) ) ) )
=> ( ( topolo5044208981011980120l_real @ T2 @ G2 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ G2 @ T2 ) @ T2 )
=> ~ ! [Y2: real] :
( ( member_real @ Y2 @ T2 )
=> ( ( G2 @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_478_invertible__fixpoint__property,axiom,
! [T2: set_int,I: int > real,S: set_real,R: real > int,G2: int > int] :
( ( topolo9130188401337238104t_real @ T2 @ I )
=> ( ( ord_less_eq_set_real @ ( image_int_real @ I @ T2 ) @ S )
=> ( ( topolo2284712892409288920al_int @ S @ R )
=> ( ( ord_less_eq_set_int @ ( image_real_int @ R @ S ) @ T2 )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F3: real > real] :
( ( topolo5044208981011980120l_real @ S @ F3 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ F3 @ S ) @ S )
=> ? [X2: real] :
( ( member_real @ X2 @ S )
& ( ( F3 @ X2 )
= X2 ) ) ) )
=> ( ( topolo2178910747331673048nt_int @ T2 @ G2 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ G2 @ T2 ) @ T2 )
=> ~ ! [Y2: int] :
( ( member_int @ Y2 @ T2 )
=> ( ( G2 @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_479_invertible__fixpoint__property,axiom,
! [T2: set_real,I: real > real,S: set_real,R: real > real,G2: real > real] :
( ( topolo5044208981011980120l_real @ T2 @ I )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ I @ T2 ) @ S )
=> ( ( topolo5044208981011980120l_real @ S @ R )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ R @ S ) @ T2 )
=> ( ! [Y2: real] :
( ( member_real @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F3: real > real] :
( ( topolo5044208981011980120l_real @ S @ F3 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ F3 @ S ) @ S )
=> ? [X2: real] :
( ( member_real @ X2 @ S )
& ( ( F3 @ X2 )
= X2 ) ) ) )
=> ( ( topolo5044208981011980120l_real @ T2 @ G2 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ G2 @ T2 ) @ T2 )
=> ~ ! [Y2: real] :
( ( member_real @ Y2 @ T2 )
=> ( ( G2 @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_480_mono__on__subset,axiom,
! [A4: set_real,F: real > real,B4: set_real] :
( ( monoto4017252874604999745l_real @ A4 @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_real @ B4 @ A4 )
=> ( monoto4017252874604999745l_real @ B4 @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ) ).
% mono_on_subset
thf(fact_481_mono__on__subset,axiom,
! [A4: set_real,F: real > nat,B4: set_real] :
( ( monotone_on_real_nat @ A4 @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_real @ B4 @ A4 )
=> ( monotone_on_real_nat @ B4 @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_482_mono__on__subset,axiom,
! [A4: set_real,F: real > int,B4: set_real] :
( ( monotone_on_real_int @ A4 @ ord_less_eq_real @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_set_real @ B4 @ A4 )
=> ( monotone_on_real_int @ B4 @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ) ).
% mono_on_subset
thf(fact_483_mono__on__subset,axiom,
! [A4: set_nat,F: nat > real,B4: set_nat] :
( ( monotone_on_nat_real @ A4 @ ord_less_eq_nat @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( monotone_on_nat_real @ B4 @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ) ).
% mono_on_subset
thf(fact_484_mono__on__subset,axiom,
! [A4: set_nat,F: nat > nat,B4: set_nat] :
( ( monotone_on_nat_nat @ A4 @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( monotone_on_nat_nat @ B4 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_485_mono__on__subset,axiom,
! [A4: set_nat,F: nat > int,B4: set_nat] :
( ( monotone_on_nat_int @ A4 @ ord_less_eq_nat @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( monotone_on_nat_int @ B4 @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ) ).
% mono_on_subset
thf(fact_486_mono__on__subset,axiom,
! [A4: set_int,F: int > real,B4: set_int] :
( ( monotone_on_int_real @ A4 @ ord_less_eq_int @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_int @ B4 @ A4 )
=> ( monotone_on_int_real @ B4 @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ) ).
% mono_on_subset
thf(fact_487_mono__on__subset,axiom,
! [A4: set_int,F: int > nat,B4: set_int] :
( ( monotone_on_int_nat @ A4 @ ord_less_eq_int @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_int @ B4 @ A4 )
=> ( monotone_on_int_nat @ B4 @ ord_less_eq_int @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_488_mono__on__subset,axiom,
! [A4: set_int,F: int > int,B4: set_int] :
( ( monotone_on_int_int @ A4 @ ord_less_eq_int @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_set_int @ B4 @ A4 )
=> ( monotone_on_int_int @ B4 @ ord_less_eq_int @ ord_less_eq_int @ F ) ) ) ).
% mono_on_subset
thf(fact_489_mono__onI,axiom,
! [A4: set_real,F: real > real] :
( ! [R2: real,S2: real] :
( ( member_real @ R2 @ A4 )
=> ( ( member_real @ S2 @ A4 )
=> ( ( ord_less_eq_real @ R2 @ S2 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A4 @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_490_mono__onI,axiom,
! [A4: set_real,F: real > nat] :
( ! [R2: real,S2: real] :
( ( member_real @ R2 @ A4 )
=> ( ( member_real @ S2 @ A4 )
=> ( ( ord_less_eq_real @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_real_nat @ A4 @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_491_mono__onI,axiom,
! [A4: set_real,F: real > int] :
( ! [R2: real,S2: real] :
( ( member_real @ R2 @ A4 )
=> ( ( member_real @ S2 @ A4 )
=> ( ( ord_less_eq_real @ R2 @ S2 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_real_int @ A4 @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_492_mono__onI,axiom,
! [A4: set_nat,F: nat > real] :
( ! [R2: nat,S2: nat] :
( ( member_nat @ R2 @ A4 )
=> ( ( member_nat @ S2 @ A4 )
=> ( ( ord_less_eq_nat @ R2 @ S2 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_nat_real @ A4 @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_493_mono__onI,axiom,
! [A4: set_nat,F: nat > nat] :
( ! [R2: nat,S2: nat] :
( ( member_nat @ R2 @ A4 )
=> ( ( member_nat @ S2 @ A4 )
=> ( ( ord_less_eq_nat @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_nat_nat @ A4 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_494_mono__onI,axiom,
! [A4: set_nat,F: nat > int] :
( ! [R2: nat,S2: nat] :
( ( member_nat @ R2 @ A4 )
=> ( ( member_nat @ S2 @ A4 )
=> ( ( ord_less_eq_nat @ R2 @ S2 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_nat_int @ A4 @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_495_mono__onI,axiom,
! [A4: set_int,F: int > real] :
( ! [R2: int,S2: int] :
( ( member_int @ R2 @ A4 )
=> ( ( member_int @ S2 @ A4 )
=> ( ( ord_less_eq_int @ R2 @ S2 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_int_real @ A4 @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_496_mono__onI,axiom,
! [A4: set_int,F: int > nat] :
( ! [R2: int,S2: int] :
( ( member_int @ R2 @ A4 )
=> ( ( member_int @ S2 @ A4 )
=> ( ( ord_less_eq_int @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_int_nat @ A4 @ ord_less_eq_int @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_497_mono__onI,axiom,
! [A4: set_int,F: int > int] :
( ! [R2: int,S2: int] :
( ( member_int @ R2 @ A4 )
=> ( ( member_int @ S2 @ A4 )
=> ( ( ord_less_eq_int @ R2 @ S2 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_int_int @ A4 @ ord_less_eq_int @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_498_mono__onD,axiom,
! [A4: set_real,F: real > real,R: real,S3: real] :
( ( monoto4017252874604999745l_real @ A4 @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( member_real @ R @ A4 )
=> ( ( member_real @ S3 @ A4 )
=> ( ( ord_less_eq_real @ R @ S3 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_499_mono__onD,axiom,
! [A4: set_real,F: real > nat,R: real,S3: real] :
( ( monotone_on_real_nat @ A4 @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( member_real @ R @ A4 )
=> ( ( member_real @ S3 @ A4 )
=> ( ( ord_less_eq_real @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_500_mono__onD,axiom,
! [A4: set_real,F: real > int,R: real,S3: real] :
( ( monotone_on_real_int @ A4 @ ord_less_eq_real @ ord_less_eq_int @ F )
=> ( ( member_real @ R @ A4 )
=> ( ( member_real @ S3 @ A4 )
=> ( ( ord_less_eq_real @ R @ S3 )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_501_mono__onD,axiom,
! [A4: set_nat,F: nat > real,R: nat,S3: nat] :
( ( monotone_on_nat_real @ A4 @ ord_less_eq_nat @ ord_less_eq_real @ F )
=> ( ( member_nat @ R @ A4 )
=> ( ( member_nat @ S3 @ A4 )
=> ( ( ord_less_eq_nat @ R @ S3 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_502_mono__onD,axiom,
! [A4: set_nat,F: nat > nat,R: nat,S3: nat] :
( ( monotone_on_nat_nat @ A4 @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R @ A4 )
=> ( ( member_nat @ S3 @ A4 )
=> ( ( ord_less_eq_nat @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_503_mono__onD,axiom,
! [A4: set_nat,F: nat > int,R: nat,S3: nat] :
( ( monotone_on_nat_int @ A4 @ ord_less_eq_nat @ ord_less_eq_int @ F )
=> ( ( member_nat @ R @ A4 )
=> ( ( member_nat @ S3 @ A4 )
=> ( ( ord_less_eq_nat @ R @ S3 )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_504_mono__onD,axiom,
! [A4: set_int,F: int > real,R: int,S3: int] :
( ( monotone_on_int_real @ A4 @ ord_less_eq_int @ ord_less_eq_real @ F )
=> ( ( member_int @ R @ A4 )
=> ( ( member_int @ S3 @ A4 )
=> ( ( ord_less_eq_int @ R @ S3 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_505_mono__onD,axiom,
! [A4: set_int,F: int > nat,R: int,S3: int] :
( ( monotone_on_int_nat @ A4 @ ord_less_eq_int @ ord_less_eq_nat @ F )
=> ( ( member_int @ R @ A4 )
=> ( ( member_int @ S3 @ A4 )
=> ( ( ord_less_eq_int @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_506_mono__onD,axiom,
! [A4: set_int,F: int > int,R: int,S3: int] :
( ( monotone_on_int_int @ A4 @ ord_less_eq_int @ ord_less_eq_int @ F )
=> ( ( member_int @ R @ A4 )
=> ( ( member_int @ S3 @ A4 )
=> ( ( ord_less_eq_int @ R @ S3 )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_507_ord_Omono__on__subset,axiom,
! [A4: set_real,Less_eq: real > real > $o,F: real > real,B4: set_real] :
( ( monoto4017252874604999745l_real @ A4 @ Less_eq @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_real @ B4 @ A4 )
=> ( monoto4017252874604999745l_real @ B4 @ Less_eq @ ord_less_eq_real @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_508_ord_Omono__on__subset,axiom,
! [A4: set_nat,Less_eq: nat > nat > $o,F: nat > nat,B4: set_nat] :
( ( monotone_on_nat_nat @ A4 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( monotone_on_nat_nat @ B4 @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_509_ord_Omono__on__def,axiom,
! [A4: set_int,Less_eq: int > int > $o,F: int > real] :
( ( monotone_on_int_real @ A4 @ Less_eq @ ord_less_eq_real @ F )
= ( ! [R3: int,S4: int] :
( ( ( member_int @ R3 @ A4 )
& ( member_int @ S4 @ A4 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_510_ord_Omono__on__def,axiom,
! [A4: set_real,Less_eq: real > real > $o,F: real > real] :
( ( monoto4017252874604999745l_real @ A4 @ Less_eq @ ord_less_eq_real @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A4 )
& ( member_real @ S4 @ A4 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_511_ord_Omono__on__def,axiom,
! [A4: set_real,Less_eq: real > real > $o,F: real > nat] :
( ( monotone_on_real_nat @ A4 @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A4 )
& ( member_real @ S4 @ A4 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_512_ord_Omono__on__def,axiom,
! [A4: set_int,Less_eq: int > int > $o,F: int > nat] :
( ( monotone_on_int_nat @ A4 @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: int,S4: int] :
( ( ( member_int @ R3 @ A4 )
& ( member_int @ S4 @ A4 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_513_ord_Omono__on__def,axiom,
! [A4: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A4 @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A4 )
& ( member_nat @ S4 @ A4 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_514_ord_Omono__on__def,axiom,
! [A4: set_real,Less_eq: real > real > $o,F: real > int] :
( ( monotone_on_real_int @ A4 @ Less_eq @ ord_less_eq_int @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A4 )
& ( member_real @ S4 @ A4 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_int @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_515_ord_Omono__on__def,axiom,
! [A4: set_int,Less_eq: int > int > $o,F: int > int] :
( ( monotone_on_int_int @ A4 @ Less_eq @ ord_less_eq_int @ F )
= ( ! [R3: int,S4: int] :
( ( ( member_int @ R3 @ A4 )
& ( member_int @ S4 @ A4 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_int @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_516_ord_Omono__onI,axiom,
! [A4: set_int,Less_eq: int > int > $o,F: int > real] :
( ! [R2: int,S2: int] :
( ( member_int @ R2 @ A4 )
=> ( ( member_int @ S2 @ A4 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_int_real @ A4 @ Less_eq @ ord_less_eq_real @ F ) ) ).
% ord.mono_onI
thf(fact_517_ord_Omono__onI,axiom,
! [A4: set_real,Less_eq: real > real > $o,F: real > real] :
( ! [R2: real,S2: real] :
( ( member_real @ R2 @ A4 )
=> ( ( member_real @ S2 @ A4 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A4 @ Less_eq @ ord_less_eq_real @ F ) ) ).
% ord.mono_onI
thf(fact_518_ord_Omono__onI,axiom,
! [A4: set_real,Less_eq: real > real > $o,F: real > nat] :
( ! [R2: real,S2: real] :
( ( member_real @ R2 @ A4 )
=> ( ( member_real @ S2 @ A4 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_real_nat @ A4 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_519_ord_Omono__onI,axiom,
! [A4: set_int,Less_eq: int > int > $o,F: int > nat] :
( ! [R2: int,S2: int] :
( ( member_int @ R2 @ A4 )
=> ( ( member_int @ S2 @ A4 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_int_nat @ A4 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_520_ord_Omono__onI,axiom,
! [A4: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ! [R2: nat,S2: nat] :
( ( member_nat @ R2 @ A4 )
=> ( ( member_nat @ S2 @ A4 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_nat_nat @ A4 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_521_ord_Omono__onI,axiom,
! [A4: set_real,Less_eq: real > real > $o,F: real > int] :
( ! [R2: real,S2: real] :
( ( member_real @ R2 @ A4 )
=> ( ( member_real @ S2 @ A4 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_real_int @ A4 @ Less_eq @ ord_less_eq_int @ F ) ) ).
% ord.mono_onI
thf(fact_522_ord_Omono__onI,axiom,
! [A4: set_int,Less_eq: int > int > $o,F: int > int] :
( ! [R2: int,S2: int] :
( ( member_int @ R2 @ A4 )
=> ( ( member_int @ S2 @ A4 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_int_int @ A4 @ Less_eq @ ord_less_eq_int @ F ) ) ).
% ord.mono_onI
thf(fact_523_ord_Omono__onD,axiom,
! [A4: set_int,Less_eq: int > int > $o,F: int > real,R: int,S3: int] :
( ( monotone_on_int_real @ A4 @ Less_eq @ ord_less_eq_real @ F )
=> ( ( member_int @ R @ A4 )
=> ( ( member_int @ S3 @ A4 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_524_ord_Omono__onD,axiom,
! [A4: set_real,Less_eq: real > real > $o,F: real > real,R: real,S3: real] :
( ( monoto4017252874604999745l_real @ A4 @ Less_eq @ ord_less_eq_real @ F )
=> ( ( member_real @ R @ A4 )
=> ( ( member_real @ S3 @ A4 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_525_ord_Omono__onD,axiom,
! [A4: set_real,Less_eq: real > real > $o,F: real > nat,R: real,S3: real] :
( ( monotone_on_real_nat @ A4 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_real @ R @ A4 )
=> ( ( member_real @ S3 @ A4 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_526_ord_Omono__onD,axiom,
! [A4: set_int,Less_eq: int > int > $o,F: int > nat,R: int,S3: int] :
( ( monotone_on_int_nat @ A4 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_int @ R @ A4 )
=> ( ( member_int @ S3 @ A4 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_527_ord_Omono__onD,axiom,
! [A4: set_nat,Less_eq: nat > nat > $o,F: nat > nat,R: nat,S3: nat] :
( ( monotone_on_nat_nat @ A4 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R @ A4 )
=> ( ( member_nat @ S3 @ A4 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_528_ord_Omono__onD,axiom,
! [A4: set_real,Less_eq: real > real > $o,F: real > int,R: real,S3: real] :
( ( monotone_on_real_int @ A4 @ Less_eq @ ord_less_eq_int @ F )
=> ( ( member_real @ R @ A4 )
=> ( ( member_real @ S3 @ A4 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_529_ord_Omono__onD,axiom,
! [A4: set_int,Less_eq: int > int > $o,F: int > int,R: int,S3: int] :
( ( monotone_on_int_int @ A4 @ Less_eq @ ord_less_eq_int @ F )
=> ( ( member_int @ R @ A4 )
=> ( ( member_int @ S3 @ A4 )
=> ( ( Less_eq @ R @ S3 )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_530_ord_Ostrict__mono__onD,axiom,
! [A4: set_int,Less: int > int > $o,F: int > real,R: int,S3: int] :
( ( monotone_on_int_real @ A4 @ Less @ ord_less_real @ F )
=> ( ( member_int @ R @ A4 )
=> ( ( member_int @ S3 @ A4 )
=> ( ( Less @ R @ S3 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_531_ord_Ostrict__mono__onD,axiom,
! [A4: set_real,Less: real > real > $o,F: real > real,R: real,S3: real] :
( ( monoto4017252874604999745l_real @ A4 @ Less @ ord_less_real @ F )
=> ( ( member_real @ R @ A4 )
=> ( ( member_real @ S3 @ A4 )
=> ( ( Less @ R @ S3 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_532_ord_Ostrict__mono__onD,axiom,
! [A4: set_real,Less: real > real > $o,F: real > nat,R: real,S3: real] :
( ( monotone_on_real_nat @ A4 @ Less @ ord_less_nat @ F )
=> ( ( member_real @ R @ A4 )
=> ( ( member_real @ S3 @ A4 )
=> ( ( Less @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_533_ord_Ostrict__mono__onD,axiom,
! [A4: set_int,Less: int > int > $o,F: int > nat,R: int,S3: int] :
( ( monotone_on_int_nat @ A4 @ Less @ ord_less_nat @ F )
=> ( ( member_int @ R @ A4 )
=> ( ( member_int @ S3 @ A4 )
=> ( ( Less @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_534_ord_Ostrict__mono__onD,axiom,
! [A4: set_nat,Less: nat > nat > $o,F: nat > nat,R: nat,S3: nat] :
( ( monotone_on_nat_nat @ A4 @ Less @ ord_less_nat @ F )
=> ( ( member_nat @ R @ A4 )
=> ( ( member_nat @ S3 @ A4 )
=> ( ( Less @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_535_ord_Ostrict__mono__onD,axiom,
! [A4: set_real,Less: real > real > $o,F: real > int,R: real,S3: real] :
( ( monotone_on_real_int @ A4 @ Less @ ord_less_int @ F )
=> ( ( member_real @ R @ A4 )
=> ( ( member_real @ S3 @ A4 )
=> ( ( Less @ R @ S3 )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_536_ord_Ostrict__mono__onD,axiom,
! [A4: set_int,Less: int > int > $o,F: int > int,R: int,S3: int] :
( ( monotone_on_int_int @ A4 @ Less @ ord_less_int @ F )
=> ( ( member_int @ R @ A4 )
=> ( ( member_int @ S3 @ A4 )
=> ( ( Less @ R @ S3 )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_537_ord_Ostrict__mono__onI,axiom,
! [A4: set_int,Less: int > int > $o,F: int > real] :
( ! [R2: int,S2: int] :
( ( member_int @ R2 @ A4 )
=> ( ( member_int @ S2 @ A4 )
=> ( ( Less @ R2 @ S2 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_int_real @ A4 @ Less @ ord_less_real @ F ) ) ).
% ord.strict_mono_onI
thf(fact_538_ord_Ostrict__mono__onI,axiom,
! [A4: set_real,Less: real > real > $o,F: real > real] :
( ! [R2: real,S2: real] :
( ( member_real @ R2 @ A4 )
=> ( ( member_real @ S2 @ A4 )
=> ( ( Less @ R2 @ S2 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A4 @ Less @ ord_less_real @ F ) ) ).
% ord.strict_mono_onI
thf(fact_539_ord_Ostrict__mono__onI,axiom,
! [A4: set_real,Less: real > real > $o,F: real > nat] :
( ! [R2: real,S2: real] :
( ( member_real @ R2 @ A4 )
=> ( ( member_real @ S2 @ A4 )
=> ( ( Less @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_real_nat @ A4 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_540_ord_Ostrict__mono__onI,axiom,
! [A4: set_int,Less: int > int > $o,F: int > nat] :
( ! [R2: int,S2: int] :
( ( member_int @ R2 @ A4 )
=> ( ( member_int @ S2 @ A4 )
=> ( ( Less @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_int_nat @ A4 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_541_ord_Ostrict__mono__onI,axiom,
! [A4: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ! [R2: nat,S2: nat] :
( ( member_nat @ R2 @ A4 )
=> ( ( member_nat @ S2 @ A4 )
=> ( ( Less @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_nat_nat @ A4 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_542_ord_Ostrict__mono__onI,axiom,
! [A4: set_real,Less: real > real > $o,F: real > int] :
( ! [R2: real,S2: real] :
( ( member_real @ R2 @ A4 )
=> ( ( member_real @ S2 @ A4 )
=> ( ( Less @ R2 @ S2 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_real_int @ A4 @ Less @ ord_less_int @ F ) ) ).
% ord.strict_mono_onI
thf(fact_543_ord_Ostrict__mono__onI,axiom,
! [A4: set_int,Less: int > int > $o,F: int > int] :
( ! [R2: int,S2: int] :
( ( member_int @ R2 @ A4 )
=> ( ( member_int @ S2 @ A4 )
=> ( ( Less @ R2 @ S2 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_int_int @ A4 @ Less @ ord_less_int @ F ) ) ).
% ord.strict_mono_onI
thf(fact_544_ord_Ostrict__mono__on__def,axiom,
! [A4: set_int,Less: int > int > $o,F: int > real] :
( ( monotone_on_int_real @ A4 @ Less @ ord_less_real @ F )
= ( ! [R3: int,S4: int] :
( ( ( member_int @ R3 @ A4 )
& ( member_int @ S4 @ A4 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_545_ord_Ostrict__mono__on__def,axiom,
! [A4: set_real,Less: real > real > $o,F: real > real] :
( ( monoto4017252874604999745l_real @ A4 @ Less @ ord_less_real @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A4 )
& ( member_real @ S4 @ A4 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_546_ord_Ostrict__mono__on__def,axiom,
! [A4: set_real,Less: real > real > $o,F: real > nat] :
( ( monotone_on_real_nat @ A4 @ Less @ ord_less_nat @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A4 )
& ( member_real @ S4 @ A4 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_547_ord_Ostrict__mono__on__def,axiom,
! [A4: set_int,Less: int > int > $o,F: int > nat] :
( ( monotone_on_int_nat @ A4 @ Less @ ord_less_nat @ F )
= ( ! [R3: int,S4: int] :
( ( ( member_int @ R3 @ A4 )
& ( member_int @ S4 @ A4 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_548_ord_Ostrict__mono__on__def,axiom,
! [A4: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A4 @ Less @ ord_less_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A4 )
& ( member_nat @ S4 @ A4 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_549_ord_Ostrict__mono__on__def,axiom,
! [A4: set_real,Less: real > real > $o,F: real > int] :
( ( monotone_on_real_int @ A4 @ Less @ ord_less_int @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A4 )
& ( member_real @ S4 @ A4 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_int @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_550_ord_Ostrict__mono__on__def,axiom,
! [A4: set_int,Less: int > int > $o,F: int > int] :
( ( monotone_on_int_int @ A4 @ Less @ ord_less_int @ F )
= ( ! [R3: int,S4: int] :
( ( ( member_int @ R3 @ A4 )
& ( member_int @ S4 @ A4 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_int @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_551_strict__mono__onD,axiom,
! [A4: set_real,F: real > real,R: real,S3: real] :
( ( monoto4017252874604999745l_real @ A4 @ ord_less_real @ ord_less_real @ F )
=> ( ( member_real @ R @ A4 )
=> ( ( member_real @ S3 @ A4 )
=> ( ( ord_less_real @ R @ S3 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_552_strict__mono__onD,axiom,
! [A4: set_real,F: real > nat,R: real,S3: real] :
( ( monotone_on_real_nat @ A4 @ ord_less_real @ ord_less_nat @ F )
=> ( ( member_real @ R @ A4 )
=> ( ( member_real @ S3 @ A4 )
=> ( ( ord_less_real @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_553_strict__mono__onD,axiom,
! [A4: set_real,F: real > int,R: real,S3: real] :
( ( monotone_on_real_int @ A4 @ ord_less_real @ ord_less_int @ F )
=> ( ( member_real @ R @ A4 )
=> ( ( member_real @ S3 @ A4 )
=> ( ( ord_less_real @ R @ S3 )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_554_strict__mono__onD,axiom,
! [A4: set_nat,F: nat > real,R: nat,S3: nat] :
( ( monotone_on_nat_real @ A4 @ ord_less_nat @ ord_less_real @ F )
=> ( ( member_nat @ R @ A4 )
=> ( ( member_nat @ S3 @ A4 )
=> ( ( ord_less_nat @ R @ S3 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_555_strict__mono__onD,axiom,
! [A4: set_nat,F: nat > nat,R: nat,S3: nat] :
( ( monotone_on_nat_nat @ A4 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ R @ A4 )
=> ( ( member_nat @ S3 @ A4 )
=> ( ( ord_less_nat @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_556_strict__mono__onD,axiom,
! [A4: set_nat,F: nat > int,R: nat,S3: nat] :
( ( monotone_on_nat_int @ A4 @ ord_less_nat @ ord_less_int @ F )
=> ( ( member_nat @ R @ A4 )
=> ( ( member_nat @ S3 @ A4 )
=> ( ( ord_less_nat @ R @ S3 )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_557_strict__mono__onD,axiom,
! [A4: set_int,F: int > real,R: int,S3: int] :
( ( monotone_on_int_real @ A4 @ ord_less_int @ ord_less_real @ F )
=> ( ( member_int @ R @ A4 )
=> ( ( member_int @ S3 @ A4 )
=> ( ( ord_less_int @ R @ S3 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_558_strict__mono__onD,axiom,
! [A4: set_int,F: int > nat,R: int,S3: int] :
( ( monotone_on_int_nat @ A4 @ ord_less_int @ ord_less_nat @ F )
=> ( ( member_int @ R @ A4 )
=> ( ( member_int @ S3 @ A4 )
=> ( ( ord_less_int @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_559_strict__mono__onD,axiom,
! [A4: set_int,F: int > int,R: int,S3: int] :
( ( monotone_on_int_int @ A4 @ ord_less_int @ ord_less_int @ F )
=> ( ( member_int @ R @ A4 )
=> ( ( member_int @ S3 @ A4 )
=> ( ( ord_less_int @ R @ S3 )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_560_strict__mono__onI,axiom,
! [A4: set_real,F: real > real] :
( ! [R2: real,S2: real] :
( ( member_real @ R2 @ A4 )
=> ( ( member_real @ S2 @ A4 )
=> ( ( ord_less_real @ R2 @ S2 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A4 @ ord_less_real @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_561_strict__mono__onI,axiom,
! [A4: set_real,F: real > nat] :
( ! [R2: real,S2: real] :
( ( member_real @ R2 @ A4 )
=> ( ( member_real @ S2 @ A4 )
=> ( ( ord_less_real @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_real_nat @ A4 @ ord_less_real @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_562_strict__mono__onI,axiom,
! [A4: set_real,F: real > int] :
( ! [R2: real,S2: real] :
( ( member_real @ R2 @ A4 )
=> ( ( member_real @ S2 @ A4 )
=> ( ( ord_less_real @ R2 @ S2 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_real_int @ A4 @ ord_less_real @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_563_strict__mono__onI,axiom,
! [A4: set_nat,F: nat > real] :
( ! [R2: nat,S2: nat] :
( ( member_nat @ R2 @ A4 )
=> ( ( member_nat @ S2 @ A4 )
=> ( ( ord_less_nat @ R2 @ S2 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_nat_real @ A4 @ ord_less_nat @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_564_strict__mono__onI,axiom,
! [A4: set_nat,F: nat > nat] :
( ! [R2: nat,S2: nat] :
( ( member_nat @ R2 @ A4 )
=> ( ( member_nat @ S2 @ A4 )
=> ( ( ord_less_nat @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_nat_nat @ A4 @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_565_strict__mono__onI,axiom,
! [A4: set_nat,F: nat > int] :
( ! [R2: nat,S2: nat] :
( ( member_nat @ R2 @ A4 )
=> ( ( member_nat @ S2 @ A4 )
=> ( ( ord_less_nat @ R2 @ S2 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_nat_int @ A4 @ ord_less_nat @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_566_strict__mono__onI,axiom,
! [A4: set_int,F: int > real] :
( ! [R2: int,S2: int] :
( ( member_int @ R2 @ A4 )
=> ( ( member_int @ S2 @ A4 )
=> ( ( ord_less_int @ R2 @ S2 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_int_real @ A4 @ ord_less_int @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_567_strict__mono__onI,axiom,
! [A4: set_int,F: int > nat] :
( ! [R2: int,S2: int] :
( ( member_int @ R2 @ A4 )
=> ( ( member_int @ S2 @ A4 )
=> ( ( ord_less_int @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_int_nat @ A4 @ ord_less_int @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_568_strict__mono__onI,axiom,
! [A4: set_int,F: int > int] :
( ! [R2: int,S2: int] :
( ( member_int @ R2 @ A4 )
=> ( ( member_int @ S2 @ A4 )
=> ( ( ord_less_int @ R2 @ S2 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_int_int @ A4 @ ord_less_int @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_569_strict__mono__on__eqD,axiom,
! [A4: set_real,F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ A4 @ ord_less_real @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_real @ X @ A4 )
=> ( ( member_real @ Y @ A4 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_570_strict__mono__on__eqD,axiom,
! [A4: set_real,F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ A4 @ ord_less_real @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_real @ X @ A4 )
=> ( ( member_real @ Y @ A4 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_571_strict__mono__on__eqD,axiom,
! [A4: set_real,F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ A4 @ ord_less_real @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_real @ X @ A4 )
=> ( ( member_real @ Y @ A4 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_572_strict__mono__on__eqD,axiom,
! [A4: set_nat,F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ A4 @ ord_less_nat @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A4 )
=> ( ( member_nat @ Y @ A4 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_573_strict__mono__on__eqD,axiom,
! [A4: set_nat,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A4 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A4 )
=> ( ( member_nat @ Y @ A4 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_574_strict__mono__on__eqD,axiom,
! [A4: set_nat,F: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ A4 @ ord_less_nat @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A4 )
=> ( ( member_nat @ Y @ A4 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_575_strict__mono__on__eqD,axiom,
! [A4: set_int,F: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ A4 @ ord_less_int @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_int @ X @ A4 )
=> ( ( member_int @ Y @ A4 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_576_strict__mono__on__eqD,axiom,
! [A4: set_int,F: int > nat,X: int,Y: int] :
( ( monotone_on_int_nat @ A4 @ ord_less_int @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_int @ X @ A4 )
=> ( ( member_int @ Y @ A4 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_577_strict__mono__on__eqD,axiom,
! [A4: set_int,F: int > int,X: int,Y: int] :
( ( monotone_on_int_int @ A4 @ ord_less_int @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_int @ X @ A4 )
=> ( ( member_int @ Y @ A4 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_578_strict__mono__image__endpoints,axiom,
! [A: real,B: real,F: real > real] :
( ( monoto4017252874604999745l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ord_less_real @ ord_less_real @ F )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% strict_mono_image_endpoints
thf(fact_579_strict__mono__image__endpoints,axiom,
! [A: real,B: real,F: real > nat] :
( ( monotone_on_real_nat @ ( set_or1222579329274155063t_real @ A @ B ) @ ord_less_real @ ord_less_nat @ F )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( image_real_nat @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1269000886237332187st_nat @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% strict_mono_image_endpoints
thf(fact_580_strict__mono__image__endpoints,axiom,
! [A: real,B: real,F: real > int] :
( ( monotone_on_real_int @ ( set_or1222579329274155063t_real @ A @ B ) @ ord_less_real @ ord_less_int @ F )
=> ( ( topolo2284712892409288920al_int @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( image_real_int @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1266510415728281911st_int @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% strict_mono_image_endpoints
thf(fact_581_strict__mono__on__imp__mono__on,axiom,
! [A4: set_real,F: real > real] :
( ( monoto4017252874604999745l_real @ A4 @ ord_less_real @ ord_less_real @ F )
=> ( monoto4017252874604999745l_real @ A4 @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_582_strict__mono__on__imp__mono__on,axiom,
! [A4: set_real,F: real > nat] :
( ( monotone_on_real_nat @ A4 @ ord_less_real @ ord_less_nat @ F )
=> ( monotone_on_real_nat @ A4 @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_583_strict__mono__on__imp__mono__on,axiom,
! [A4: set_real,F: real > int] :
( ( monotone_on_real_int @ A4 @ ord_less_real @ ord_less_int @ F )
=> ( monotone_on_real_int @ A4 @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_584_strict__mono__on__imp__mono__on,axiom,
! [A4: set_nat,F: nat > real] :
( ( monotone_on_nat_real @ A4 @ ord_less_nat @ ord_less_real @ F )
=> ( monotone_on_nat_real @ A4 @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_585_strict__mono__on__imp__mono__on,axiom,
! [A4: set_nat,F: nat > nat] :
( ( monotone_on_nat_nat @ A4 @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ A4 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_586_strict__mono__on__imp__mono__on,axiom,
! [A4: set_nat,F: nat > int] :
( ( monotone_on_nat_int @ A4 @ ord_less_nat @ ord_less_int @ F )
=> ( monotone_on_nat_int @ A4 @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_587_strict__mono__on__imp__mono__on,axiom,
! [A4: set_int,F: int > real] :
( ( monotone_on_int_real @ A4 @ ord_less_int @ ord_less_real @ F )
=> ( monotone_on_int_real @ A4 @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_588_strict__mono__on__imp__mono__on,axiom,
! [A4: set_int,F: int > nat] :
( ( monotone_on_int_nat @ A4 @ ord_less_int @ ord_less_nat @ F )
=> ( monotone_on_int_nat @ A4 @ ord_less_eq_int @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_589_strict__mono__on__imp__mono__on,axiom,
! [A4: set_int,F: int > int] :
( ( monotone_on_int_int @ A4 @ ord_less_int @ ord_less_int @ F )
=> ( monotone_on_int_int @ A4 @ ord_less_eq_int @ ord_less_eq_int @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_590_strict__mono__on__leD,axiom,
! [A4: set_real,F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ A4 @ ord_less_real @ ord_less_real @ F )
=> ( ( member_real @ X @ A4 )
=> ( ( member_real @ Y @ A4 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_591_strict__mono__on__leD,axiom,
! [A4: set_real,F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ A4 @ ord_less_real @ ord_less_nat @ F )
=> ( ( member_real @ X @ A4 )
=> ( ( member_real @ Y @ A4 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_592_strict__mono__on__leD,axiom,
! [A4: set_real,F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ A4 @ ord_less_real @ ord_less_int @ F )
=> ( ( member_real @ X @ A4 )
=> ( ( member_real @ Y @ A4 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_593_strict__mono__on__leD,axiom,
! [A4: set_nat,F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ A4 @ ord_less_nat @ ord_less_real @ F )
=> ( ( member_nat @ X @ A4 )
=> ( ( member_nat @ Y @ A4 )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_594_strict__mono__on__leD,axiom,
! [A4: set_nat,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A4 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ X @ A4 )
=> ( ( member_nat @ Y @ A4 )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_595_strict__mono__on__leD,axiom,
! [A4: set_nat,F: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ A4 @ ord_less_nat @ ord_less_int @ F )
=> ( ( member_nat @ X @ A4 )
=> ( ( member_nat @ Y @ A4 )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_596_strict__mono__on__leD,axiom,
! [A4: set_int,F: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ A4 @ ord_less_int @ ord_less_real @ F )
=> ( ( member_int @ X @ A4 )
=> ( ( member_int @ Y @ A4 )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_597_strict__mono__on__leD,axiom,
! [A4: set_int,F: int > nat,X: int,Y: int] :
( ( monotone_on_int_nat @ A4 @ ord_less_int @ ord_less_nat @ F )
=> ( ( member_int @ X @ A4 )
=> ( ( member_int @ Y @ A4 )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_598_strict__mono__on__leD,axiom,
! [A4: set_int,F: int > int,X: int,Y: int] :
( ( monotone_on_int_int @ A4 @ ord_less_int @ ord_less_int @ F )
=> ( ( member_int @ X @ A4 )
=> ( ( member_int @ Y @ A4 )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_599_mono__on__greaterD,axiom,
! [A4: set_real,G2: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ A4 @ ord_less_eq_real @ ord_less_eq_real @ G2 )
=> ( ( member_real @ X @ A4 )
=> ( ( member_real @ Y @ A4 )
=> ( ( ord_less_real @ ( G2 @ Y ) @ ( G2 @ X ) )
=> ( ord_less_real @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_600_mono__on__greaterD,axiom,
! [A4: set_real,G2: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ A4 @ ord_less_eq_real @ ord_less_eq_nat @ G2 )
=> ( ( member_real @ X @ A4 )
=> ( ( member_real @ Y @ A4 )
=> ( ( ord_less_nat @ ( G2 @ Y ) @ ( G2 @ X ) )
=> ( ord_less_real @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_601_mono__on__greaterD,axiom,
! [A4: set_real,G2: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ A4 @ ord_less_eq_real @ ord_less_eq_int @ G2 )
=> ( ( member_real @ X @ A4 )
=> ( ( member_real @ Y @ A4 )
=> ( ( ord_less_int @ ( G2 @ Y ) @ ( G2 @ X ) )
=> ( ord_less_real @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_602_mono__on__greaterD,axiom,
! [A4: set_nat,G2: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ A4 @ ord_less_eq_nat @ ord_less_eq_real @ G2 )
=> ( ( member_nat @ X @ A4 )
=> ( ( member_nat @ Y @ A4 )
=> ( ( ord_less_real @ ( G2 @ Y ) @ ( G2 @ X ) )
=> ( ord_less_nat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_603_mono__on__greaterD,axiom,
! [A4: set_nat,G2: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A4 @ ord_less_eq_nat @ ord_less_eq_nat @ G2 )
=> ( ( member_nat @ X @ A4 )
=> ( ( member_nat @ Y @ A4 )
=> ( ( ord_less_nat @ ( G2 @ Y ) @ ( G2 @ X ) )
=> ( ord_less_nat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_604_mono__on__greaterD,axiom,
! [A4: set_nat,G2: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ A4 @ ord_less_eq_nat @ ord_less_eq_int @ G2 )
=> ( ( member_nat @ X @ A4 )
=> ( ( member_nat @ Y @ A4 )
=> ( ( ord_less_int @ ( G2 @ Y ) @ ( G2 @ X ) )
=> ( ord_less_nat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_605_mono__on__greaterD,axiom,
! [A4: set_int,G2: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ A4 @ ord_less_eq_int @ ord_less_eq_real @ G2 )
=> ( ( member_int @ X @ A4 )
=> ( ( member_int @ Y @ A4 )
=> ( ( ord_less_real @ ( G2 @ Y ) @ ( G2 @ X ) )
=> ( ord_less_int @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_606_mono__on__greaterD,axiom,
! [A4: set_int,G2: int > nat,X: int,Y: int] :
( ( monotone_on_int_nat @ A4 @ ord_less_eq_int @ ord_less_eq_nat @ G2 )
=> ( ( member_int @ X @ A4 )
=> ( ( member_int @ Y @ A4 )
=> ( ( ord_less_nat @ ( G2 @ Y ) @ ( G2 @ X ) )
=> ( ord_less_int @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_607_mono__on__greaterD,axiom,
! [A4: set_int,G2: int > int,X: int,Y: int] :
( ( monotone_on_int_int @ A4 @ ord_less_eq_int @ ord_less_eq_int @ G2 )
=> ( ( member_int @ X @ A4 )
=> ( ( member_int @ Y @ A4 )
=> ( ( ord_less_int @ ( G2 @ Y ) @ ( G2 @ X ) )
=> ( ord_less_int @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_608_monotone__on__o,axiom,
! [A4: set_int,Orda: int > int > $o,Ordb: nat > nat > $o,F: int > nat,B4: set_nat,Ordc: nat > nat > $o,G2: nat > int] :
( ( monotone_on_int_nat @ A4 @ Orda @ Ordb @ F )
=> ( ( monotone_on_nat_int @ B4 @ Ordc @ Orda @ G2 )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ G2 @ B4 ) @ A4 )
=> ( monotone_on_nat_nat @ B4 @ Ordc @ Ordb @ ( comp_int_nat_nat @ F @ G2 ) ) ) ) ) ).
% monotone_on_o
thf(fact_609_monotone__on__o,axiom,
! [A4: set_real,Orda: real > real > $o,Ordb: real > real > $o,F: real > real,B4: set_real,Ordc: real > real > $o,G2: real > real] :
( ( monoto4017252874604999745l_real @ A4 @ Orda @ Ordb @ F )
=> ( ( monoto4017252874604999745l_real @ B4 @ Ordc @ Orda @ G2 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ G2 @ B4 ) @ A4 )
=> ( monoto4017252874604999745l_real @ B4 @ Ordc @ Ordb @ ( comp_real_real_real @ F @ G2 ) ) ) ) ) ).
% monotone_on_o
thf(fact_610_monotone__on__o,axiom,
! [A4: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,B4: set_nat,Ordc: nat > nat > $o,G2: nat > nat] :
( ( monotone_on_nat_nat @ A4 @ Orda @ Ordb @ F )
=> ( ( monotone_on_nat_nat @ B4 @ Ordc @ Orda @ G2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G2 @ B4 ) @ A4 )
=> ( monotone_on_nat_nat @ B4 @ Ordc @ Ordb @ ( comp_nat_nat_nat @ F @ G2 ) ) ) ) ) ).
% monotone_on_o
thf(fact_611_imageI,axiom,
! [X: nat,A4: set_nat,F: nat > int] :
( ( member_nat @ X @ A4 )
=> ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ A4 ) ) ) ).
% imageI
thf(fact_612_imageI,axiom,
! [X: real,A4: set_real,F: real > real] :
( ( member_real @ X @ A4 )
=> ( member_real @ ( F @ X ) @ ( image_real_real @ F @ A4 ) ) ) ).
% imageI
thf(fact_613_imageI,axiom,
! [X: real,A4: set_real,F: real > int] :
( ( member_real @ X @ A4 )
=> ( member_int @ ( F @ X ) @ ( image_real_int @ F @ A4 ) ) ) ).
% imageI
thf(fact_614_imageI,axiom,
! [X: int,A4: set_int,F: int > real] :
( ( member_int @ X @ A4 )
=> ( member_real @ ( F @ X ) @ ( image_int_real @ F @ A4 ) ) ) ).
% imageI
thf(fact_615_imageI,axiom,
! [X: int,A4: set_int,F: int > int] :
( ( member_int @ X @ A4 )
=> ( member_int @ ( F @ X ) @ ( image_int_int @ F @ A4 ) ) ) ).
% imageI
thf(fact_616_image__iff,axiom,
! [Z2: real,F: real > real,A4: set_real] :
( ( member_real @ Z2 @ ( image_real_real @ F @ A4 ) )
= ( ? [X4: real] :
( ( member_real @ X4 @ A4 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_617_image__iff,axiom,
! [Z2: int,F: nat > int,A4: set_nat] :
( ( member_int @ Z2 @ ( image_nat_int @ F @ A4 ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_618_image__iff,axiom,
! [Z2: int,F: int > int,A4: set_int] :
( ( member_int @ Z2 @ ( image_int_int @ F @ A4 ) )
= ( ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_619_bex__imageD,axiom,
! [F: real > real,A4: set_real,P: real > $o] :
( ? [X2: real] :
( ( member_real @ X2 @ ( image_real_real @ F @ A4 ) )
& ( P @ X2 ) )
=> ? [X3: real] :
( ( member_real @ X3 @ A4 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_620_bex__imageD,axiom,
! [F: nat > int,A4: set_nat,P: int > $o] :
( ? [X2: int] :
( ( member_int @ X2 @ ( image_nat_int @ F @ A4 ) )
& ( P @ X2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_621_bex__imageD,axiom,
! [F: int > int,A4: set_int,P: int > $o] :
( ? [X2: int] :
( ( member_int @ X2 @ ( image_int_int @ F @ A4 ) )
& ( P @ X2 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ A4 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_622_image__cong,axiom,
! [M3: set_nat,N3: set_nat,F: nat > int,G2: nat > int] :
( ( M3 = N3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( image_nat_int @ F @ M3 )
= ( image_nat_int @ G2 @ N3 ) ) ) ) ).
% image_cong
thf(fact_623_image__cong,axiom,
! [M3: set_real,N3: set_real,F: real > real,G2: real > real] :
( ( M3 = N3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( image_real_real @ F @ M3 )
= ( image_real_real @ G2 @ N3 ) ) ) ) ).
% image_cong
thf(fact_624_image__cong,axiom,
! [M3: set_int,N3: set_int,F: int > int,G2: int > int] :
( ( M3 = N3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( image_int_int @ F @ M3 )
= ( image_int_int @ G2 @ N3 ) ) ) ) ).
% image_cong
thf(fact_625_ball__imageD,axiom,
! [F: real > real,A4: set_real,P: real > $o] :
( ! [X3: real] :
( ( member_real @ X3 @ ( image_real_real @ F @ A4 ) )
=> ( P @ X3 ) )
=> ! [X2: real] :
( ( member_real @ X2 @ A4 )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_626_ball__imageD,axiom,
! [F: nat > int,A4: set_nat,P: int > $o] :
( ! [X3: int] :
( ( member_int @ X3 @ ( image_nat_int @ F @ A4 ) )
=> ( P @ X3 ) )
=> ! [X2: nat] :
( ( member_nat @ X2 @ A4 )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_627_ball__imageD,axiom,
! [F: int > int,A4: set_int,P: int > $o] :
( ! [X3: int] :
( ( member_int @ X3 @ ( image_int_int @ F @ A4 ) )
=> ( P @ X3 ) )
=> ! [X2: int] :
( ( member_int @ X2 @ A4 )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_628_rev__image__eqI,axiom,
! [X: nat,A4: set_nat,B: int,F: nat > int] :
( ( member_nat @ X @ A4 )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_nat_int @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_629_rev__image__eqI,axiom,
! [X: real,A4: set_real,B: real,F: real > real] :
( ( member_real @ X @ A4 )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_real_real @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_630_rev__image__eqI,axiom,
! [X: real,A4: set_real,B: int,F: real > int] :
( ( member_real @ X @ A4 )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_real_int @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_631_rev__image__eqI,axiom,
! [X: int,A4: set_int,B: real,F: int > real] :
( ( member_int @ X @ A4 )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_int_real @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_632_rev__image__eqI,axiom,
! [X: int,A4: set_int,B: int,F: int > int] :
( ( member_int @ X @ A4 )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_int_int @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_633_subset__image__iff,axiom,
! [B4: set_real,F: real > real,A4: set_real] :
( ( ord_less_eq_set_real @ B4 @ ( image_real_real @ F @ A4 ) )
= ( ? [AA: set_real] :
( ( ord_less_eq_set_real @ AA @ A4 )
& ( B4
= ( image_real_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_634_subset__image__iff,axiom,
! [B4: set_int,F: nat > int,A4: set_nat] :
( ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A4 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A4 )
& ( B4
= ( image_nat_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_635_subset__image__iff,axiom,
! [B4: set_int,F: int > int,A4: set_int] :
( ( ord_less_eq_set_int @ B4 @ ( image_int_int @ F @ A4 ) )
= ( ? [AA: set_int] :
( ( ord_less_eq_set_int @ AA @ A4 )
& ( B4
= ( image_int_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_636_image__subset__iff,axiom,
! [F: real > real,A4: set_real,B4: set_real] :
( ( ord_less_eq_set_real @ ( image_real_real @ F @ A4 ) @ B4 )
= ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( member_real @ ( F @ X4 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_637_image__subset__iff,axiom,
! [F: nat > int,A4: set_nat,B4: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ B4 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( member_int @ ( F @ X4 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_638_image__subset__iff,axiom,
! [F: int > int,A4: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ B4 )
= ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( member_int @ ( F @ X4 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_639_subset__imageE,axiom,
! [B4: set_real,F: real > real,A4: set_real] :
( ( ord_less_eq_set_real @ B4 @ ( image_real_real @ F @ A4 ) )
=> ~ ! [C2: set_real] :
( ( ord_less_eq_set_real @ C2 @ A4 )
=> ( B4
!= ( image_real_real @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_640_subset__imageE,axiom,
! [B4: set_int,F: nat > int,A4: set_nat] :
( ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A4 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A4 )
=> ( B4
!= ( image_nat_int @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_641_subset__imageE,axiom,
! [B4: set_int,F: int > int,A4: set_int] :
( ( ord_less_eq_set_int @ B4 @ ( image_int_int @ F @ A4 ) )
=> ~ ! [C2: set_int] :
( ( ord_less_eq_set_int @ C2 @ A4 )
=> ( B4
!= ( image_int_int @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_642_image__subsetI,axiom,
! [A4: set_nat,F: nat > int,B4: set_int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_int @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_643_image__subsetI,axiom,
! [A4: set_real,F: real > real,B4: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( member_real @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_644_image__subsetI,axiom,
! [A4: set_real,F: real > int,B4: set_int] :
( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( member_int @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_int @ ( image_real_int @ F @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_645_image__subsetI,axiom,
! [A4: set_int,F: int > real,B4: set_real] :
( ! [X3: int] :
( ( member_int @ X3 @ A4 )
=> ( member_real @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_real @ ( image_int_real @ F @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_646_image__subsetI,axiom,
! [A4: set_int,F: int > int,B4: set_int] :
( ! [X3: int] :
( ( member_int @ X3 @ A4 )
=> ( member_int @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_647_image__mono,axiom,
! [A4: set_real,B4: set_real,F: real > real] :
( ( ord_less_eq_set_real @ A4 @ B4 )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A4 ) @ ( image_real_real @ F @ B4 ) ) ) ).
% image_mono
thf(fact_648_image__mono,axiom,
! [A4: set_nat,B4: set_nat,F: nat > int] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ ( image_nat_int @ F @ B4 ) ) ) ).
% image_mono
thf(fact_649_image__mono,axiom,
! [A4: set_int,B4: set_int,F: int > int] :
( ( ord_less_eq_set_int @ A4 @ B4 )
=> ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ ( image_int_int @ F @ B4 ) ) ) ).
% image_mono
thf(fact_650_Equivalence__Measurable__On__Borel_Ointegrable__on__mono__on,axiom,
! [A: real,B: real,F: real > real] :
( ( monoto4017252874604999745l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( hensto5963834015518849588l_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% Equivalence_Measurable_On_Borel.integrable_on_mono_on
thf(fact_651_continuous__image__closed__interval,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [C3: real,D3: real] :
( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ C3 @ D3 ) )
& ( ord_less_eq_real @ C3 @ D3 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_652_integrable__continuous__real,axiom,
! [A: real,B: real,F: real > real] :
( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( hensto5963834015518849588l_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% integrable_continuous_real
thf(fact_653_Henstock__Kurzweil__Integration_Ointegrable__combine,axiom,
! [A: real,C: real,B: real,F: real > real] :
( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ( hensto5963834015518849588l_real @ F @ ( set_or1222579329274155063t_real @ A @ C ) )
=> ( ( hensto5963834015518849588l_real @ F @ ( set_or1222579329274155063t_real @ C @ B ) )
=> ( hensto5963834015518849588l_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ) ) ) ).
% Henstock_Kurzweil_Integration.integrable_combine
thf(fact_654_integrable__continuous__interval,axiom,
! [A: real,B: real,F: real > real] :
( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( hensto5963834015518849588l_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% integrable_continuous_interval
thf(fact_655_integrable__on__subinterval,axiom,
! [F: real > real,S: set_real,A: real,B: real] :
( ( hensto5963834015518849588l_real @ F @ S )
=> ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ S )
=> ( hensto5963834015518849588l_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ) ).
% integrable_on_subinterval
thf(fact_656_continuous__on__compose,axiom,
! [S3: set_real,F: real > real,G2: real > real] :
( ( topolo5044208981011980120l_real @ S3 @ F )
=> ( ( topolo5044208981011980120l_real @ ( image_real_real @ F @ S3 ) @ G2 )
=> ( topolo5044208981011980120l_real @ S3 @ ( comp_real_real_real @ G2 @ F ) ) ) ) ).
% continuous_on_compose
thf(fact_657_integrable__on__superset,axiom,
! [F: real > real,S: set_real,T3: set_real] :
( ( hensto5963834015518849588l_real @ F @ S )
=> ( ! [X3: real] :
( ~ ( member_real @ X3 @ S )
=> ( ( F @ X3 )
= zero_zero_real ) )
=> ( ( ord_less_eq_set_real @ S @ T3 )
=> ( hensto5963834015518849588l_real @ F @ T3 ) ) ) ) ).
% integrable_on_superset
thf(fact_658_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_659_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_660_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_661_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_662_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_663_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_664_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_665_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_666_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_667_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_668_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_669_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_670_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_671_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_672_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_673_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_674_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_675_continuous__on__cong,axiom,
! [S3: set_real,T3: set_real,F: real > real,G2: real > real] :
( ( S3 = T3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ T3 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( topolo5044208981011980120l_real @ S3 @ F )
= ( topolo5044208981011980120l_real @ T3 @ G2 ) ) ) ) ).
% continuous_on_cong
thf(fact_676_Henstock__Kurzweil__Integration_Ointegrable__cong,axiom,
! [A4: set_real,F: real > real,G2: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( hensto5963834015518849588l_real @ F @ A4 )
= ( hensto5963834015518849588l_real @ G2 @ A4 ) ) ) ).
% Henstock_Kurzweil_Integration.integrable_cong
thf(fact_677_integrable__eq,axiom,
! [F: real > real,S3: set_real,G2: real > real] :
( ( hensto5963834015518849588l_real @ F @ S3 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( hensto5963834015518849588l_real @ G2 @ S3 ) ) ) ).
% integrable_eq
thf(fact_678_strict__mono__continuous__invD,axiom,
! [A: real,F: real > real,G2: real > real] :
( ( monoto4017252874604999745l_real @ ( set_ord_atLeast_real @ A ) @ ord_less_real @ ord_less_real @ F )
=> ( ( topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ A ) @ F )
=> ( ( ( image_real_real @ F @ ( set_ord_atLeast_real @ A ) )
= ( set_ord_atLeast_real @ ( F @ A ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( G2 @ ( F @ X3 ) )
= X3 ) )
=> ( topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ ( F @ A ) ) @ G2 ) ) ) ) ) ).
% strict_mono_continuous_invD
thf(fact_679_continuous__on__subset,axiom,
! [S3: set_real,F: real > real,T3: set_real] :
( ( topolo5044208981011980120l_real @ S3 @ F )
=> ( ( ord_less_eq_set_real @ T3 @ S3 )
=> ( topolo5044208981011980120l_real @ T3 @ F ) ) ) ).
% continuous_on_subset
thf(fact_680_uniformly__continuous__imp__continuous,axiom,
! [S3: set_real,F: real > real] :
( ( topolo8845477368217174713l_real @ S3 @ F )
=> ( topolo5044208981011980120l_real @ S3 @ F ) ) ).
% uniformly_continuous_imp_continuous
thf(fact_681_integrable__subinterval__real,axiom,
! [F: real > real,A: real,B: real,C: real,D: real] :
( ( hensto5963834015518849588l_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ C @ D ) @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( hensto5963834015518849588l_real @ F @ ( set_or1222579329274155063t_real @ C @ D ) ) ) ) ).
% integrable_subinterval_real
thf(fact_682_IVT_H,axiom,
! [F: real > nat,A: real,Y: nat,B: real] :
( ( ord_less_eq_nat @ ( F @ A ) @ Y )
=> ( ( ord_less_eq_nat @ Y @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B )
& ( ( F @ X3 )
= Y ) ) ) ) ) ) ).
% IVT'
thf(fact_683_IVT_H,axiom,
! [F: real > int,A: real,Y: int,B: real] :
( ( ord_less_eq_int @ ( F @ A ) @ Y )
=> ( ( ord_less_eq_int @ Y @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2284712892409288920al_int @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B )
& ( ( F @ X3 )
= Y ) ) ) ) ) ) ).
% IVT'
thf(fact_684_IVT_H,axiom,
! [F: real > real,A: real,Y: real,B: real] :
( ( ord_less_eq_real @ ( F @ A ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( F @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B )
& ( ( F @ X3 )
= Y ) ) ) ) ) ) ).
% IVT'
thf(fact_685_IVT2_H,axiom,
! [F: real > nat,B: real,Y: nat,A: real] :
( ( ord_less_eq_nat @ ( F @ B ) @ Y )
=> ( ( ord_less_eq_nat @ Y @ ( F @ A ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B )
& ( ( F @ X3 )
= Y ) ) ) ) ) ) ).
% IVT2'
thf(fact_686_IVT2_H,axiom,
! [F: real > int,B: real,Y: int,A: real] :
( ( ord_less_eq_int @ ( F @ B ) @ Y )
=> ( ( ord_less_eq_int @ Y @ ( F @ A ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo2284712892409288920al_int @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B )
& ( ( F @ X3 )
= Y ) ) ) ) ) ) ).
% IVT2'
thf(fact_687_IVT2_H,axiom,
! [F: real > real,B: real,Y: real,A: real] :
( ( ord_less_eq_real @ ( F @ B ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( F @ A ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B )
& ( ( F @ X3 )
= Y ) ) ) ) ) ) ).
% IVT2'
thf(fact_688_Sup_OSUP__image,axiom,
! [Sup: set_real > real,G2: real > real,F: real > real,A4: set_real] :
( ( Sup @ ( image_real_real @ G2 @ ( image_real_real @ F @ A4 ) ) )
= ( Sup @ ( image_real_real @ ( comp_real_real_real @ G2 @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_689_Sup_OSUP__image,axiom,
! [Sup: set_int > int,G2: nat > int,F: nat > nat,A4: set_nat] :
( ( Sup @ ( image_nat_int @ G2 @ ( image_nat_nat @ F @ A4 ) ) )
= ( Sup @ ( image_nat_int @ ( comp_nat_int_nat @ G2 @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_690_Sup_OSUP__image,axiom,
! [Sup: set_int > int,G2: nat > int,F: int > nat,A4: set_int] :
( ( Sup @ ( image_nat_int @ G2 @ ( image_int_nat @ F @ A4 ) ) )
= ( Sup @ ( image_int_int @ ( comp_nat_int_int @ G2 @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_691_Sup_OSUP__image,axiom,
! [Sup: set_int > int,G2: int > int,F: nat > int,A4: set_nat] :
( ( Sup @ ( image_int_int @ G2 @ ( image_nat_int @ F @ A4 ) ) )
= ( Sup @ ( image_nat_int @ ( comp_int_int_nat @ G2 @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_692_Sup_OSUP__image,axiom,
! [Sup: set_int > int,G2: int > int,F: int > int,A4: set_int] :
( ( Sup @ ( image_int_int @ G2 @ ( image_int_int @ F @ A4 ) ) )
= ( Sup @ ( image_int_int @ ( comp_int_int_int @ G2 @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_693_Inf_OINF__image,axiom,
! [Inf: set_real > real,G2: real > real,F: real > real,A4: set_real] :
( ( Inf @ ( image_real_real @ G2 @ ( image_real_real @ F @ A4 ) ) )
= ( Inf @ ( image_real_real @ ( comp_real_real_real @ G2 @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_694_Inf_OINF__image,axiom,
! [Inf: set_int > int,G2: nat > int,F: nat > nat,A4: set_nat] :
( ( Inf @ ( image_nat_int @ G2 @ ( image_nat_nat @ F @ A4 ) ) )
= ( Inf @ ( image_nat_int @ ( comp_nat_int_nat @ G2 @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_695_Inf_OINF__image,axiom,
! [Inf: set_int > int,G2: nat > int,F: int > nat,A4: set_int] :
( ( Inf @ ( image_nat_int @ G2 @ ( image_int_nat @ F @ A4 ) ) )
= ( Inf @ ( image_int_int @ ( comp_nat_int_int @ G2 @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_696_Inf_OINF__image,axiom,
! [Inf: set_int > int,G2: int > int,F: nat > int,A4: set_nat] :
( ( Inf @ ( image_int_int @ G2 @ ( image_nat_int @ F @ A4 ) ) )
= ( Inf @ ( image_nat_int @ ( comp_int_int_nat @ G2 @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_697_Inf_OINF__image,axiom,
! [Inf: set_int > int,G2: int > int,F: int > int,A4: set_int] :
( ( Inf @ ( image_int_int @ G2 @ ( image_int_int @ F @ A4 ) ) )
= ( Inf @ ( image_int_int @ ( comp_int_int_int @ G2 @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_698_del,axiom,
! [X6: real,X: real,E: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X6 @ X ) ) @ ( del @ E ) )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ( ( member_real @ X @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
=> ( ( member_real @ X6 @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( f @ X6 ) @ ( f @ X ) ) ) @ E ) ) ) ) ) ).
% del
thf(fact_699_seq__mono__lemma,axiom,
! [M2: nat,D: nat > real,E: nat > real] :
( ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_real @ ( D @ N4 ) @ ( E @ N4 ) ) )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_eq_real @ ( E @ N4 ) @ ( E @ M2 ) ) )
=> ! [N5: nat] :
( ( ord_less_eq_nat @ M2 @ N5 )
=> ( ord_less_real @ ( D @ N5 ) @ ( E @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_700_bgauge__existence__lemma,axiom,
! [S3: set_real,Q2: real > real > $o] :
( ( ! [X4: real] :
( ( member_real @ X4 @ S3 )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ( Q2 @ D4 @ X4 ) ) ) )
= ( ! [X4: real] :
? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ( ( member_real @ X4 @ S3 )
=> ( Q2 @ D4 @ X4 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_701_bgauge__existence__lemma,axiom,
! [S3: set_int,Q2: real > int > $o] :
( ( ! [X4: int] :
( ( member_int @ X4 @ S3 )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ( Q2 @ D4 @ X4 ) ) ) )
= ( ! [X4: int] :
? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ( ( member_int @ X4 @ S3 )
=> ( Q2 @ D4 @ X4 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_702_all__subset__image,axiom,
! [F: real > real,A4: set_real,P: set_real > $o] :
( ( ! [B6: set_real] :
( ( ord_less_eq_set_real @ B6 @ ( image_real_real @ F @ A4 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_real] :
( ( ord_less_eq_set_real @ B6 @ A4 )
=> ( P @ ( image_real_real @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_703_all__subset__image,axiom,
! [F: nat > int,A4: set_nat,P: set_int > $o] :
( ( ! [B6: set_int] :
( ( ord_less_eq_set_int @ B6 @ ( image_nat_int @ F @ A4 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A4 )
=> ( P @ ( image_nat_int @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_704_all__subset__image,axiom,
! [F: int > int,A4: set_int,P: set_int > $o] :
( ( ! [B6: set_int] :
( ( ord_less_eq_set_int @ B6 @ ( image_int_int @ F @ A4 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_int] :
( ( ord_less_eq_set_int @ B6 @ A4 )
=> ( P @ ( image_int_int @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_705_abs__idempotent,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_idempotent
thf(fact_706_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_707_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_708_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_709_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_710_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_711_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_712_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_713_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_714_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_715_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_716_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_717_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_718_abs__0__eq,axiom,
! [A: real] :
( ( zero_zero_real
= ( abs_abs_real @ A ) )
= ( A = zero_zero_real ) ) ).
% abs_0_eq
thf(fact_719_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_720_abs__eq__0,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0
thf(fact_721_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_722_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_zero
thf(fact_723_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_724__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062del_O_A_092_060lbrakk_062_092_060And_062e_O_A0_A_060_Ae_A_092_060Longrightarrow_062_A0_A_060_Adel_Ae_059_A_092_060And_062e_Ax_Ax_H_O_A_092_060lbrakk_062_092_060bar_062x_H_A_N_Ax_092_060bar_062_A_060_Adel_Ae_059_A0_A_060_Ae_059_Ax_A_092_060in_062_A_1230_O_Oa_125_059_Ax_H_A_092_060in_062_A_1230_O_Oa_125_092_060rbrakk_062_A_092_060Longrightarrow_062_A_092_060bar_062f_Ax_H_A_N_Af_Ax_092_060bar_062_A_060_Ae_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Del: real > real] :
( ! [E3: real] :
( ( ord_less_real @ zero_zero_real @ E3 )
=> ( ord_less_real @ zero_zero_real @ ( Del @ E3 ) ) )
=> ~ ! [E3: real,X2: real,X7: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X7 @ X2 ) ) @ ( Del @ E3 ) )
=> ( ( ord_less_real @ zero_zero_real @ E3 )
=> ( ( member_real @ X2 @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
=> ( ( member_real @ X7 @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( f @ X7 ) @ ( f @ X2 ) ) ) @ E3 ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>del. \<lbrakk>\<And>e. 0 < e \<Longrightarrow> 0 < del e; \<And>e x x'. \<lbrakk>\<bar>x' - x\<bar> < del e; 0 < e; x \<in> {0..a}; x' \<in> {0..a}\<rbrakk> \<Longrightarrow> \<bar>f x' - f x\<bar> < e\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_725_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_726_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_727_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_728_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_729_abs__le__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_le_zero_iff
thf(fact_730_abs__le__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_731_abs__le__self__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% abs_le_self_iff
thf(fact_732_abs__le__self__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% abs_le_self_iff
thf(fact_733_abs__of__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_734_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_735_zero__less__abs__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
= ( A != zero_zero_real ) ) ).
% zero_less_abs_iff
thf(fact_736_zero__less__abs__iff,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
= ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_737_image__diff__atLeastAtMost,axiom,
! [D: real,A: real,B: real] :
( ( image_real_real @ ( minus_minus_real @ D ) @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ ( minus_minus_real @ D @ B ) @ ( minus_minus_real @ D @ A ) ) ) ).
% image_diff_atLeastAtMost
thf(fact_738_image__diff__atLeastAtMost,axiom,
! [D: int,A: int,B: int] :
( ( image_int_int @ ( minus_minus_int @ D ) @ ( set_or1266510415728281911st_int @ A @ B ) )
= ( set_or1266510415728281911st_int @ ( minus_minus_int @ D @ B ) @ ( minus_minus_int @ D @ A ) ) ) ).
% image_diff_atLeastAtMost
thf(fact_739_abs__minus__commute,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
= ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_740_abs__minus__commute,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
= ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_741_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_742_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_743_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_744_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_745_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_746_diff__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_747_diff__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_748_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M3: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_749_abs__triangle__ineq2__sym,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_750_abs__triangle__ineq2__sym,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_751_abs__triangle__ineq3,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_752_abs__triangle__ineq3,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_753_abs__triangle__ineq2,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_754_abs__triangle__ineq2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_755_abs__ge__self,axiom,
! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% abs_ge_self
thf(fact_756_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_757_abs__le__D1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% abs_le_D1
thf(fact_758_abs__le__D1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% abs_le_D1
thf(fact_759_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_760_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_761_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_762_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_763_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_764_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_765_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_766_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_767_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [A2: real,B2: real] :
( ( minus_minus_real @ A2 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_768_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z3: int] : ( Y3 = Z3 ) )
= ( ^ [A2: int,B2: int] :
( ( minus_minus_int @ A2 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_769_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_770_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_771_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_772_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_773_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_774_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_775_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_776_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_777_continuous__on__op__minus,axiom,
! [S3: set_real,X: real] : ( topolo5044208981011980120l_real @ S3 @ ( minus_minus_real @ X ) ) ).
% continuous_on_op_minus
thf(fact_778_abs__ge__zero,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% abs_ge_zero
thf(fact_779_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_780_abs__not__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% abs_not_less_zero
thf(fact_781_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_782_abs__of__pos,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_pos
thf(fact_783_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_784_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_785_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_786_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A2: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_787_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_788_dense__eq0__I,axiom,
! [X: real] :
( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E2 ) )
=> ( X = zero_zero_real ) ) ).
% dense_eq0_I
thf(fact_789_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A3: real,B3: real,C3: real] :
( ( P @ A3 @ B3 )
=> ( ( P @ B3 @ C3 )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( P @ A3 @ C3 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [A3: real,B3: real] :
( ( ( ord_less_eq_real @ A3 @ X3 )
& ( ord_less_eq_real @ X3 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D5 ) )
=> ( P @ A3 @ B3 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_790_Inf_OINF__cong,axiom,
! [A4: set_nat,B4: set_nat,C4: nat > int,D6: nat > int,Inf: set_int > int] :
( ( A4 = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( C4 @ X3 )
= ( D6 @ X3 ) ) )
=> ( ( Inf @ ( image_nat_int @ C4 @ A4 ) )
= ( Inf @ ( image_nat_int @ D6 @ B4 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_791_Inf_OINF__cong,axiom,
! [A4: set_real,B4: set_real,C4: real > real,D6: real > real,Inf: set_real > real] :
( ( A4 = B4 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B4 )
=> ( ( C4 @ X3 )
= ( D6 @ X3 ) ) )
=> ( ( Inf @ ( image_real_real @ C4 @ A4 ) )
= ( Inf @ ( image_real_real @ D6 @ B4 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_792_Inf_OINF__cong,axiom,
! [A4: set_int,B4: set_int,C4: int > int,D6: int > int,Inf: set_int > int] :
( ( A4 = B4 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B4 )
=> ( ( C4 @ X3 )
= ( D6 @ X3 ) ) )
=> ( ( Inf @ ( image_int_int @ C4 @ A4 ) )
= ( Inf @ ( image_int_int @ D6 @ B4 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_793_Sup_OSUP__cong,axiom,
! [A4: set_nat,B4: set_nat,C4: nat > int,D6: nat > int,Sup: set_int > int] :
( ( A4 = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( C4 @ X3 )
= ( D6 @ X3 ) ) )
=> ( ( Sup @ ( image_nat_int @ C4 @ A4 ) )
= ( Sup @ ( image_nat_int @ D6 @ B4 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_794_Sup_OSUP__cong,axiom,
! [A4: set_real,B4: set_real,C4: real > real,D6: real > real,Sup: set_real > real] :
( ( A4 = B4 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B4 )
=> ( ( C4 @ X3 )
= ( D6 @ X3 ) ) )
=> ( ( Sup @ ( image_real_real @ C4 @ A4 ) )
= ( Sup @ ( image_real_real @ D6 @ B4 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_795_Sup_OSUP__cong,axiom,
! [A4: set_int,B4: set_int,C4: int > int,D6: int > int,Sup: set_int > int] :
( ( A4 = B4 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B4 )
=> ( ( C4 @ X3 )
= ( D6 @ X3 ) ) )
=> ( ( Sup @ ( image_int_int @ C4 @ A4 ) )
= ( Sup @ ( image_int_int @ D6 @ B4 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_796_lemma__interval,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D3 )
=> ( ( ord_less_eq_real @ A @ Y5 )
& ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).
% lemma_interval
thf(fact_797_abs__0,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_0
thf(fact_798_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_799_lemma__interval__lt,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D3 )
=> ( ( ord_less_real @ A @ Y5 )
& ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_800_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B2 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_801_ge__iff__diff__ge__0,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_802_abs__abs,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_abs
thf(fact_803_abs__abs,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_abs
thf(fact_804_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_805_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_806_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_807_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_808_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_809_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_810_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_811_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_812_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_813_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_814_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_815_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_816_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_817_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_818_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_819_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_820_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_821_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_822_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_823_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_824_psubset__imp__ex__mem,axiom,
! [A4: set_real,B4: set_real] :
( ( ord_less_set_real @ A4 @ B4 )
=> ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_825_psubset__imp__ex__mem,axiom,
! [A4: set_int,B4: set_int] :
( ( ord_less_set_int @ A4 @ B4 )
=> ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B4 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_826_image__diff__subset,axiom,
! [F: real > real,A4: set_real,B4: set_real] : ( ord_less_eq_set_real @ ( minus_minus_set_real @ ( image_real_real @ F @ A4 ) @ ( image_real_real @ F @ B4 ) ) @ ( image_real_real @ F @ ( minus_minus_set_real @ A4 @ B4 ) ) ) ).
% image_diff_subset
thf(fact_827_image__diff__subset,axiom,
! [F: nat > int,A4: set_nat,B4: set_nat] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ ( image_nat_int @ F @ A4 ) @ ( image_nat_int @ F @ B4 ) ) @ ( image_nat_int @ F @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).
% image_diff_subset
thf(fact_828_image__diff__subset,axiom,
! [F: int > int,A4: set_int,B4: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ ( image_int_int @ F @ A4 ) @ ( image_int_int @ F @ B4 ) ) @ ( image_int_int @ F @ ( minus_minus_set_int @ A4 @ B4 ) ) ) ).
% image_diff_subset
thf(fact_829_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_830_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_831_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_832_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_833_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_834_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_835_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_836_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_837_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_838_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_839_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_840_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_841_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_842_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_843_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 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_844_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_845_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_846_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_847_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_848_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_849_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_850_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_851_abs__eq__0__iff,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0_iff
thf(fact_852_abs__eq__0__iff,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_853__092_060open_0620_A_060_An_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% \<open>0 < n\<close>
thf(fact_854_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( P @ M ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_855_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( P @ M ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_856_Diff__iff,axiom,
! [C: real,A4: set_real,B4: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A4 @ B4 ) )
= ( ( member_real @ C @ A4 )
& ~ ( member_real @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_857_Diff__iff,axiom,
! [C: int,A4: set_int,B4: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A4 @ B4 ) )
= ( ( member_int @ C @ A4 )
& ~ ( member_int @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_858_DiffI,axiom,
! [C: real,A4: set_real,B4: set_real] :
( ( member_real @ C @ A4 )
=> ( ~ ( member_real @ C @ B4 )
=> ( member_real @ C @ ( minus_minus_set_real @ A4 @ B4 ) ) ) ) ).
% DiffI
thf(fact_859_DiffI,axiom,
! [C: int,A4: set_int,B4: set_int] :
( ( member_int @ C @ A4 )
=> ( ~ ( member_int @ C @ B4 )
=> ( member_int @ C @ ( minus_minus_set_int @ A4 @ B4 ) ) ) ) ).
% DiffI
thf(fact_860_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_861_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_862_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_863_DiffD2,axiom,
! [C: real,A4: set_real,B4: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A4 @ B4 ) )
=> ~ ( member_real @ C @ B4 ) ) ).
% DiffD2
thf(fact_864_DiffD2,axiom,
! [C: int,A4: set_int,B4: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A4 @ B4 ) )
=> ~ ( member_int @ C @ B4 ) ) ).
% DiffD2
thf(fact_865_DiffD1,axiom,
! [C: real,A4: set_real,B4: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A4 @ B4 ) )
=> ( member_real @ C @ A4 ) ) ).
% DiffD1
thf(fact_866_DiffD1,axiom,
! [C: int,A4: set_int,B4: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A4 @ B4 ) )
=> ( member_int @ C @ A4 ) ) ).
% DiffD1
thf(fact_867_DiffE,axiom,
! [C: real,A4: set_real,B4: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A4 @ B4 ) )
=> ~ ( ( member_real @ C @ A4 )
=> ( member_real @ C @ B4 ) ) ) ).
% DiffE
thf(fact_868_DiffE,axiom,
! [C: int,A4: set_int,B4: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A4 @ B4 ) )
=> ~ ( ( member_int @ C @ A4 )
=> ( member_int @ C @ B4 ) ) ) ).
% DiffE
thf(fact_869_an__less__del,axiom,
ord_less_real @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) @ ( del @ ( divide_divide_real @ epsilon @ a ) ) ).
% an_less_del
thf(fact_870_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_871_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: real] :
( ( ord_less_eq_real @ A @ C3 )
& ( ord_less_eq_real @ C3 @ B )
& ! [X2: real] :
( ( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_real @ X2 @ C3 ) )
=> ( P @ X2 ) )
& ! [D5: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ D5 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D5 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_872_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ C3 ) )
=> ( P @ X2 ) )
& ! [D5: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D5 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D5 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_873_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: int] :
( ( ord_less_eq_int @ A @ C3 )
& ( ord_less_eq_int @ C3 @ B )
& ! [X2: int] :
( ( ( ord_less_eq_int @ A @ X2 )
& ( ord_less_int @ X2 @ C3 ) )
=> ( P @ X2 ) )
& ! [D5: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D5 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D5 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_874_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_875_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_876_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_877_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_878_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_879_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_880_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_881_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% abs_of_nat
thf(fact_882_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_883_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_884_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_885_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_886_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_887_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_888_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_889_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_890_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_891_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_892_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_893_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_894_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_895_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_896_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_897_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_898_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_899_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_900_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_901_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_902_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_903_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_904_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_905_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_906_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_907_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_908_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_909_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_910_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_911_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_912_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_913_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_914_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_915_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_916_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_917_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_918_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_919_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_920_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% of_nat_diff
thf(fact_921_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_922_ex__gt__or__lt,axiom,
! [A: real] :
? [B3: real] :
( ( ord_less_real @ A @ B3 )
| ( ord_less_real @ B3 @ A ) ) ).
% ex_gt_or_lt
thf(fact_923_comp__cong,axiom,
! [F: real > real,G2: real > real,X: real,F4: real > real,G3: real > real,X6: real] :
( ( ( F @ ( G2 @ X ) )
= ( F4 @ ( G3 @ X6 ) ) )
=> ( ( comp_real_real_real @ F @ G2 @ X )
= ( comp_real_real_real @ F4 @ G3 @ X6 ) ) ) ).
% comp_cong
thf(fact_924_divide__le__0__abs__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
= ( ( ord_less_eq_real @ A @ zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_le_0_abs_iff
thf(fact_925_zero__le__divide__abs__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
= ( ( ord_less_eq_real @ zero_zero_real @ A )
| ( B = zero_zero_real ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_926_fa__eq__b,axiom,
( ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ n ) ) )
= b ) ).
% fa_eq_b
thf(fact_927_a__seg__eq__a__iff,axiom,
! [X: real] :
( ( ( a_seg @ X )
= a )
= ( X
= ( semiri5074537144036343181t_real @ n ) ) ) ).
% a_seg_eq_a_iff
thf(fact_928_abs__divide,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_divide
thf(fact_929_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_930_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_931_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_932_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_933_a__seg__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( a_seg @ X ) @ ( a_seg @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% a_seg_le_iff
thf(fact_934_a__seg__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( a_seg @ X ) @ ( a_seg @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% a_seg_less_iff
thf(fact_935_a__seg__ge__0,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( a_seg @ X ) )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% a_seg_ge_0
thf(fact_936_a__seg__def,axiom,
( a_seg
= ( ^ [U2: real] : ( divide_divide_real @ ( times_times_real @ U2 @ a ) @ ( semiri5074537144036343181t_real @ n ) ) ) ) ).
% a_seg_def
thf(fact_937_a__seg__le__a,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( a_seg @ X ) @ a )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ n ) ) ) ).
% a_seg_le_a
thf(fact_938_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_939_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_940_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_941_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_942_image__int__atLeastAtMost,axiom,
! [A: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastAtMost
thf(fact_943_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_944_linordered__field__no__lb,axiom,
! [X2: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X2 ) ).
% linordered_field_no_lb
thf(fact_945_linordered__field__no__ub,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_946_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_947_diff__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_948_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_949_divide__nonpos__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_950_divide__nonpos__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_951_divide__nonneg__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_952_divide__nonneg__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_953_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_954_divide__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_955_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_956_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_957_divide__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_958_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_959_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_960_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_961_divide__pos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_962_divide__pos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_963_divide__neg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_964_divide__neg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_965_nonzero__abs__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% nonzero_abs_divide
thf(fact_966_frac__le,axiom,
! [Y: real,X: real,W2: real,Z2: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_le
thf(fact_967_frac__less,axiom,
! [X: real,Y: real,W2: real,Z2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_less
thf(fact_968_frac__less2,axiom,
! [X: real,Y: real,W2: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_real @ W2 @ Z2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_less2
thf(fact_969_divide__le__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_970_divide__nonneg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_971_divide__nonneg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_pos
thf(fact_972_divide__nonpos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_neg
thf(fact_973_divide__nonpos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_974_abs__div__pos,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
= ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% abs_div_pos
thf(fact_975_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_976_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_977_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_978_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_979__092_060open_062strict__mono_Aa__seg_092_060close_062,axiom,
monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ a_seg ).
% \<open>strict_mono a_seg\<close>
thf(fact_980_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_981_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ N @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_982_UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_I
thf(fact_983_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_984_UNIV__I,axiom,
! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% UNIV_I
thf(fact_985_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_986_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_987_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_988_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_989_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_990_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_991_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_992_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_993_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_994_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_995_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_996_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_997_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_998_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_999_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1000_abs__mult__self__eq,axiom,
! [A: real] :
( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
= ( times_times_real @ A @ A ) ) ).
% abs_mult_self_eq
thf(fact_1001_abs__mult__self__eq,axiom,
! [A: int] :
( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
= ( times_times_int @ A @ A ) ) ).
% abs_mult_self_eq
thf(fact_1002_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1003_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1004_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_1005_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_1006_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1007_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1008_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1009_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1010_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1011_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1012_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1013_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1014_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1015_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1016_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1017_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1018_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1019_range__diff,axiom,
! [A: real] :
( ( image_real_real @ ( minus_minus_real @ A ) @ top_top_set_real )
= top_top_set_real ) ).
% range_diff
thf(fact_1020_range__diff,axiom,
! [A: int] :
( ( image_int_int @ ( minus_minus_int @ A ) @ top_top_set_int )
= top_top_set_int ) ).
% range_diff
thf(fact_1021_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1022_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1023_image__mult__atLeastAtMost,axiom,
! [D: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ( image_real_real @ ( times_times_real @ D ) @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ ( times_times_real @ D @ A ) @ ( times_times_real @ D @ B ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_1024_zdiv__mono1,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1025_zdiv__mono2,axiom,
! [A: int,B5: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B5 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1026_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1027_zdiv__mono1__neg,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1028_zdiv__mono2__neg,axiom,
! [A: int,B5: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B5 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1029_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1030_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1031_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1032_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1033_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1034_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1035_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1036_int__ops_I8_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_1037_nat__int__comparison_I1_J,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A2 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1038_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_1039_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1040_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1041_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1042_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1043_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1044_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1045_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_1046_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_1047_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_1048_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_1049_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_1050_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_1051_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_1052_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_1053_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_1054_inf__period_I2_J,axiom,
! [P: real > $o,D6: real,Q: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D6 ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D6 ) ) ) )
=> ! [X2: real,K3: real] :
( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P @ ( minus_minus_real @ X2 @ ( times_times_real @ K3 @ D6 ) ) )
| ( Q @ ( minus_minus_real @ X2 @ ( times_times_real @ K3 @ D6 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1055_inf__period_I2_J,axiom,
! [P: int > $o,D6: int,Q: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D6 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D6 ) ) ) )
=> ! [X2: int,K3: int] :
( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D6 ) ) )
| ( Q @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D6 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1056_inf__period_I1_J,axiom,
! [P: real > $o,D6: real,Q: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D6 ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D6 ) ) ) )
=> ! [X2: real,K3: real] :
( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P @ ( minus_minus_real @ X2 @ ( times_times_real @ K3 @ D6 ) ) )
& ( Q @ ( minus_minus_real @ X2 @ ( times_times_real @ K3 @ D6 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1057_inf__period_I1_J,axiom,
! [P: int > $o,D6: int,Q: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D6 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D6 ) ) ) )
=> ! [X2: int,K3: int] :
( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D6 ) ) )
& ( Q @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D6 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1058_top_Oextremum__uniqueI,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A )
=> ( A = top_top_set_real ) ) ).
% top.extremum_uniqueI
thf(fact_1059_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_1060_top_Oextremum__uniqueI,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ top_top_set_int @ A )
=> ( A = top_top_set_int ) ) ).
% top.extremum_uniqueI
thf(fact_1061_top_Oextremum__unique,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A )
= ( A = top_top_set_real ) ) ).
% top.extremum_unique
thf(fact_1062_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_1063_top_Oextremum__unique,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ top_top_set_int @ A )
= ( A = top_top_set_int ) ) ).
% top.extremum_unique
thf(fact_1064_top__greatest,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).
% top_greatest
thf(fact_1065_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_1066_top__greatest,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ top_top_set_int ) ).
% top_greatest
thf(fact_1067_top_Onot__eq__extremum,axiom,
! [A: set_real] :
( ( A != top_top_set_real )
= ( ord_less_set_real @ A @ top_top_set_real ) ) ).
% top.not_eq_extremum
thf(fact_1068_top_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != top_top_set_nat )
= ( ord_less_set_nat @ A @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_1069_top_Onot__eq__extremum,axiom,
! [A: set_int] :
( ( A != top_top_set_int )
= ( ord_less_set_int @ A @ top_top_set_int ) ) ).
% top.not_eq_extremum
thf(fact_1070_top_Oextremum__strict,axiom,
! [A: set_real] :
~ ( ord_less_set_real @ top_top_set_real @ A ) ).
% top.extremum_strict
thf(fact_1071_top_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A ) ).
% top.extremum_strict
thf(fact_1072_top_Oextremum__strict,axiom,
! [A: set_int] :
~ ( ord_less_set_int @ top_top_set_int @ A ) ).
% top.extremum_strict
thf(fact_1073_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_1074_UNIV__eq__I,axiom,
! [A4: set_real] :
( ! [X3: real] : ( member_real @ X3 @ A4 )
=> ( top_top_set_real = A4 ) ) ).
% UNIV_eq_I
thf(fact_1075_UNIV__eq__I,axiom,
! [A4: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A4 )
=> ( top_top_set_nat = A4 ) ) ).
% UNIV_eq_I
thf(fact_1076_UNIV__eq__I,axiom,
! [A4: set_int] :
( ! [X3: int] : ( member_int @ X3 @ A4 )
=> ( top_top_set_int = A4 ) ) ).
% UNIV_eq_I
thf(fact_1077_UNIV__witness,axiom,
? [X3: real] : ( member_real @ X3 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_1078_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_1079_UNIV__witness,axiom,
? [X3: int] : ( member_int @ X3 @ top_top_set_int ) ).
% UNIV_witness
thf(fact_1080_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1081_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1082_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1083_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1084_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1085_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1086_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A2: real,B2: real] : ( times_times_real @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_1087_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_1088_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A2: int,B2: int] : ( times_times_int @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_1089_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1090_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1091_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1092_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_1093_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_1094_left__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1095_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1096_right__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1097_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1098_left__diff__distrib_H,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1099_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1100_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1101_right__diff__distrib_H,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1102_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1103_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1104_abs__mult,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_mult
thf(fact_1105_abs__mult,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( times_times_int @ A @ B ) )
= ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_mult
thf(fact_1106_continuous__on__mult__const,axiom,
! [S3: set_real,C: real] : ( topolo5044208981011980120l_real @ S3 @ ( times_times_real @ C ) ) ).
% continuous_on_mult_const
thf(fact_1107_surj__def,axiom,
! [F: real > real] :
( ( ( image_real_real @ F @ top_top_set_real )
= top_top_set_real )
= ( ! [Y4: real] :
? [X4: real] :
( Y4
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1108_surj__def,axiom,
! [F: real > nat] :
( ( ( image_real_nat @ F @ top_top_set_real )
= top_top_set_nat )
= ( ! [Y4: nat] :
? [X4: real] :
( Y4
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1109_surj__def,axiom,
! [F: real > int] :
( ( ( image_real_int @ F @ top_top_set_real )
= top_top_set_int )
= ( ! [Y4: int] :
? [X4: real] :
( Y4
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1110_surj__def,axiom,
! [F: nat > real] :
( ( ( image_nat_real @ F @ top_top_set_nat )
= top_top_set_real )
= ( ! [Y4: real] :
? [X4: nat] :
( Y4
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1111_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y4: nat] :
? [X4: nat] :
( Y4
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1112_surj__def,axiom,
! [F: nat > int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
= ( ! [Y4: int] :
? [X4: nat] :
( Y4
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1113_surj__def,axiom,
! [F: int > real] :
( ( ( image_int_real @ F @ top_top_set_int )
= top_top_set_real )
= ( ! [Y4: real] :
? [X4: int] :
( Y4
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1114_surj__def,axiom,
! [F: int > nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
= ( ! [Y4: nat] :
? [X4: int] :
( Y4
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1115_surj__def,axiom,
! [F: int > int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
= ( ! [Y4: int] :
? [X4: int] :
( Y4
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1116_surjI,axiom,
! [G2: real > real,F: real > real] :
( ! [X3: real] :
( ( G2 @ ( F @ X3 ) )
= X3 )
=> ( ( image_real_real @ G2 @ top_top_set_real )
= top_top_set_real ) ) ).
% surjI
thf(fact_1117_surjI,axiom,
! [G2: real > nat,F: nat > real] :
( ! [X3: nat] :
( ( G2 @ ( F @ X3 ) )
= X3 )
=> ( ( image_real_nat @ G2 @ top_top_set_real )
= top_top_set_nat ) ) ).
% surjI
thf(fact_1118_surjI,axiom,
! [G2: real > int,F: int > real] :
( ! [X3: int] :
( ( G2 @ ( F @ X3 ) )
= X3 )
=> ( ( image_real_int @ G2 @ top_top_set_real )
= top_top_set_int ) ) ).
% surjI
thf(fact_1119_surjI,axiom,
! [G2: nat > real,F: real > nat] :
( ! [X3: real] :
( ( G2 @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_real @ G2 @ top_top_set_nat )
= top_top_set_real ) ) ).
% surjI
thf(fact_1120_surjI,axiom,
! [G2: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( G2 @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_nat @ G2 @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_1121_surjI,axiom,
! [G2: nat > int,F: int > nat] :
( ! [X3: int] :
( ( G2 @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_int @ G2 @ top_top_set_nat )
= top_top_set_int ) ) ).
% surjI
thf(fact_1122_surjI,axiom,
! [G2: int > real,F: real > int] :
( ! [X3: real] :
( ( G2 @ ( F @ X3 ) )
= X3 )
=> ( ( image_int_real @ G2 @ top_top_set_int )
= top_top_set_real ) ) ).
% surjI
thf(fact_1123_surjI,axiom,
! [G2: int > nat,F: nat > int] :
( ! [X3: nat] :
( ( G2 @ ( F @ X3 ) )
= X3 )
=> ( ( image_int_nat @ G2 @ top_top_set_int )
= top_top_set_nat ) ) ).
% surjI
thf(fact_1124_surjI,axiom,
! [G2: int > int,F: int > int] :
( ! [X3: int] :
( ( G2 @ ( F @ X3 ) )
= X3 )
=> ( ( image_int_int @ G2 @ top_top_set_int )
= top_top_set_int ) ) ).
% surjI
thf(fact_1125_surjE,axiom,
! [F: real > real,Y: real] :
( ( ( image_real_real @ F @ top_top_set_real )
= top_top_set_real )
=> ~ ! [X3: real] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1126_surjE,axiom,
! [F: real > nat,Y: nat] :
( ( ( image_real_nat @ F @ top_top_set_real )
= top_top_set_nat )
=> ~ ! [X3: real] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1127_surjE,axiom,
! [F: real > int,Y: int] :
( ( ( image_real_int @ F @ top_top_set_real )
= top_top_set_int )
=> ~ ! [X3: real] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1128_surjE,axiom,
! [F: nat > real,Y: real] :
( ( ( image_nat_real @ F @ top_top_set_nat )
= top_top_set_real )
=> ~ ! [X3: nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1129_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X3: nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1130_surjE,axiom,
! [F: nat > int,Y: int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ~ ! [X3: nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1131_surjE,axiom,
! [F: int > real,Y: real] :
( ( ( image_int_real @ F @ top_top_set_int )
= top_top_set_real )
=> ~ ! [X3: int] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1132_surjE,axiom,
! [F: int > nat,Y: nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
=> ~ ! [X3: int] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1133_surjE,axiom,
! [F: int > int,Y: int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ~ ! [X3: int] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1134_surjD,axiom,
! [F: real > real,Y: real] :
( ( ( image_real_real @ F @ top_top_set_real )
= top_top_set_real )
=> ? [X3: real] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1135_surjD,axiom,
! [F: real > nat,Y: nat] :
( ( ( image_real_nat @ F @ top_top_set_real )
= top_top_set_nat )
=> ? [X3: real] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1136_surjD,axiom,
! [F: real > int,Y: int] :
( ( ( image_real_int @ F @ top_top_set_real )
= top_top_set_int )
=> ? [X3: real] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1137_surjD,axiom,
! [F: nat > real,Y: real] :
( ( ( image_nat_real @ F @ top_top_set_nat )
= top_top_set_real )
=> ? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1138_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1139_surjD,axiom,
! [F: nat > int,Y: int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1140_surjD,axiom,
! [F: int > real,Y: real] :
( ( ( image_int_real @ F @ top_top_set_int )
= top_top_set_real )
=> ? [X3: int] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1141_surjD,axiom,
! [F: int > nat,Y: nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
=> ? [X3: int] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1142_surjD,axiom,
! [F: int > int,Y: int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ? [X3: int] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1143_range__eqI,axiom,
! [B: real,F: real > real,X: real] :
( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_real_real @ F @ top_top_set_real ) ) ) ).
% range_eqI
thf(fact_1144_range__eqI,axiom,
! [B: int,F: real > int,X: real] :
( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_real_int @ F @ top_top_set_real ) ) ) ).
% range_eqI
thf(fact_1145_range__eqI,axiom,
! [B: real,F: nat > real,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_nat_real @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1146_range__eqI,axiom,
! [B: int,F: nat > int,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_nat_int @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1147_range__eqI,axiom,
! [B: real,F: int > real,X: int] :
( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_int_real @ F @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_1148_range__eqI,axiom,
! [B: int,F: int > int,X: int] :
( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_int_int @ F @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_1149_rangeI,axiom,
! [F: real > real,X: real] : ( member_real @ ( F @ X ) @ ( image_real_real @ F @ top_top_set_real ) ) ).
% rangeI
thf(fact_1150_rangeI,axiom,
! [F: real > int,X: real] : ( member_int @ ( F @ X ) @ ( image_real_int @ F @ top_top_set_real ) ) ).
% rangeI
thf(fact_1151_rangeI,axiom,
! [F: nat > real,X: nat] : ( member_real @ ( F @ X ) @ ( image_nat_real @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1152_rangeI,axiom,
! [F: nat > int,X: nat] : ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1153_rangeI,axiom,
! [F: int > real,X: int] : ( member_real @ ( F @ X ) @ ( image_int_real @ F @ top_top_set_int ) ) ).
% rangeI
thf(fact_1154_rangeI,axiom,
! [F: int > int,X: int] : ( member_int @ ( F @ X ) @ ( image_int_int @ F @ top_top_set_int ) ) ).
% rangeI
thf(fact_1155_subset__UNIV,axiom,
! [A4: set_real] : ( ord_less_eq_set_real @ A4 @ top_top_set_real ) ).
% subset_UNIV
thf(fact_1156_subset__UNIV,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_1157_subset__UNIV,axiom,
! [A4: set_int] : ( ord_less_eq_set_int @ A4 @ top_top_set_int ) ).
% subset_UNIV
thf(fact_1158_not__UNIV__eq__Icc,axiom,
! [L2: real,H2: real] :
( top_top_set_real
!= ( set_or1222579329274155063t_real @ L2 @ H2 ) ) ).
% not_UNIV_eq_Icc
thf(fact_1159_not__UNIV__eq__Icc,axiom,
! [L2: nat,H2: nat] :
( top_top_set_nat
!= ( set_or1269000886237332187st_nat @ L2 @ H2 ) ) ).
% not_UNIV_eq_Icc
thf(fact_1160_not__UNIV__eq__Icc,axiom,
! [L2: int,H2: int] :
( top_top_set_int
!= ( set_or1266510415728281911st_int @ L2 @ H2 ) ) ).
% not_UNIV_eq_Icc
thf(fact_1161_monotoneD,axiom,
! [Orda: real > real > $o,Ordb: real > real > $o,F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ Orda @ Ordb @ F )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monotoneD
thf(fact_1162_monotoneD,axiom,
! [Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ Orda @ Ordb @ F )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monotoneD
thf(fact_1163_monotoneI,axiom,
! [Orda: real > real > $o,Ordb: real > real > $o,F: real > real] :
( ! [X3: real,Y2: real] :
( ( Orda @ X3 @ Y2 )
=> ( Ordb @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( monoto4017252874604999745l_real @ top_top_set_real @ Orda @ Ordb @ F ) ) ).
% monotoneI
thf(fact_1164_monotoneI,axiom,
! [Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat] :
( ! [X3: nat,Y2: nat] :
( ( Orda @ X3 @ Y2 )
=> ( Ordb @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ Orda @ Ordb @ F ) ) ).
% monotoneI
thf(fact_1165_not__UNIV__eq__Ici,axiom,
! [L2: int] :
( top_top_set_int
!= ( set_ord_atLeast_int @ L2 ) ) ).
% not_UNIV_eq_Ici
thf(fact_1166_not__UNIV__eq__Ici,axiom,
! [L2: real] :
( top_top_set_real
!= ( set_ord_atLeast_real @ L2 ) ) ).
% not_UNIV_eq_Ici
thf(fact_1167_Rings_Omono__mult,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ ( times_times_real @ A ) ) ) ).
% Rings.mono_mult
thf(fact_1168_Rings_Omono__mult,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( times_times_nat @ A ) ) ) ).
% Rings.mono_mult
thf(fact_1169_Rings_Omono__mult,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( monotone_on_int_int @ top_top_set_int @ ord_less_eq_int @ ord_less_eq_int @ ( times_times_int @ A ) ) ) ).
% Rings.mono_mult
thf(fact_1170_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1171_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1172_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1173_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1174_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1175_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1176_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_1177_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_1178_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_1179_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_1180_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1181_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1182_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1183_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1184_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1185_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1186_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1187_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1188_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1189_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1190_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1191_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1192_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_1193_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_1194_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_1195_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_1196_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_1197_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1198_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y5: real] :
? [N4: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1199_div__le__mono,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1200_div__le__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ).
% div_le_dividend
thf(fact_1201_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1202_div__le__mono2,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1203_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1204_square__continuous,axiom,
! [E: real,X: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ Y5 @ X ) ) @ D3 )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( times_times_real @ Y5 @ Y5 ) @ ( times_times_real @ X @ X ) ) ) @ E ) ) ) ) ).
% square_continuous
thf(fact_1205_atLeast__0,axiom,
( ( set_ord_atLeast_nat @ zero_zero_nat )
= top_top_set_nat ) ).
% atLeast_0
thf(fact_1206_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1207_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1208_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1209_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1210_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1211_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1212_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1213_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1214_diff__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1215_mono__times__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( times_times_nat @ N ) ) ) ).
% mono_times_nat
thf(fact_1216_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1217_div__times__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_1218_times__div__less__eq__dividend,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_1219_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_1220_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1221_plusinfinity,axiom,
! [D: int,P4: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z: int] :
! [X3: int] :
( ( ord_less_int @ Z @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1222_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N @ Q2 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q2 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1223_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1224_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1225_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1226_lower__def,axiom,
( lower
= ( ^ [X4: real] : ( a_seg @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ n ) @ X4 ) @ a ) ) ) ) ) ) ).
% lower_def
thf(fact_1227_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N4: nat] :
( K
!= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% nonneg_int_cases
thf(fact_1228_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N4: nat] :
( K
= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1229_n__def,axiom,
( n
= ( nat2 @ ( archim6058952711729229775r_real @ ( divide_divide_real @ a @ delta ) ) ) ) ).
% n_def
thf(fact_1230_floor__divide__real__eq__div,axiom,
! [B: int,A: real] :
( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
= ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% floor_divide_real_eq_div
thf(fact_1231_real__of__int__div4,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).
% real_of_int_div4
thf(fact_1232_int__in__range__abs,axiom,
! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% int_in_range_abs
thf(fact_1233_real__of__int__div2,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) ) ).
% real_of_int_div2
thf(fact_1234_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1235_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1236_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M4: nat,N4: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% int_diff_cases
thf(fact_1237_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1238_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_1239_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_1240_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1241_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1242_int__nat__eq,axiom,
! [Z2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_1243_nat__abs__mult__distrib,axiom,
! [W2: int,Z2: int] :
( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W2 @ Z2 ) ) )
= ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W2 ) ) @ ( nat2 @ ( abs_abs_int @ Z2 ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_1244_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1245_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
& ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% ex_nat
thf(fact_1246_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
! [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% all_nat
thf(fact_1247_eq__nat__nat__iff,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z5 ) )
= ( Z2 = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1248_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1249_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_1250_nat__0__le,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) ) ).
% nat_0_le
thf(fact_1251_int__eq__iff,axiom,
! [M2: nat,Z2: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z2 )
= ( ( M2
= ( nat2 @ Z2 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).
% int_eq_iff
thf(fact_1252_int__minus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M2 ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ) ).
% int_minus
thf(fact_1253_div__abs__eq__div__nat,axiom,
! [K: int,L: int] :
( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% div_abs_eq_div_nat
thf(fact_1254_nat__eq__iff,axiom,
! [W2: int,M2: nat] :
( ( ( nat2 @ W2 )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1255_nat__eq__iff2,axiom,
! [M2: nat,W2: int] :
( ( M2
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1256_nat__le__eq__zle,axiom,
! [W2: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_1257_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1258_nat__less__eq__zless,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_1259_nat__mult__distrib,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( nat2 @ ( times_times_int @ Z2 @ Z5 ) )
= ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1260_nat__abs__int__diff,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ B @ A ) ) )
& ( ~ ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ A @ B ) ) ) ) ).
% nat_abs_int_diff
thf(fact_1261_nat__floor__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= zero_zero_nat ) ) ).
% nat_floor_neg
thf(fact_1262_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_1263_nat__diff__distrib,axiom,
! [Z5: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_eq_int @ Z5 @ Z2 )
=> ( ( nat2 @ ( minus_minus_int @ Z2 @ Z5 ) )
= ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1264_le__nat__floor,axiom,
! [X: nat,A: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
=> ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% le_nat_floor
thf(fact_1265_nat__div__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_1266_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_1267_nat__less__iff,axiom,
! [W2: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M2 )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_1268_real__norm__def,axiom,
real_V7735802525324610683m_real = abs_abs_real ).
% real_norm_def
thf(fact_1269_integral__eq__0__iff,axiom,
! [A: real,B: real,F: real > real] :
( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ( ord_less_real @ A @ B )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( ( hensto2714581292692559302l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
= zero_zero_real )
= ( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ( F @ X4 )
= zero_zero_real ) ) ) ) ) ) ) ).
% integral_eq_0_iff
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_real @ ( f1 @ x ) @ ( f @ x ) ).
%------------------------------------------------------------------------------