TPTP Problem File: SLH0287^1.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_00697_030537__13221874_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1410 ( 568 unt; 134 typ; 0 def)
% Number of atoms : 3729 (1324 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 12671 ( 292 ~; 114 |; 307 &;10395 @)
% ( 0 <=>;1563 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 809 ( 809 >; 0 *; 0 +; 0 <<)
% Number of symbols : 130 ( 127 usr; 15 con; 0-4 aty)
% Number of variables : 3774 ( 272 ^;3367 !; 135 ?;3774 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:32:08.733
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
set_set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (127)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
comple4887499456419720421f_real: set_real > real ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
comple1385675409528146559p_real: set_real > real ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Real__Oreal_J,type,
comple3096694443085538997t_real: set_set_real > set_real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
comp_real_real_real: ( real > real ) > ( real > real ) > real > real ).
thf(sy_c_Fun_Omonotone__on_001t__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_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Real__Oreal_J_M_Eo_J,type,
minus_2011193488284532564real_o: ( set_real > $o ) > ( set_real > $o ) > set_real > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_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_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
minus_5467046032205032049t_real: set_set_real > set_set_real > set_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_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
groups1932886352136224148al_int: ( real > int ) > set_real > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Real__Oreal_J_001t__Int__Oint,type,
groups3009712052913938890al_int: ( set_real > int ) > set_set_real > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Real__Oreal_J_001t__Nat__Onat,type,
groups3012202523422989166al_nat: ( set_real > nat ) > set_set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Real__Oreal_J_001t__Real__Oreal,type,
groups8702937949983641418l_real: ( set_real > real ) > set_set_real > real ).
thf(sy_c_Henstock__Kurzweil__Integration_Ohas__integral_001t__Real__Oreal_001t__Real__Oreal,type,
hensto240673015341029504l_real: ( 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_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
bot_bot_set_set_real: set_set_real ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Real__Oreal_J_M_Eo_J,type,
ord_less_set_real_o: ( set_real > $o ) > ( set_real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
ord_le7926960851185191020t_real: set_set_real > set_set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Real__Oreal_J_M_Eo_J,type,
ord_le2392157289819280397real_o: ( set_real > $o ) > ( set_real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
ord_le3558479182127378552t_real: set_set_real > set_set_real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Real__Oreal_J,type,
collect_set_real: ( set_real > $o ) > set_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__Set__Oset_It__Real__Oreal_J,type,
image_int_set_real: ( int > set_real ) > set_int > set_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__Nat__Onat_001t__Set__Oset_It__Real__Oreal_J,type,
image_nat_set_real: ( nat > set_real ) > set_nat > set_set_real ).
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_Oimage_001t__Real__Oreal_001t__Set__Oset_It__Real__Oreal_J,type,
image_real_set_real: ( real > set_real ) > set_real > set_set_real ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Int__Oint,type,
image_set_real_int: ( set_real > int ) > set_set_real > set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Nat__Onat,type,
image_set_real_nat: ( set_real > nat ) > set_set_real > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Real__Oreal,type,
image_set_real_real: ( set_real > real ) > set_set_real > set_real ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
image_2436557299294012491t_real: ( set_real > set_real ) > set_set_real > set_set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Real__Oreal_J,type,
set_or7743017856606604397t_real: set_real > set_real > set_set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
set_or66887138388493659n_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Real__Oreal_J,type,
set_or5046967147999637905t_real: set_real > set_real > set_set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
set_ord_atLeast_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
set_or5984915006950818249n_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Real__Oreal_J,type,
set_or3940062689191130623t_real: set_real > set_set_real ).
thf(sy_c_Tagged__Division_Odivision__of_001t__Real__Oreal,type,
tagged6100619406677346166f_real: set_set_real > set_real > $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_Youngs_Oregular__division,type,
regular_division: real > real > nat > set_set_real ).
thf(sy_c_Youngs_Osegment,type,
segment: nat > nat > set_real ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
member_set_real: set_real > set_set_real > $o ).
thf(sy_v__092_060delta_062____,type,
delta: real ).
thf(sy_v__092_060epsilon_062____,type,
epsilon: real ).
thf(sy_v_a,type,
a: real ).
thf(sy_v_a__seg____,type,
a_seg: real > real ).
thf(sy_v_b,type,
b: real ).
thf(sy_v_del____,type,
del: real > real ).
thf(sy_v_f,type,
f: real > real ).
thf(sy_v_f1____,type,
f1: real > real ).
thf(sy_v_f2____,type,
f2: real > real ).
thf(sy_v_g,type,
g: real > real ).
thf(sy_v_g1____,type,
g1: real > real ).
thf(sy_v_g2____,type,
g2: real > real ).
thf(sy_v_lower____,type,
lower: real > real ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_upper____,type,
upper: real > real ).
thf(sy_v_yidx____,type,
yidx: real > nat ).
% Relevant facts (1270)
thf(fact_0_f_I1_J,axiom,
( ( f @ zero_zero_real )
= zero_zero_real ) ).
% f(1)
thf(fact_1_False,axiom,
a != zero_zero_real ).
% False
thf(fact_2_f_I2_J,axiom,
( ( f @ a )
= b ) ).
% f(2)
thf(fact_3_a__seg__def,axiom,
( a_seg
= ( ^ [U: real] : ( divide_divide_real @ ( times_times_real @ U @ a ) @ ( semiri5074537144036343181t_real @ n ) ) ) ) ).
% a_seg_def
thf(fact_4_a__seg__eq__a__iff,axiom,
! [X: real] :
( ( ( a_seg @ X )
= a )
= ( X
= ( semiri5074537144036343181t_real @ n ) ) ) ).
% a_seg_eq_a_iff
thf(fact_5_int__g1,axiom,
( hensto240673015341029504l_real @ g1
@ ( groups6591440286371151544t_real
@ ^ [K: nat] : ( times_times_real @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( minus_minus_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K ) ) ) ) )
@ ( set_ord_lessThan_nat @ n ) )
@ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) ) ).
% int_g1
thf(fact_6_g1,axiom,
! [Y: real] :
( ( member_real @ Y @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) )
=> ( member_real @ ( g1 @ Y ) @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ) ) ).
% g1
thf(fact_7_fa__eq__b,axiom,
( ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ n ) ) )
= b ) ).
% fa_eq_b
thf(fact_8_a,axiom,
ord_less_eq_real @ zero_zero_real @ a ).
% a
thf(fact_9__092_060open_062_Ireal_An_A_K_Af_A_Ia__seg_A_Ireal_An_J_J_A_N_A_I_092_060Sum_062k_060n_O_Af_A_Ia__seg_A_Ireal_Ak_J_J_J_J_A_K_Aa_A_P_Areal_An_A_061_Aa_A_K_Ab_A_N_A_I_092_060Sum_062k_060n_O_Af_A_Ia__seg_A_Ireal_Ak_J_J_J_A_K_Aa_A_P_Areal_An_092_060close_062,axiom,
( ( divide_divide_real
@ ( times_times_real
@ ( minus_minus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ n ) @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ n ) ) ) )
@ ( groups6591440286371151544t_real
@ ^ [K: nat] : ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K ) ) )
@ ( set_ord_lessThan_nat @ n ) ) )
@ a )
@ ( semiri5074537144036343181t_real @ n ) )
= ( minus_minus_real @ ( times_times_real @ a @ b )
@ ( divide_divide_real
@ ( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [K: nat] : ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K ) ) )
@ ( set_ord_lessThan_nat @ n ) )
@ a )
@ ( semiri5074537144036343181t_real @ n ) ) ) ) ).
% \<open>(real n * f (a_seg (real n)) - (\<Sum>k<n. f (a_seg (real k)))) * a / real n = a * b - (\<Sum>k<n. f (a_seg (real k))) * a / real n\<close>
thf(fact_10__092_060open_062_I_092_060Sum_062k_060n_O_Aa__seg_A_Ireal_A_ISuc_Ak_J_J_A_K_A_If_A_Ia__seg_A_Ireal_A_ISuc_Ak_J_J_J_A_N_Af_A_Ia__seg_A_Ireal_Ak_J_J_J_J_A_061_A_I_092_060Sum_062k_060n_O_Areal_A_ISuc_Ak_J_A_K_A_If_A_Ia__seg_A_Ireal_A_ISuc_Ak_J_J_J_A_N_Af_A_Ia__seg_A_Ireal_Ak_J_J_J_J_A_K_A_Ia_A_P_Areal_An_J_092_060close_062,axiom,
( ( groups6591440286371151544t_real
@ ^ [K: nat] : ( times_times_real @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( minus_minus_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K ) ) ) ) )
@ ( set_ord_lessThan_nat @ n ) )
= ( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [K: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( minus_minus_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K ) ) ) ) )
@ ( set_ord_lessThan_nat @ n ) )
@ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) ) ) ).
% \<open>(\<Sum>k<n. a_seg (real (Suc k)) * (f (a_seg (real (Suc k))) - f (a_seg (real k)))) = (\<Sum>k<n. real (Suc k) * (f (a_seg (real (Suc k))) - f (a_seg (real k)))) * (a / real n)\<close>
thf(fact_11__092_060open_062_I_092_060Sum_062k_060n_O_Areal_A_ISuc_Ak_J_A_K_A_If_A_Ia__seg_A_Ireal_A_ISuc_Ak_J_J_J_A_N_Af_A_Ia__seg_A_Ireal_Ak_J_J_J_J_A_K_A_Ia_A_P_Areal_An_J_A_061_A_Ireal_An_A_K_Af_A_Ia__seg_A_Ireal_An_J_J_A_N_A_I_092_060Sum_062k_060n_O_Af_A_Ia__seg_A_Ireal_Ak_J_J_J_J_A_K_Aa_A_P_Areal_An_092_060close_062,axiom,
( ( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [K: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( minus_minus_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K ) ) ) ) )
@ ( set_ord_lessThan_nat @ n ) )
@ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) )
= ( divide_divide_real
@ ( times_times_real
@ ( minus_minus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ n ) @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ n ) ) ) )
@ ( groups6591440286371151544t_real
@ ^ [K: nat] : ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K ) ) )
@ ( set_ord_lessThan_nat @ n ) ) )
@ a )
@ ( semiri5074537144036343181t_real @ n ) ) ) ).
% \<open>(\<Sum>k<n. real (Suc k) * (f (a_seg (real (Suc k))) - f (a_seg (real k)))) * (a / real n) = (real n * f (a_seg (real n)) - (\<Sum>k<n. f (a_seg (real k)))) * a / real n\<close>
thf(fact_12_g1__def,axiom,
( g1
= ( ^ [Y2: real] : ( if_real @ ( Y2 = b ) @ a @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ ( yidx @ Y2 ) ) ) ) ) ) ) ).
% g1_def
thf(fact_13_g2,axiom,
! [Y: real] :
( ( member_real @ Y @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) )
=> ( member_real @ ( g2 @ Y ) @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ) ) ).
% g2
thf(fact_14_int__f1,axiom,
( hensto240673015341029504l_real @ f1
@ ( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [K: nat] : ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K ) ) )
@ ( set_ord_lessThan_nat @ n ) )
@ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) )
@ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ) ).
% int_f1
thf(fact_15_int__g2,axiom,
( hensto240673015341029504l_real @ g2
@ ( groups6591440286371151544t_real
@ ^ [K: nat] : ( times_times_real @ ( a_seg @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K ) ) ) ) )
@ ( set_ord_lessThan_nat @ n ) )
@ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) ) ).
% int_g2
thf(fact_16_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_17_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_18_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_19_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_20_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_21_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_22_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_23_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_24_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_25_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_26_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_27_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_28_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_29_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_30_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_31__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_32_g2__def,axiom,
( g2
= ( ^ [Y2: real] : ( if_real @ ( Y2 = zero_zero_real ) @ zero_zero_real @ ( a_seg @ ( semiri5074537144036343181t_real @ ( yidx @ Y2 ) ) ) ) ) ) ).
% g2_def
thf(fact_33_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_34__092_060open_062yidx_Ab_A_061_An_092_060close_062,axiom,
( ( yidx @ b )
= n ) ).
% \<open>yidx b = n\<close>
thf(fact_35_f1__lower,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ a )
=> ( ord_less_eq_real @ ( f1 @ X ) @ ( f @ X ) ) ) ) ).
% f1_lower
thf(fact_36_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_37_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_38_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_39_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_40_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_41_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_42_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_43_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_44_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_45_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_46_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_47_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_48_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_49_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_50_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_51_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_52_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_53_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_54_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_55_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_56_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_57_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_58_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_59_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_60_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_61_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_62_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_63_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_64_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_65_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
! [A: set_real,P: set_real > $o] :
( ( member_set_real @ A @ ( collect_set_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A2: set_set_real] :
( ( collect_set_real
@ ^ [X2: set_real] : ( member_set_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_71_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_72_fa__yidx__le,axiom,
! [Y: real] :
( ( member_real @ Y @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) )
=> ( ord_less_eq_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( yidx @ Y ) ) ) ) @ Y ) ) ).
% fa_yidx_le
thf(fact_73_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_74_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_75_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_76_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_77_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_78__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_79_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_80_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_81_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_82_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_83_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_84_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_85_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_86_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_87_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_88_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_89_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_90_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_91_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_92_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_93_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_94_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_95_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_96_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_97_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_98_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_99_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_100_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_101_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_102_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_103_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_104_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_105_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_106_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_107_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_108_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_109_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_110_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_111_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_112_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_113_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_114_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_115_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_116_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_117_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_118_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_119_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_120_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_121_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_122_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_123_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_124_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_125_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_126_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_127_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_128_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_129_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_130_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_131_frac__le__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% frac_le_eq
thf(fact_132_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_133_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_134_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_135_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_136_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_137_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_138_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_139_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_140_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_141_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_142_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_143_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_144_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_145_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_146_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_147_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_148_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_149_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_150_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_151_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_152_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_153_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_154_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_155_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_156_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_157_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_158_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_159_times__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_160_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_161_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_162_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_163_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_164_nonzero__eq__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_165_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A )
= ( B
= ( times_times_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_166_eq__divide__imp,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= B )
=> ( A
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_167_divide__eq__imp,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_168_eq__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_169_divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( divide_divide_real @ B @ C )
= A )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_170_frac__eq__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X @ Z )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_171_divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_172_diff__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_173_diff__frac__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_174_add__divide__eq__if__simps_I4_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_175_g2__le__g,axiom,
! [Y: real] :
( ( member_real @ Y @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) )
=> ( ord_less_eq_real @ ( g2 @ Y ) @ ( g @ Y ) ) ) ).
% g2_le_g
thf(fact_176_g__le__g1,axiom,
! [Y: real] :
( ( member_real @ Y @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) )
=> ( ord_less_eq_real @ ( g @ Y ) @ ( g1 @ Y ) ) ) ).
% g_le_g1
thf(fact_177_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_178_f2__upper,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ a )
=> ( ord_less_eq_real @ ( f @ X ) @ ( f2 @ X ) ) ) ) ).
% f2_upper
thf(fact_179_int__g2__D,axiom,
! [K2: nat] :
( ( ord_less_nat @ K2 @ n )
=> ( hensto240673015341029504l_real @ g2 @ ( times_times_real @ ( a_seg @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( minus_minus_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) ) ) @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) @ ( set_or1222579329274155063t_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K2 ) ) ) @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) ) ) ) ) ) ).
% int_g2_D
thf(fact_180_int__g1__D,axiom,
! [K2: nat] :
( ( ord_less_nat @ K2 @ n )
=> ( hensto240673015341029504l_real @ g1 @ ( times_times_real @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) ) @ ( minus_minus_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) ) ) @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) @ ( set_or1222579329274155063t_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K2 ) ) ) @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) ) ) ) ) ) ).
% int_g1_D
thf(fact_181_yidx__gt,axiom,
! [Y: real] :
( ( member_real @ Y @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) )
=> ( ord_less_real @ Y @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ ( yidx @ Y ) ) ) ) ) ) ) ).
% yidx_gt
thf(fact_182_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_183_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_184_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_185_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_186_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_187_yidx__equality,axiom,
! [Y: real,K2: nat] :
( ( member_real @ Y @ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) )
=> ( ( member_real @ Y @ ( set_or66887138388493659n_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K2 ) ) ) @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) ) ) ) )
=> ( ( yidx @ Y )
= K2 ) ) ) ).
% yidx_equality
thf(fact_188_has__integral__0__eq,axiom,
! [I: real,S: set_real] :
( ( hensto240673015341029504l_real
@ ^ [X2: real] : zero_zero_real
@ I
@ S )
= ( I = zero_zero_real ) ) ).
% has_integral_0_eq
thf(fact_189_that,axiom,
ord_less_real @ zero_zero_real @ epsilon ).
% that
thf(fact_190__092_060open_0620_A_060_Aa_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ a ).
% \<open>0 < a\<close>
thf(fact_191__092_060open_0620_A_060_An_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% \<open>0 < n\<close>
thf(fact_192__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_193_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_194_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_195_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_196_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_197_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_198_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_199_mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_200_mult__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ( times_times_nat @ M @ K2 )
= ( times_times_nat @ N @ K2 ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_201_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_202_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_203_Suc__diff__diff,axiom,
! [M: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_204_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_205_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_206_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_207_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_208_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_209_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_210_yidx__le__n,axiom,
! [Y: real] :
( ( ord_less_eq_real @ Y @ b )
=> ( ord_less_eq_nat @ ( yidx @ Y ) @ n ) ) ).
% yidx_le_n
thf(fact_211_yidx__less__n,axiom,
! [Y: real] :
( ( ord_less_real @ Y @ b )
=> ( ord_less_nat @ ( yidx @ Y ) @ n ) ) ).
% yidx_less_n
thf(fact_212_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_213_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_214_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_215_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_216_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_217_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_218_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_219_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_220_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_221_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_222_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_223_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_224_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_225_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_226_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_227_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_228_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_229_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_230_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_231_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_232_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_233_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_234_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_235_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_236_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_237_mult__le__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_238_mult__less__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_239_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_240_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_241_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_242_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_243_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_244_has__integral__restrict,axiom,
! [S: set_real,T: set_real,F: real > real,I: real] :
( ( ord_less_eq_set_real @ S @ T )
=> ( ( hensto240673015341029504l_real
@ ^ [X2: real] : ( if_real @ ( member_real @ X2 @ S ) @ ( F @ X2 ) @ zero_zero_real )
@ I
@ T )
= ( hensto240673015341029504l_real @ F @ I @ S ) ) ) ).
% has_integral_restrict
thf(fact_245_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_246_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_247_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_248_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_249_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_250_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_251_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_252_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_253_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_254_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_255_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_256_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_257_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_258_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_259_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_260_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_261_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_262_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_263_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_264_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_265_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_266_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_267_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_268_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_269_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_270_yidx__def,axiom,
( yidx
= ( ^ [Y2: real] :
( ord_Least_nat
@ ^ [K: nat] : ( ord_less_real @ Y2 @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ) ) ) ).
% yidx_def
thf(fact_271_del__gt0,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_real @ zero_zero_real @ ( del @ E ) ) ) ).
% del_gt0
thf(fact_272_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_273_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_274_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_275_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_276_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_277_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_278_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_279_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_280_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_281_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_282_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_283_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_284_linordered__field__no__lb,axiom,
! [X3: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X3 ) ).
% linordered_field_no_lb
thf(fact_285_linordered__field__no__ub,axiom,
! [X3: real] :
? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_286_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_287_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_288_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_289_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_290_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_291_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_292_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_293_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_294_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_295_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_296_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_297_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_298_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_299_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
& ( M2 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_300_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_301_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_302_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_303_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_304_less__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_305_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_306_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_307_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_308_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_309_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K3 )
=> ~ ( P @ I2 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_310_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_311_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_312_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_313_mult__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_314_mult__less__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_315_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_316_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
| ( M2 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_317_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ K3 )
=> ~ ( P @ I2 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_318_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_319_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_320_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_321_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_322_zdiv__int,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_323_le__div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ M @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_324_div__le__mono,axiom,
! [M: nat,N: nat,K2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K2 ) @ ( divide_divide_nat @ N @ K2 ) ) ) ).
% div_le_mono
thf(fact_325_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_326_div__le__mono2,axiom,
! [M: nat,N: nat,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K2 @ N ) @ ( divide_divide_nat @ K2 @ M ) ) ) ) ).
% div_le_mono2
thf(fact_327_div__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) ).
% div_mult2_eq
thf(fact_328_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_329_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_330_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_331_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_332_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_333_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_334_less__eq__div__iff__mult__less__eq,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_335_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_336_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_337_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_338_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_339_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_340_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_341_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_342_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_343_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_344_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_345_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_346_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_347_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_348_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_349_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_350_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_351_div__if,axiom,
( divide_divide_nat
= ( ^ [M2: nat,N4: nat] :
( if_nat
@ ( ( ord_less_nat @ M2 @ N4 )
| ( N4 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N4 ) @ N4 ) ) ) ) ) ).
% div_if
thf(fact_352_div__nat__eqI,axiom,
! [N: nat,Q: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q ) ) ) ).
% div_nat_eqI
thf(fact_353_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
& ( P @ Q2 ) ) ) ) ).
% split_div'
thf(fact_354_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_355_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_356_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_357_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_358_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_359_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_360_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_361_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_362_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_363_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_364_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_365_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_366_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I: nat] :
( ( P @ K2 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_367_Suc__mult__less__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_368_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_369_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_370_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_371_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_372_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_373_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I3 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_374_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_375_less__trans__Suc,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_376_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_377_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_378_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_379_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_380_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_381_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_382_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_383_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_384_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_385_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_386_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_387_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_388_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_389_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_390_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_391_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_392_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_393_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z2: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z2 )
=> ( R @ X4 @ Z2 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_394_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_395_Suc__mult__le__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_396_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_397_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_398_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_399_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_400_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_401_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_402_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_403_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_404_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_405_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_406_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_407_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_408_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_409_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_410_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_411_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% of_nat_diff
thf(fact_412_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_413_has__integral__on__superset,axiom,
! [F: real > real,I: real,S: set_real,T: set_real] :
( ( hensto240673015341029504l_real @ F @ I @ S )
=> ( ! [X4: real] :
( ~ ( member_real @ X4 @ S )
=> ( ( F @ X4 )
= zero_zero_real ) )
=> ( ( ord_less_eq_set_real @ S @ T )
=> ( hensto240673015341029504l_real @ F @ I @ T ) ) ) ) ).
% has_integral_on_superset
thf(fact_414_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_415_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_416_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_417_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_418_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_419_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_420_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_421_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_422_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_423_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_424_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_425_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_426_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_427_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_428_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_429_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_430_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_431_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_432_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_433_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_434_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_435_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_436_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_437_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_438_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_439_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_440_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_441_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_442_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_443_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_444_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_445_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_446_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_447_mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_448_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_449_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_450_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_451_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_452_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_453_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_454_mult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_455_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_456_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_457_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_458_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_459_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_460_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_461_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_462_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_463_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_464_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_465_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_466_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_467_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_468_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_469_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_470_has__integral__subset__le,axiom,
! [S2: set_real,T2: set_real,F: real > real,I: real,J: real] :
( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ( hensto240673015341029504l_real @ F @ I @ S2 )
=> ( ( hensto240673015341029504l_real @ F @ J @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ T2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ I @ J ) ) ) ) ) ).
% has_integral_subset_le
thf(fact_471_mult__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_472_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_473_mult__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_474_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_475_mult__left__less__imp__less,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_476_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_477_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_478_mult__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_479_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_480_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_481_mult__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_482_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_483_mult__right__less__imp__less,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_484_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_485_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_486_mult__strict__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_487_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_488_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_489_mult__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_490_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_491_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_492_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_493_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_494_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_495_mult__left__le__imp__le,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_496_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_497_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_498_mult__right__le__imp__le,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_499_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_500_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_501_mult__le__less__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_502_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_503_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_504_mult__less__le__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_505_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_506_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_507_frac__le,axiom,
! [Y: real,X: real,W: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% frac_le
thf(fact_508_frac__less,axiom,
! [X: real,Y: real,W: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% frac_less
thf(fact_509_frac__less2,axiom,
! [X: real,Y: real,W: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_real @ W @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% frac_less2
thf(fact_510_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_511_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_512_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_513_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_514_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_515_divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_516_less__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_517_neg__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_518_neg__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_519_pos__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_520_pos__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_521_mult__imp__div__pos__less,axiom,
! [Y: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_522_mult__imp__less__div__pos,axiom,
! [Y: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
=> ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_523_divide__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_524_divide__strict__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_525_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_526_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_527_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_528_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_529_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_530_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_531_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_532_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_533_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_534_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_535_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_536_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_537_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_538_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_539_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_540_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_541_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_542_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_543_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_544_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_545_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_546_Suc__mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K2 ) @ M )
= ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_547_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_548_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_549_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_550_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_551_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_552_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P @ X4 @ Y3 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_553_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_554_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_555_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_556_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_557_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_558_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_559_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_560_has__integral__eq,axiom,
! [S2: set_real,F: real > real,G: real > real,K2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( hensto240673015341029504l_real @ F @ K2 @ S2 )
=> ( hensto240673015341029504l_real @ G @ K2 @ S2 ) ) ) ).
% has_integral_eq
thf(fact_561_has__integral__cong,axiom,
! [S2: set_real,F: real > real,G: real > real,I: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( hensto240673015341029504l_real @ F @ I @ S2 )
= ( hensto240673015341029504l_real @ G @ I @ S2 ) ) ) ).
% has_integral_cong
thf(fact_562_has__integral__eq__rhs,axiom,
! [F: real > real,J: real,S: set_real,I: real] :
( ( hensto240673015341029504l_real @ F @ J @ S )
=> ( ( I = J )
=> ( hensto240673015341029504l_real @ F @ I @ S ) ) ) ).
% has_integral_eq_rhs
thf(fact_563_has__integral__unique,axiom,
! [F: real > real,K1: real,I: set_real,K22: real] :
( ( hensto240673015341029504l_real @ F @ K1 @ I )
=> ( ( hensto240673015341029504l_real @ F @ K22 @ I )
=> ( K1 = K22 ) ) ) ).
% has_integral_unique
thf(fact_564_divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_565_le__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_566_divide__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono
thf(fact_567_neg__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_le_eq
thf(fact_568_neg__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_le_divide_eq
thf(fact_569_pos__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_le_eq
thf(fact_570_pos__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_le_divide_eq
thf(fact_571_mult__imp__div__pos__le,axiom,
! [Y: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_le
thf(fact_572_mult__imp__le__div__pos,axiom,
! [Y: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X )
=> ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_573_divide__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_574_frac__less__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_575_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_576_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_577_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_578_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_579_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_580_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_581_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_582_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_583_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_584_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_585_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_586_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_587_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_588_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_589_lift__Suc__antimono__le,axiom,
! [F: nat > set_real,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_real @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_590_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_591_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_592_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_593_lift__Suc__mono__le,axiom,
! [F: nat > set_real,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_real @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_594_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_595_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_596_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_597_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_598_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_599_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_600_has__integral__is__0,axiom,
! [S: set_real,F: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ( F @ X4 )
= zero_zero_real ) )
=> ( hensto240673015341029504l_real @ F @ zero_zero_real @ S ) ) ).
% has_integral_is_0
thf(fact_601_has__integral__le,axiom,
! [F: real > real,I: real,S: set_real,G: real > real,J: real] :
( ( hensto240673015341029504l_real @ F @ I @ S )
=> ( ( hensto240673015341029504l_real @ G @ J @ S )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_real @ I @ J ) ) ) ) ).
% has_integral_le
thf(fact_602_has__integral__0,axiom,
! [S: set_real] :
( hensto240673015341029504l_real
@ ^ [X2: real] : zero_zero_real
@ zero_zero_real
@ S ) ).
% has_integral_0
thf(fact_603_has__integral__mult__left,axiom,
! [F: real > real,Y: real,S: set_real,C: real] :
( ( hensto240673015341029504l_real @ F @ Y @ S )
=> ( hensto240673015341029504l_real
@ ^ [X2: real] : ( times_times_real @ ( F @ X2 ) @ C )
@ ( times_times_real @ Y @ C )
@ S ) ) ).
% has_integral_mult_left
thf(fact_604_has__integral__mult__right,axiom,
! [F: real > real,Y: real,I: set_real,C: real] :
( ( hensto240673015341029504l_real @ F @ Y @ I )
=> ( hensto240673015341029504l_real
@ ^ [X2: real] : ( times_times_real @ C @ ( F @ X2 ) )
@ ( times_times_real @ C @ Y )
@ I ) ) ).
% has_integral_mult_right
thf(fact_605_has__integral__diff,axiom,
! [F: real > real,K2: real,S: set_real,G: real > real,L: real] :
( ( hensto240673015341029504l_real @ F @ K2 @ S )
=> ( ( hensto240673015341029504l_real @ G @ L @ S )
=> ( hensto240673015341029504l_real
@ ^ [X2: real] : ( minus_minus_real @ ( F @ X2 ) @ ( G @ X2 ) )
@ ( minus_minus_real @ K2 @ L )
@ S ) ) ) ).
% has_integral_diff
thf(fact_606_has__integral__divide,axiom,
! [F: real > real,Y: real,S: set_real,C: real] :
( ( hensto240673015341029504l_real @ F @ Y @ S )
=> ( hensto240673015341029504l_real
@ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ C )
@ ( divide_divide_real @ Y @ C )
@ S ) ) ).
% has_integral_divide
thf(fact_607_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_608_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_609_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_610_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_611_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_612_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_613_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_614_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_615_has__integral__nonneg,axiom,
! [F: real > real,I: real,S: set_real] :
( ( hensto240673015341029504l_real @ F @ I @ S )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ I ) ) ) ).
% has_integral_nonneg
thf(fact_616_has__integral__cmult__real,axiom,
! [C: real,F: real > real,X: real,A2: set_real] :
( ( ( C != zero_zero_real )
=> ( hensto240673015341029504l_real @ F @ X @ A2 ) )
=> ( hensto240673015341029504l_real
@ ^ [X2: real] : ( times_times_real @ C @ ( F @ X2 ) )
@ ( times_times_real @ C @ X )
@ A2 ) ) ).
% has_integral_cmult_real
thf(fact_617_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_618_atLeastLessThan__iff,axiom,
! [I: set_real,L: set_real,U2: set_real] :
( ( member_set_real @ I @ ( set_or5046967147999637905t_real @ L @ U2 ) )
= ( ( ord_less_eq_set_real @ L @ I )
& ( ord_less_set_real @ I @ U2 ) ) ) ).
% atLeastLessThan_iff
thf(fact_619_atLeastLessThan__iff,axiom,
! [I: real,L: real,U2: real] :
( ( member_real @ I @ ( set_or66887138388493659n_real @ L @ U2 ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_real @ I @ U2 ) ) ) ).
% atLeastLessThan_iff
thf(fact_620_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U2: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U2 ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U2 ) ) ) ).
% atLeastLessThan_iff
thf(fact_621_atLeastLessThan__iff,axiom,
! [I: int,L: int,U2: int] :
( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U2 ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_int @ I @ U2 ) ) ) ).
% atLeastLessThan_iff
thf(fact_622_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > int,K4: int,K2: nat] :
( ! [P2: nat] :
( ( ord_less_nat @ P2 @ N )
=> ( ord_less_eq_int @ ( F @ P2 ) @ K4 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K4 )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K4 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_623_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > real,K4: real,K2: nat] :
( ! [P2: nat] :
( ( ord_less_nat @ P2 @ N )
=> ( ord_less_eq_real @ ( F @ P2 ) @ K4 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ K4 )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K4 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_624_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > nat,K4: nat,K2: nat] :
( ! [P2: nat] :
( ( ord_less_nat @ P2 @ N )
=> ( ord_less_eq_nat @ ( F @ P2 ) @ K4 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ K4 )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K4 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_625_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_626_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_627_of__nat__sum,axiom,
! [F: set_real > nat,A2: set_set_real] :
( ( semiri5074537144036343181t_real @ ( groups3012202523422989166al_nat @ F @ A2 ) )
= ( groups8702937949983641418l_real
@ ^ [X2: set_real] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_628_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3539618377306564664at_int
@ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_629_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [X2: nat] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_630_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( semiri1316708129612266289at_nat @ ( F @ X2 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_631_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu: nat] : zero_zero_real
@ A2 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_632_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A2 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_633_sum_Oneutral__const,axiom,
! [A2: set_set_real] :
( ( groups8702937949983641418l_real
@ ^ [Uu: set_real] : zero_zero_real
@ A2 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_634__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] :
( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_real @ zero_zero_real @ ( Del @ E2 ) ) )
=> ~ ! [E2: real,X3: real,X5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X5 @ X3 ) ) @ ( Del @ E2 ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
=> ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( f @ X5 ) @ ( f @ X3 ) ) ) @ E2 ) ) ) ) ) ) ).
% \<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_635_sum__lessThan__telescope,axiom,
! [F: nat > int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [N4: nat] : ( minus_minus_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_636_sum__lessThan__telescope,axiom,
! [F: nat > real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( minus_minus_real @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_637_abs__idempotent,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_idempotent
thf(fact_638_abs__abs,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_abs
thf(fact_639_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_640_div__neg__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ( ( ord_less_int @ L @ K2 )
=> ( ( divide_divide_int @ K2 @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_641_div__pos__pos__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ K2 @ L )
=> ( ( divide_divide_int @ K2 @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_642_atLeastAtMost__iff,axiom,
! [I: set_real,L: set_real,U2: set_real] :
( ( member_set_real @ I @ ( set_or7743017856606604397t_real @ L @ U2 ) )
= ( ( ord_less_eq_set_real @ L @ I )
& ( ord_less_eq_set_real @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_643_atLeastAtMost__iff,axiom,
! [I: real,L: real,U2: real] :
( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U2 ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_eq_real @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_644_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U2: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U2 ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_645_atLeastAtMost__iff,axiom,
! [I: int,L: int,U2: int] :
( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U2 ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_eq_int @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_646_Icc__eq__Icc,axiom,
! [L: set_real,H2: set_real,L2: set_real,H3: set_real] :
( ( ( set_or7743017856606604397t_real @ L @ H2 )
= ( set_or7743017856606604397t_real @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_set_real @ L @ H2 )
& ~ ( ord_less_eq_set_real @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_647_Icc__eq__Icc,axiom,
! [L: real,H2: real,L2: real,H3: real] :
( ( ( set_or1222579329274155063t_real @ L @ H2 )
= ( set_or1222579329274155063t_real @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_real @ L @ H2 )
& ~ ( ord_less_eq_real @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_648_Icc__eq__Icc,axiom,
! [L: nat,H2: nat,L2: nat,H3: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H2 )
= ( set_or1269000886237332187st_nat @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_nat @ L @ H2 )
& ~ ( ord_less_eq_nat @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_649_Icc__eq__Icc,axiom,
! [L: int,H2: int,L2: int,H3: int] :
( ( ( set_or1266510415728281911st_int @ L @ H2 )
= ( set_or1266510415728281911st_int @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_int @ L @ H2 )
& ~ ( ord_less_eq_int @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_650_abs__0,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_0
thf(fact_651_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_652_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_zero
thf(fact_653_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_654_abs__eq__0,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0
thf(fact_655_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_656_abs__0__eq,axiom,
! [A: real] :
( ( zero_zero_real
= ( abs_abs_real @ A ) )
= ( A = zero_zero_real ) ) ).
% abs_0_eq
thf(fact_657_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_658_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_659_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_660_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_661_ivl__diff,axiom,
! [I: real,N: real,M: real] :
( ( ord_less_eq_real @ I @ N )
=> ( ( minus_minus_set_real @ ( set_or66887138388493659n_real @ I @ M ) @ ( set_or66887138388493659n_real @ I @ N ) )
= ( set_or66887138388493659n_real @ N @ M ) ) ) ).
% ivl_diff
thf(fact_662_ivl__diff,axiom,
! [I: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_663_ivl__diff,axiom,
! [I: int,N: int,M: int] :
( ( ord_less_eq_int @ I @ N )
=> ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I @ M ) @ ( set_or4662586982721622107an_int @ I @ N ) )
= ( set_or4662586982721622107an_int @ N @ M ) ) ) ).
% ivl_diff
thf(fact_664_lessThan__iff,axiom,
! [I: set_real,K2: set_real] :
( ( member_set_real @ I @ ( set_or3940062689191130623t_real @ K2 ) )
= ( ord_less_set_real @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_665_lessThan__iff,axiom,
! [I: real,K2: real] :
( ( member_real @ I @ ( set_or5984915006950818249n_real @ K2 ) )
= ( ord_less_real @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_666_lessThan__iff,axiom,
! [I: int,K2: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K2 ) )
= ( ord_less_int @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_667_lessThan__iff,axiom,
! [I: nat,K2: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K2 ) )
= ( ord_less_nat @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_668_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% abs_of_nat
thf(fact_669_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_670_real__divide__square__eq,axiom,
! [R2: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
= ( divide_divide_real @ A @ R2 ) ) ).
% real_divide_square_eq
thf(fact_671_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_672_lessThan__minus__lessThan,axiom,
! [N: real,M: real] :
( ( minus_minus_set_real @ ( set_or5984915006950818249n_real @ N ) @ ( set_or5984915006950818249n_real @ M ) )
= ( set_or66887138388493659n_real @ M @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_673_lessThan__minus__lessThan,axiom,
! [N: nat,M: nat] :
( ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( set_ord_lessThan_nat @ M ) )
= ( set_or4665077453230672383an_nat @ M @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_674_lessThan__minus__lessThan,axiom,
! [N: int,M: int] :
( ( minus_minus_set_int @ ( set_ord_lessThan_int @ N ) @ ( set_ord_lessThan_int @ M ) )
= ( set_or4662586982721622107an_int @ M @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_675_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= zero_zero_nat ) ) ).
% Least_eq_0
thf(fact_676_abs__sum__abs,axiom,
! [F: nat > real,A2: set_nat] :
( ( abs_abs_real
@ ( groups6591440286371151544t_real
@ ^ [A3: nat] : ( abs_abs_real @ ( F @ A3 ) )
@ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [A3: nat] : ( abs_abs_real @ ( F @ A3 ) )
@ A2 ) ) ).
% abs_sum_abs
thf(fact_677_abs__sum__abs,axiom,
! [F: set_real > real,A2: set_set_real] :
( ( abs_abs_real
@ ( groups8702937949983641418l_real
@ ^ [A3: set_real] : ( abs_abs_real @ ( F @ A3 ) )
@ A2 ) )
= ( groups8702937949983641418l_real
@ ^ [A3: set_real] : ( abs_abs_real @ ( F @ A3 ) )
@ A2 ) ) ).
% abs_sum_abs
thf(fact_678_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_679_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_680_abs__of__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_681_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_682_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_683_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_684_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_685_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_686_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_687_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_688_atLeastatMost__subset__iff,axiom,
! [A: set_real,B: set_real,C: set_real,D: set_real] :
( ( ord_le3558479182127378552t_real @ ( set_or7743017856606604397t_real @ A @ B ) @ ( set_or7743017856606604397t_real @ C @ D ) )
= ( ~ ( ord_less_eq_set_real @ A @ B )
| ( ( ord_less_eq_set_real @ C @ A )
& ( ord_less_eq_set_real @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_689_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_690_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_691_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_692_lessThan__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_693_lessThan__subset__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_694_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_695_ivl__subset,axiom,
! [I: real,J: real,M: real,N: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ I @ J ) @ ( set_or66887138388493659n_real @ M @ N ) )
= ( ( ord_less_eq_real @ J @ I )
| ( ( ord_less_eq_real @ M @ I )
& ( ord_less_eq_real @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_696_ivl__subset,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_697_ivl__subset,axiom,
! [I: int,J: int,M: int,N: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J ) @ ( set_or4662586982721622107an_int @ M @ N ) )
= ( ( ord_less_eq_int @ J @ I )
| ( ( ord_less_eq_int @ M @ I )
& ( ord_less_eq_int @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_698_sum__abs,axiom,
! [F: nat > real,A2: set_nat] :
( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
@ A2 ) ) ).
% sum_abs
thf(fact_699_sum__abs,axiom,
! [F: set_real > real,A2: set_set_real] :
( ord_less_eq_real @ ( abs_abs_real @ ( groups8702937949983641418l_real @ F @ A2 ) )
@ ( groups8702937949983641418l_real
@ ^ [I4: set_real] : ( abs_abs_real @ ( F @ I4 ) )
@ A2 ) ) ).
% sum_abs
thf(fact_700_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_701_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_702_sum__abs__ge__zero,axiom,
! [F: nat > real,A2: set_nat] :
( ord_less_eq_real @ zero_zero_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
@ A2 ) ) ).
% sum_abs_ge_zero
thf(fact_703_sum__abs__ge__zero,axiom,
! [F: set_real > real,A2: set_set_real] :
( ord_less_eq_real @ zero_zero_real
@ ( groups8702937949983641418l_real
@ ^ [I4: set_real] : ( abs_abs_real @ ( F @ I4 ) )
@ A2 ) ) ).
% sum_abs_ge_zero
thf(fact_704_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_705_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_706_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_707_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_708_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_709_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_710_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_711_eq__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_712_le__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_713_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_714_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_715_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_716_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_717_mult__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_718_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_719_mult__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_720_mult__le__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_721_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_722_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ~ ! [N3: nat] :
( K2
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_723_diff__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_724_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ? [N3: nat] :
( K2
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_725_diff__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_726_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_727_sum__subtractf__nat,axiom,
! [A2: set_real,G: real > nat,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups1935376822645274424al_nat
@ ^ [X2: real] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_728_sum__subtractf__nat,axiom,
! [A2: set_set_real,G: set_real > nat,F: set_real > nat] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups3012202523422989166al_nat
@ ^ [X2: set_real] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups3012202523422989166al_nat @ F @ A2 ) @ ( groups3012202523422989166al_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_729_sum__subtractf__nat,axiom,
! [A2: set_nat,G: nat > nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_730_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M7 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_731_zdiv__mono1,axiom,
! [A: int,A4: int,B: int] :
( ( ord_less_eq_int @ A @ A4 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A4 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_732_zdiv__mono2,axiom,
! [A: int,B3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ B3 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B3 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_733_zdiv__eq__0__iff,axiom,
! [I: int,K2: int] :
( ( ( divide_divide_int @ I @ K2 )
= zero_zero_int )
= ( ( K2 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K2 ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K2 @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_734_zdiv__mono1__neg,axiom,
! [A: int,A4: int,B: int] :
( ( ord_less_eq_int @ A @ A4 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_735_zdiv__mono2__neg,axiom,
! [A: int,B3: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ B3 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B3 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_736_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_737_div__int__pos__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K2 @ L ) )
= ( ( K2 = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K2 )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K2 @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_738_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_739_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_740_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_741_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_742_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_743_pos__imp__zdiv__pos__iff,axiom,
! [K2: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K2 ) )
= ( ord_less_eq_int @ K2 @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_744_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_745_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_746_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_747_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N3: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_748_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_749_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_750_int__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3539618377306564664at_int
@ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
@ A2 ) ) ).
% int_sum
thf(fact_751_sum__diff__distrib,axiom,
! [Q3: nat > nat,P: nat > nat,N: nat] :
( ! [X4: nat] : ( ord_less_eq_nat @ ( Q3 @ X4 ) @ ( P @ X4 ) )
=> ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q3 @ ( set_ord_lessThan_nat @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q3 @ X2 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_752_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_753_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_754_abs__ge__self,axiom,
! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% abs_ge_self
thf(fact_755_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_756_abs__eq__0__iff,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0_iff
thf(fact_757_abs__eq__0__iff,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_758_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_759_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_760_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_761_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_762_sum__diff__nat__ivl,axiom,
! [M: nat,N: nat,P3: nat,F: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ P3 )
=> ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ M @ N ) ) )
= ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ N @ P3 ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_763_sum__diff__nat__ivl,axiom,
! [M: nat,N: nat,P3: nat,F: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ P3 )
=> ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ M @ N ) ) )
= ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ N @ P3 ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_764_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_765_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K2: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_766_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K2
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_767_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N3: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_768_sum_Oshift__bounds__Suc__ivl,axiom,
! [G: nat > real,M: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_769_sum_Oshift__bounds__Suc__ivl,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_770_abs__ge__zero,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% abs_ge_zero
thf(fact_771_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_772_abs__not__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% abs_not_less_zero
thf(fact_773_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_774_abs__of__pos,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_pos
thf(fact_775_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_776_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_777_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_778_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_779_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_780_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_781_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_782_abs__mult__less,axiom,
! [A: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
=> ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
=> ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_783_abs__mult__less,axiom,
! [A: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
=> ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
=> ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_784_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_785_sum__SucD,axiom,
! [F: nat > nat,A2: set_nat,N: nat] :
( ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( suc @ N ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ).
% sum_SucD
thf(fact_786_sum__Suc__diff_H,axiom,
! [M: nat,N: nat,F: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% sum_Suc_diff'
thf(fact_787_sum__Suc__diff_H,axiom,
! [M: nat,N: nat,F: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% sum_Suc_diff'
thf(fact_788_Least__Suc2,axiom,
! [P: nat > $o,N: nat,Q3: nat > $o,M: nat] :
( ( P @ N )
=> ( ( Q3 @ M )
=> ( ~ ( P @ zero_zero_nat )
=> ( ! [K3: nat] :
( ( P @ ( suc @ K3 ) )
= ( Q3 @ K3 ) )
=> ( ( ord_Least_nat @ P )
= ( suc @ ( ord_Least_nat @ Q3 ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_789_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > int,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_int )
=> ( ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_790_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > real,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_791_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > nat,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_792_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= ( suc
@ ( ord_Least_nat
@ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_793_dense__eq0__I,axiom,
! [X: real] :
( ! [E3: real] :
( ( ord_less_real @ zero_zero_real @ E3 )
=> ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E3 ) )
=> ( X = zero_zero_real ) ) ).
% dense_eq0_I
thf(fact_794_abs__mult__pos_H,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( times_times_real @ X @ ( abs_abs_real @ Y ) )
= ( abs_abs_real @ ( times_times_real @ X @ Y ) ) ) ) ).
% abs_mult_pos'
thf(fact_795_abs__mult__pos_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ X @ ( abs_abs_int @ Y ) )
= ( abs_abs_int @ ( times_times_int @ X @ Y ) ) ) ) ).
% abs_mult_pos'
thf(fact_796_abs__eq__mult,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
| ( ord_less_eq_real @ A @ zero_zero_real ) )
& ( ( ord_less_eq_real @ zero_zero_real @ B )
| ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_797_abs__eq__mult,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
| ( ord_less_eq_int @ A @ zero_zero_int ) )
& ( ( ord_less_eq_int @ zero_zero_int @ B )
| ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
= ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_798_abs__mult__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
= ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_799_abs__mult__pos,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
= ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_800_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_801_complete__real,axiom,
! [S: set_real] :
( ? [X3: real] : ( member_real @ X3 @ S )
=> ( ? [Z4: real] :
! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Z4 ) )
=> ? [Y3: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Y3 ) )
& ! [Z4: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Z4 ) )
=> ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_802_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I: nat > real,J: real > nat,T: set_nat,H2: nat > real,G: real > real] :
( ! [A5: real] :
( ( member_real @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: real] :
( ( member_real @ A5 @ S )
=> ( member_nat @ ( J @ A5 ) @ T ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T )
=> ( member_real @ ( I @ B4 ) @ S ) )
=> ( ! [A5: real] :
( ( member_real @ A5 @ S )
=> ( ( H2 @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups6591440286371151544t_real @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_803_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I: nat > real,J: real > nat,T: set_nat,H2: nat > nat,G: real > nat] :
( ! [A5: real] :
( ( member_real @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: real] :
( ( member_real @ A5 @ S )
=> ( member_nat @ ( J @ A5 ) @ T ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T )
=> ( member_real @ ( I @ B4 ) @ S ) )
=> ( ! [A5: real] :
( ( member_real @ A5 @ S )
=> ( ( H2 @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_804_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: real > nat,J: nat > real,T: set_real,H2: real > real,G: nat > real] :
( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( member_real @ ( J @ A5 ) @ T ) )
=> ( ! [B4: real] :
( ( member_real @ B4 @ T )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: real] :
( ( member_real @ B4 @ T )
=> ( member_nat @ ( I @ B4 ) @ S ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( ( H2 @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups8097168146408367636l_real @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_805_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: nat > nat,J: nat > nat,T: set_nat,H2: nat > real,G: nat > real] :
( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( member_nat @ ( J @ A5 ) @ T ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T )
=> ( member_nat @ ( I @ B4 ) @ S ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( ( H2 @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups6591440286371151544t_real @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_806_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: real > nat,J: nat > real,T: set_real,H2: real > nat,G: nat > nat] :
( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( member_real @ ( J @ A5 ) @ T ) )
=> ( ! [B4: real] :
( ( member_real @ B4 @ T )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: real] :
( ( member_real @ B4 @ T )
=> ( member_nat @ ( I @ B4 ) @ S ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( ( H2 @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups1935376822645274424al_nat @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_807_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: nat > nat,J: nat > nat,T: set_nat,H2: nat > nat,G: nat > nat] :
( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( member_nat @ ( J @ A5 ) @ T ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T )
=> ( member_nat @ ( I @ B4 ) @ S ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( ( H2 @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_808_sum_Oreindex__bij__witness,axiom,
! [S: set_set_real,I: nat > set_real,J: set_real > nat,T: set_nat,H2: nat > nat,G: set_real > nat] :
( ! [A5: set_real] :
( ( member_set_real @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: set_real] :
( ( member_set_real @ A5 @ S )
=> ( member_nat @ ( J @ A5 ) @ T ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T )
=> ( member_set_real @ ( I @ B4 ) @ S ) )
=> ( ! [A5: set_real] :
( ( member_set_real @ A5 @ S )
=> ( ( H2 @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups3012202523422989166al_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_809_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I: set_real > real,J: real > set_real,T: set_set_real,H2: set_real > real,G: real > real] :
( ! [A5: real] :
( ( member_real @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: real] :
( ( member_real @ A5 @ S )
=> ( member_set_real @ ( J @ A5 ) @ T ) )
=> ( ! [B4: set_real] :
( ( member_set_real @ B4 @ T )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: set_real] :
( ( member_set_real @ B4 @ T )
=> ( member_real @ ( I @ B4 ) @ S ) )
=> ( ! [A5: real] :
( ( member_real @ A5 @ S )
=> ( ( H2 @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups8702937949983641418l_real @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_810_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: set_real > nat,J: nat > set_real,T: set_set_real,H2: set_real > real,G: nat > real] :
( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( member_set_real @ ( J @ A5 ) @ T ) )
=> ( ! [B4: set_real] :
( ( member_set_real @ B4 @ T )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: set_real] :
( ( member_set_real @ B4 @ T )
=> ( member_nat @ ( I @ B4 ) @ S ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( ( H2 @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups8702937949983641418l_real @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_811_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: set_real > nat,J: nat > set_real,T: set_set_real,H2: set_real > nat,G: nat > nat] :
( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( member_set_real @ ( J @ A5 ) @ T ) )
=> ( ! [B4: set_real] :
( ( member_set_real @ B4 @ T )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: set_real] :
( ( member_set_real @ B4 @ T )
=> ( member_nat @ ( I @ B4 ) @ S ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( ( H2 @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups3012202523422989166al_nat @ H2 @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_812_sum_Oeq__general__inverses,axiom,
! [B5: set_nat,K2: nat > real,A2: set_real,H2: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
=> ( ( member_real @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ( member_nat @ ( H2 @ X4 ) @ B5 )
& ( ( K2 @ ( H2 @ X4 ) )
= X4 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_813_sum_Oeq__general__inverses,axiom,
! [B5: set_nat,K2: nat > real,A2: set_real,H2: real > nat,Gamma: nat > nat,Phi: real > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
=> ( ( member_real @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ( member_nat @ ( H2 @ X4 ) @ B5 )
& ( ( K2 @ ( H2 @ X4 ) )
= X4 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups1935376822645274424al_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_814_sum_Oeq__general__inverses,axiom,
! [B5: set_real,K2: real > nat,A2: set_nat,H2: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B5 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_real @ ( H2 @ X4 ) @ B5 )
& ( ( K2 @ ( H2 @ X4 ) )
= X4 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8097168146408367636l_real @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_815_sum_Oeq__general__inverses,axiom,
! [B5: set_nat,K2: nat > nat,A2: set_nat,H2: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_nat @ ( H2 @ X4 ) @ B5 )
& ( ( K2 @ ( H2 @ X4 ) )
= X4 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_816_sum_Oeq__general__inverses,axiom,
! [B5: set_real,K2: real > nat,A2: set_nat,H2: nat > real,Gamma: real > nat,Phi: nat > nat] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B5 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_real @ ( H2 @ X4 ) @ B5 )
& ( ( K2 @ ( H2 @ X4 ) )
= X4 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups1935376822645274424al_nat @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_817_sum_Oeq__general__inverses,axiom,
! [B5: set_nat,K2: nat > nat,A2: set_nat,H2: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_nat @ ( H2 @ X4 ) @ B5 )
& ( ( K2 @ ( H2 @ X4 ) )
= X4 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_818_sum_Oeq__general__inverses,axiom,
! [B5: set_nat,K2: nat > set_real,A2: set_set_real,H2: set_real > nat,Gamma: nat > nat,Phi: set_real > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
=> ( ( member_set_real @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
=> ( ( member_nat @ ( H2 @ X4 ) @ B5 )
& ( ( K2 @ ( H2 @ X4 ) )
= X4 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3012202523422989166al_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_819_sum_Oeq__general__inverses,axiom,
! [B5: set_set_real,K2: set_real > real,A2: set_real,H2: real > set_real,Gamma: set_real > real,Phi: real > real] :
( ! [Y3: set_real] :
( ( member_set_real @ Y3 @ B5 )
=> ( ( member_real @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ( member_set_real @ ( H2 @ X4 ) @ B5 )
& ( ( K2 @ ( H2 @ X4 ) )
= X4 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups8702937949983641418l_real @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_820_sum_Oeq__general__inverses,axiom,
! [B5: set_set_real,K2: set_real > nat,A2: set_nat,H2: nat > set_real,Gamma: set_real > real,Phi: nat > real] :
( ! [Y3: set_real] :
( ( member_set_real @ Y3 @ B5 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_set_real @ ( H2 @ X4 ) @ B5 )
& ( ( K2 @ ( H2 @ X4 ) )
= X4 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8702937949983641418l_real @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_821_sum_Oeq__general__inverses,axiom,
! [B5: set_set_real,K2: set_real > nat,A2: set_nat,H2: nat > set_real,Gamma: set_real > nat,Phi: nat > nat] :
( ! [Y3: set_real] :
( ( member_set_real @ Y3 @ B5 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A2 )
& ( ( H2 @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_set_real @ ( H2 @ X4 ) @ B5 )
& ( ( K2 @ ( H2 @ X4 ) )
= X4 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups3012202523422989166al_nat @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_822_sum_Oeq__general,axiom,
! [B5: set_nat,A2: set_real,H2: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ( H2 @ X3 )
= Y3 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ( member_nat @ ( H2 @ X4 ) @ B5 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general
thf(fact_823_sum_Oeq__general,axiom,
! [B5: set_nat,A2: set_real,H2: real > nat,Gamma: nat > nat,Phi: real > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ( H2 @ X3 )
= Y3 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ( member_nat @ ( H2 @ X4 ) @ B5 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups1935376822645274424al_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general
thf(fact_824_sum_Oeq__general,axiom,
! [B5: set_real,A2: set_nat,H2: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B5 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ( H2 @ X3 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_real @ ( H2 @ X4 ) @ B5 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8097168146408367636l_real @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general
thf(fact_825_sum_Oeq__general,axiom,
! [B5: set_nat,A2: set_nat,H2: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ( H2 @ X3 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_nat @ ( H2 @ X4 ) @ B5 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general
thf(fact_826_sum_Oeq__general,axiom,
! [B5: set_real,A2: set_nat,H2: nat > real,Gamma: real > nat,Phi: nat > nat] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B5 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ( H2 @ X3 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_real @ ( H2 @ X4 ) @ B5 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups1935376822645274424al_nat @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general
thf(fact_827_sum_Oeq__general,axiom,
! [B5: set_nat,A2: set_nat,H2: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ( H2 @ X3 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_nat @ ( H2 @ X4 ) @ B5 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general
thf(fact_828_sum_Oeq__general,axiom,
! [B5: set_nat,A2: set_set_real,H2: set_real > nat,Gamma: nat > nat,Phi: set_real > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B5 )
=> ? [X3: set_real] :
( ( member_set_real @ X3 @ A2 )
& ( ( H2 @ X3 )
= Y3 )
& ! [Ya: set_real] :
( ( ( member_set_real @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
=> ( ( member_nat @ ( H2 @ X4 ) @ B5 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3012202523422989166al_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general
thf(fact_829_sum_Oeq__general,axiom,
! [B5: set_set_real,A2: set_real,H2: real > set_real,Gamma: set_real > real,Phi: real > real] :
( ! [Y3: set_real] :
( ( member_set_real @ Y3 @ B5 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ( H2 @ X3 )
= Y3 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ( member_set_real @ ( H2 @ X4 ) @ B5 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups8702937949983641418l_real @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general
thf(fact_830_sum_Oeq__general,axiom,
! [B5: set_set_real,A2: set_nat,H2: nat > set_real,Gamma: set_real > real,Phi: nat > real] :
( ! [Y3: set_real] :
( ( member_set_real @ Y3 @ B5 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ( H2 @ X3 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_set_real @ ( H2 @ X4 ) @ B5 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8702937949983641418l_real @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general
thf(fact_831_sum_Oeq__general,axiom,
! [B5: set_set_real,A2: set_nat,H2: nat > set_real,Gamma: set_real > nat,Phi: nat > nat] :
( ! [Y3: set_real] :
( ( member_set_real @ Y3 @ B5 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ( H2 @ X3 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H2 @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_set_real @ ( H2 @ X4 ) @ B5 )
& ( ( Gamma @ ( H2 @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups3012202523422989166al_nat @ Gamma @ B5 ) ) ) ) ).
% sum.eq_general
thf(fact_832_sum_Ocong,axiom,
! [A2: set_nat,B5: set_nat,G: nat > real,H2: nat > real] :
( ( A2 = B5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B5 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= ( groups6591440286371151544t_real @ H2 @ B5 ) ) ) ) ).
% sum.cong
thf(fact_833_sum_Ocong,axiom,
! [A2: set_nat,B5: set_nat,G: nat > nat,H2: nat > nat] :
( ( A2 = B5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B5 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= ( groups3542108847815614940at_nat @ H2 @ B5 ) ) ) ) ).
% sum.cong
thf(fact_834_sum_Ocong,axiom,
! [A2: set_set_real,B5: set_set_real,G: set_real > real,H2: set_real > real] :
( ( A2 = B5 )
=> ( ! [X4: set_real] :
( ( member_set_real @ X4 @ B5 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups8702937949983641418l_real @ G @ A2 )
= ( groups8702937949983641418l_real @ H2 @ B5 ) ) ) ) ).
% sum.cong
thf(fact_835_sum_Oswap,axiom,
! [G: nat > nat > real,B5: set_nat,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( groups6591440286371151544t_real @ ( G @ I4 ) @ B5 )
@ A2 )
= ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ I4 @ J3 )
@ A2 )
@ B5 ) ) ).
% sum.swap
thf(fact_836_sum_Oswap,axiom,
! [G: nat > set_real > real,B5: set_set_real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( groups8702937949983641418l_real @ ( G @ I4 ) @ B5 )
@ A2 )
= ( groups8702937949983641418l_real
@ ^ [J3: set_real] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ I4 @ J3 )
@ A2 )
@ B5 ) ) ).
% sum.swap
thf(fact_837_sum_Oswap,axiom,
! [G: nat > nat > nat,B5: set_nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( groups3542108847815614940at_nat @ ( G @ I4 ) @ B5 )
@ A2 )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ I4 @ J3 )
@ A2 )
@ B5 ) ) ).
% sum.swap
thf(fact_838_sum_Oswap,axiom,
! [G: set_real > nat > real,B5: set_nat,A2: set_set_real] :
( ( groups8702937949983641418l_real
@ ^ [I4: set_real] : ( groups6591440286371151544t_real @ ( G @ I4 ) @ B5 )
@ A2 )
= ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( groups8702937949983641418l_real
@ ^ [I4: set_real] : ( G @ I4 @ J3 )
@ A2 )
@ B5 ) ) ).
% sum.swap
thf(fact_839_sum_Oswap,axiom,
! [G: set_real > set_real > real,B5: set_set_real,A2: set_set_real] :
( ( groups8702937949983641418l_real
@ ^ [I4: set_real] : ( groups8702937949983641418l_real @ ( G @ I4 ) @ B5 )
@ A2 )
= ( groups8702937949983641418l_real
@ ^ [J3: set_real] :
( groups8702937949983641418l_real
@ ^ [I4: set_real] : ( G @ I4 @ J3 )
@ A2 )
@ B5 ) ) ).
% sum.swap
thf(fact_840_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E3: real] :
( ( ord_less_real @ zero_zero_real @ E3 )
& ( ord_less_real @ E3 @ D1 )
& ( ord_less_real @ E3 @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_841_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A2: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A5: real] :
( ( member_real @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_842_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A2: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A5: real] :
( ( member_real @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_843_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: set_real > nat,A2: set_set_real] :
( ( ( groups3012202523422989166al_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A5: set_real] :
( ( member_set_real @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_844_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > int,A2: set_real] :
( ( ( groups1932886352136224148al_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A5: real] :
( ( member_real @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_845_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > int,A2: set_nat] :
( ( ( groups3539618377306564664at_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_846_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: set_real > int,A2: set_set_real] :
( ( ( groups3009712052913938890al_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A5: set_real] :
( ( member_set_real @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_847_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A2: set_nat] :
( ( ( groups6591440286371151544t_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_848_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A2: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_849_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: set_real > real,A2: set_set_real] :
( ( ( groups8702937949983641418l_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A5: set_real] :
( ( member_set_real @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_850_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_851_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_852_sum_Oneutral,axiom,
! [A2: set_set_real,G: set_real > real] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups8702937949983641418l_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_853_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% less_eq_real_def
thf(fact_854_atLeastLessThan__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A @ B ) @ ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_eq_real @ B @ A )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_855_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_856_atLeastLessThan__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_eq_int @ B @ A )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_857_lessThan__strict__subset__iff,axiom,
! [M: real,N: real] :
( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
= ( ord_less_real @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_858_lessThan__strict__subset__iff,axiom,
! [M: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_859_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_860_atLeastLessThan__inj_I2_J,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_861_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_862_atLeastLessThan__inj_I2_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_863_atLeastLessThan__inj_I1_J,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_864_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_865_atLeastLessThan__inj_I1_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_866_Ico__eq__Ico,axiom,
! [L: real,H2: real,L2: real,H3: real] :
( ( ( set_or66887138388493659n_real @ L @ H2 )
= ( set_or66887138388493659n_real @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_real @ L @ H2 )
& ~ ( ord_less_real @ L2 @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_867_Ico__eq__Ico,axiom,
! [L: nat,H2: nat,L2: nat,H3: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H2 )
= ( set_or4665077453230672383an_nat @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_nat @ L @ H2 )
& ~ ( ord_less_nat @ L2 @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_868_Ico__eq__Ico,axiom,
! [L: int,H2: int,L2: int,H3: int] :
( ( ( set_or4662586982721622107an_int @ L @ H2 )
= ( set_or4662586982721622107an_int @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_int @ L @ H2 )
& ~ ( ord_less_int @ L2 @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_869_atLeastLessThan__eq__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_870_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_871_atLeastLessThan__eq__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_872_sum__mono,axiom,
! [K4: set_real,F: real > real,G: real > real] :
( ! [I3: real] :
( ( member_real @ I3 @ K4 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ K4 ) @ ( groups8097168146408367636l_real @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_873_sum__mono,axiom,
! [K4: set_real,F: real > nat,G: real > nat] :
( ! [I3: real] :
( ( member_real @ I3 @ K4 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K4 ) @ ( groups1935376822645274424al_nat @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_874_sum__mono,axiom,
! [K4: set_set_real,F: set_real > nat,G: set_real > nat] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ K4 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups3012202523422989166al_nat @ F @ K4 ) @ ( groups3012202523422989166al_nat @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_875_sum__mono,axiom,
! [K4: set_real,F: real > int,G: real > int] :
( ! [I3: real] :
( ( member_real @ I3 @ K4 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K4 ) @ ( groups1932886352136224148al_int @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_876_sum__mono,axiom,
! [K4: set_nat,F: nat > int,G: nat > int] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K4 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K4 ) @ ( groups3539618377306564664at_int @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_877_sum__mono,axiom,
! [K4: set_set_real,F: set_real > int,G: set_real > int] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ K4 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups3009712052913938890al_int @ F @ K4 ) @ ( groups3009712052913938890al_int @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_878_sum__mono,axiom,
! [K4: set_nat,F: nat > real,G: nat > real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K4 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K4 ) @ ( groups6591440286371151544t_real @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_879_sum__mono,axiom,
! [K4: set_nat,F: nat > nat,G: nat > nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K4 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K4 ) @ ( groups3542108847815614940at_nat @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_880_sum__mono,axiom,
! [K4: set_set_real,F: set_real > real,G: set_real > real] :
( ! [I3: set_real] :
( ( member_set_real @ I3 @ K4 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups8702937949983641418l_real @ F @ K4 ) @ ( groups8702937949983641418l_real @ G @ K4 ) ) ) ).
% sum_mono
thf(fact_881_sum__product,axiom,
! [F: nat > real,A2: set_nat,G: nat > real,B5: set_nat] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B5 ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] :
( groups6591440286371151544t_real
@ ^ [J3: nat] : ( times_times_real @ ( F @ I4 ) @ ( G @ J3 ) )
@ B5 )
@ A2 ) ) ).
% sum_product
thf(fact_882_sum__product,axiom,
! [F: nat > real,A2: set_nat,G: set_real > real,B5: set_set_real] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups8702937949983641418l_real @ G @ B5 ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] :
( groups8702937949983641418l_real
@ ^ [J3: set_real] : ( times_times_real @ ( F @ I4 ) @ ( G @ J3 ) )
@ B5 )
@ A2 ) ) ).
% sum_product
thf(fact_883_sum__product,axiom,
! [F: nat > nat,A2: set_nat,G: nat > nat,B5: set_nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B5 ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] :
( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( times_times_nat @ ( F @ I4 ) @ ( G @ J3 ) )
@ B5 )
@ A2 ) ) ).
% sum_product
thf(fact_884_sum__product,axiom,
! [F: set_real > real,A2: set_set_real,G: nat > real,B5: set_nat] :
( ( times_times_real @ ( groups8702937949983641418l_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B5 ) )
= ( groups8702937949983641418l_real
@ ^ [I4: set_real] :
( groups6591440286371151544t_real
@ ^ [J3: nat] : ( times_times_real @ ( F @ I4 ) @ ( G @ J3 ) )
@ B5 )
@ A2 ) ) ).
% sum_product
thf(fact_885_sum__product,axiom,
! [F: set_real > real,A2: set_set_real,G: set_real > real,B5: set_set_real] :
( ( times_times_real @ ( groups8702937949983641418l_real @ F @ A2 ) @ ( groups8702937949983641418l_real @ G @ B5 ) )
= ( groups8702937949983641418l_real
@ ^ [I4: set_real] :
( groups8702937949983641418l_real
@ ^ [J3: set_real] : ( times_times_real @ ( F @ I4 ) @ ( G @ J3 ) )
@ B5 )
@ A2 ) ) ).
% sum_product
thf(fact_886_sum__distrib__right,axiom,
! [F: nat > real,A2: set_nat,R2: real] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
= ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ R2 )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_887_sum__distrib__right,axiom,
! [F: nat > nat,A2: set_nat,R2: nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R2 )
= ( groups3542108847815614940at_nat
@ ^ [N4: nat] : ( times_times_nat @ ( F @ N4 ) @ R2 )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_888_sum__distrib__right,axiom,
! [F: set_real > real,A2: set_set_real,R2: real] :
( ( times_times_real @ ( groups8702937949983641418l_real @ F @ A2 ) @ R2 )
= ( groups8702937949983641418l_real
@ ^ [N4: set_real] : ( times_times_real @ ( F @ N4 ) @ R2 )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_889_sum__distrib__left,axiom,
! [R2: real,F: nat > real,A2: set_nat] :
( ( times_times_real @ R2 @ ( groups6591440286371151544t_real @ F @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( times_times_real @ R2 @ ( F @ N4 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_890_sum__distrib__left,axiom,
! [R2: nat,F: nat > nat,A2: set_nat] :
( ( times_times_nat @ R2 @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [N4: nat] : ( times_times_nat @ R2 @ ( F @ N4 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_891_sum__distrib__left,axiom,
! [R2: real,F: set_real > real,A2: set_set_real] :
( ( times_times_real @ R2 @ ( groups8702937949983641418l_real @ F @ A2 ) )
= ( groups8702937949983641418l_real
@ ^ [N4: set_real] : ( times_times_real @ R2 @ ( F @ N4 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_892_sum__subtractf,axiom,
! [F: nat > real,G: nat > real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X2: nat] : ( minus_minus_real @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_893_sum__subtractf,axiom,
! [F: set_real > real,G: set_real > real,A2: set_set_real] :
( ( groups8702937949983641418l_real
@ ^ [X2: set_real] : ( minus_minus_real @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_real @ ( groups8702937949983641418l_real @ F @ A2 ) @ ( groups8702937949983641418l_real @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_894_sum__divide__distrib,axiom,
! [F: nat > real,A2: set_nat,R2: real] :
( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
= ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ R2 )
@ A2 ) ) ).
% sum_divide_distrib
thf(fact_895_sum__divide__distrib,axiom,
! [F: set_real > real,A2: set_set_real,R2: real] :
( ( divide_divide_real @ ( groups8702937949983641418l_real @ F @ A2 ) @ R2 )
= ( groups8702937949983641418l_real
@ ^ [N4: set_real] : ( divide_divide_real @ ( F @ N4 ) @ R2 )
@ A2 ) ) ).
% sum_divide_distrib
thf(fact_896_lessThan__def,axiom,
( set_or5984915006950818249n_real
= ( ^ [U: real] :
( collect_real
@ ^ [X2: real] : ( ord_less_real @ X2 @ U ) ) ) ) ).
% lessThan_def
thf(fact_897_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U: int] :
( collect_int
@ ^ [X2: int] : ( ord_less_int @ X2 @ U ) ) ) ) ).
% lessThan_def
thf(fact_898_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ X2 @ U ) ) ) ) ).
% lessThan_def
thf(fact_899_sum__nonpos,axiom,
! [A2: set_real,F: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_900_sum__nonpos,axiom,
! [A2: set_real,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_901_sum__nonpos,axiom,
! [A2: set_set_real,F: set_real > nat] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3012202523422989166al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_902_sum__nonpos,axiom,
! [A2: set_real,F: real > int] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_903_sum__nonpos,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_904_sum__nonpos,axiom,
! [A2: set_set_real,F: set_real > int] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups3009712052913938890al_int @ F @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_905_sum__nonpos,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_906_sum__nonpos,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_907_sum__nonpos,axiom,
! [A2: set_set_real,F: set_real > real] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8702937949983641418l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_908_sum__nonneg,axiom,
! [A2: set_real,F: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_909_sum__nonneg,axiom,
! [A2: set_real,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_910_sum__nonneg,axiom,
! [A2: set_set_real,F: set_real > nat] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3012202523422989166al_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_911_sum__nonneg,axiom,
! [A2: set_real,F: real > int] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_912_sum__nonneg,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_913_sum__nonneg,axiom,
! [A2: set_set_real,F: set_real > int] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups3009712052913938890al_int @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_914_sum__nonneg,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_915_sum__nonneg,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_916_sum__nonneg,axiom,
! [A2: set_set_real,F: set_real > real] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8702937949983641418l_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_917_atLeastatMost__psubset__iff,axiom,
! [A: set_real,B: set_real,C: set_real,D: set_real] :
( ( ord_le7926960851185191020t_real @ ( set_or7743017856606604397t_real @ A @ B ) @ ( set_or7743017856606604397t_real @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_real @ A @ B )
| ( ( ord_less_eq_set_real @ C @ A )
& ( ord_less_eq_set_real @ B @ D )
& ( ( ord_less_set_real @ C @ A )
| ( ord_less_set_real @ B @ D ) ) ) )
& ( ord_less_eq_set_real @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_918_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_919_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_920_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_921_sum__cong__Suc,axiom,
! [A2: set_nat,F: nat > real,G: nat > real] :
( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ! [X4: nat] :
( ( member_nat @ ( suc @ X4 ) @ A2 )
=> ( ( F @ ( suc @ X4 ) )
= ( G @ ( suc @ X4 ) ) ) )
=> ( ( groups6591440286371151544t_real @ F @ A2 )
= ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).
% sum_cong_Suc
thf(fact_922_sum__cong__Suc,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ! [X4: nat] :
( ( member_nat @ ( suc @ X4 ) @ A2 )
=> ( ( F @ ( suc @ X4 ) )
= ( G @ ( suc @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% sum_cong_Suc
thf(fact_923_sum_Onat__diff__reindex,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nat_diff_reindex
thf(fact_924_sum_Onat__diff__reindex,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nat_diff_reindex
thf(fact_925_sum_Oivl__cong,axiom,
! [A: set_real,C: set_real,B: set_real,D: set_real,G: set_real > real,H2: set_real > real] :
( ( A = C )
=> ( ( B = D )
=> ( ! [X4: set_real] :
( ( ord_less_eq_set_real @ C @ X4 )
=> ( ( ord_less_set_real @ X4 @ D )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) ) )
=> ( ( groups8702937949983641418l_real @ G @ ( set_or5046967147999637905t_real @ A @ B ) )
= ( groups8702937949983641418l_real @ H2 @ ( set_or5046967147999637905t_real @ C @ D ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_926_sum_Oivl__cong,axiom,
! [A: nat,C: nat,B: nat,D: nat,G: nat > real,H2: nat > real] :
( ( A = C )
=> ( ( B = D )
=> ( ! [X4: nat] :
( ( ord_less_eq_nat @ C @ X4 )
=> ( ( ord_less_nat @ X4 @ D )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
= ( groups6591440286371151544t_real @ H2 @ ( set_or4665077453230672383an_nat @ C @ D ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_927_sum_Oivl__cong,axiom,
! [A: nat,C: nat,B: nat,D: nat,G: nat > nat,H2: nat > nat] :
( ( A = C )
=> ( ( B = D )
=> ( ! [X4: nat] :
( ( ord_less_eq_nat @ C @ X4 )
=> ( ( ord_less_nat @ X4 @ D )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
= ( groups3542108847815614940at_nat @ H2 @ ( set_or4665077453230672383an_nat @ C @ D ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_928_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: set_real,B: set_real,C: set_real,D: set_real] :
( ( ord_le3558479182127378552t_real @ ( set_or7743017856606604397t_real @ A @ B ) @ ( set_or5046967147999637905t_real @ C @ D ) )
= ( ( ord_less_eq_set_real @ A @ B )
=> ( ( ord_less_eq_set_real @ C @ A )
& ( ord_less_set_real @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_929_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or66887138388493659n_real @ C @ D ) )
= ( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_real @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_930_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_nat @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_931_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D ) )
= ( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ A )
& ( ord_less_int @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_932_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_933_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y5: real] :
? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_934_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_935_sumr__diff__mult__const2,axiom,
! [F: nat > int,N: nat,R2: int] :
( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ R2 ) )
= ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ R2 )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sumr_diff_mult_const2
thf(fact_936_sumr__diff__mult__const2,axiom,
! [F: nat > real,N: nat,R2: real] :
( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ R2 ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( minus_minus_real @ ( F @ I4 ) @ R2 )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sumr_diff_mult_const2
thf(fact_937_sum__lessThan__telescope_H,axiom,
! [F: nat > int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [N4: nat] : ( minus_minus_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_938_sum__lessThan__telescope_H,axiom,
! [F: nat > real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_939_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_940_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_941_f2__near__f1,axiom,
( ord_less_real
@ ( hensto2714581292692559302l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ a )
@ ^ [X2: real] : ( minus_minus_real @ ( f2 @ X2 ) @ ( f1 @ X2 ) ) )
@ epsilon ) ).
% f2_near_f1
thf(fact_942_Chebyshev__sum__upper,axiom,
! [N: nat,A: nat > int,B: nat > int] :
( ! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_int @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_int @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
=> ( ord_less_eq_int
@ ( times_times_int @ ( semiri1314217659103216013at_int @ N )
@ ( groups3539618377306564664at_int
@ ^ [K: nat] : ( times_times_int @ ( A @ K ) @ ( B @ K ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
@ ( times_times_int @ ( groups3539618377306564664at_int @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3539618377306564664at_int @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_943_Chebyshev__sum__upper,axiom,
! [N: nat,A: nat > real,B: nat > real] :
( ! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_real @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_real @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
=> ( ord_less_eq_real
@ ( times_times_real @ ( semiri5074537144036343181t_real @ N )
@ ( groups6591440286371151544t_real
@ ^ [K: nat] : ( times_times_real @ ( A @ K ) @ ( B @ K ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
@ ( times_times_real @ ( groups6591440286371151544t_real @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups6591440286371151544t_real @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_944_fim,axiom,
( ( image_real_real @ f @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) )
= ( set_or1222579329274155063t_real @ zero_zero_real @ b ) ) ).
% fim
thf(fact_945_integral__unique,axiom,
! [F: real > real,Y: real,K2: set_real] :
( ( hensto240673015341029504l_real @ F @ Y @ K2 )
=> ( ( hensto2714581292692559302l_real @ K2 @ F )
= Y ) ) ).
% integral_unique
thf(fact_946_integral__0,axiom,
! [S: set_real] :
( ( hensto2714581292692559302l_real @ S
@ ^ [X2: real] : zero_zero_real )
= zero_zero_real ) ).
% integral_0
thf(fact_947_Henstock__Kurzweil__Integration_Ointegral__mult__left,axiom,
! [S: set_real,F: real > real,C: real] :
( ( hensto2714581292692559302l_real @ S
@ ^ [X2: real] : ( times_times_real @ ( F @ X2 ) @ C ) )
= ( times_times_real @ ( hensto2714581292692559302l_real @ S @ F ) @ C ) ) ).
% Henstock_Kurzweil_Integration.integral_mult_left
thf(fact_948_Henstock__Kurzweil__Integration_Ointegral__mult__right,axiom,
! [S: set_real,C: real,F: real > real] :
( ( hensto2714581292692559302l_real @ S
@ ^ [X2: real] : ( times_times_real @ C @ ( F @ X2 ) ) )
= ( times_times_real @ C @ ( hensto2714581292692559302l_real @ S @ F ) ) ) ).
% Henstock_Kurzweil_Integration.integral_mult_right
thf(fact_949_Henstock__Kurzweil__Integration_Ointegral__divide,axiom,
! [S: set_real,F: real > real,Z: real] :
( ( hensto2714581292692559302l_real @ S
@ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ Z ) )
= ( divide_divide_real @ ( hensto2714581292692559302l_real @ S @ F ) @ Z ) ) ).
% Henstock_Kurzweil_Integration.integral_divide
thf(fact_950_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_951_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_952_image__minus__const__atLeastAtMost_H,axiom,
! [D: real,A: real,B: real] :
( ( image_real_real
@ ^ [T3: real] : ( minus_minus_real @ T3 @ D )
@ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ ( minus_minus_real @ A @ D ) @ ( minus_minus_real @ B @ D ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_953_image__minus__const__atLeastAtMost_H,axiom,
! [D: int,A: int,B: int] :
( ( image_int_int
@ ^ [T3: int] : ( minus_minus_int @ T3 @ D )
@ ( set_or1266510415728281911st_int @ A @ B ) )
= ( set_or1266510415728281911st_int @ ( minus_minus_int @ A @ D ) @ ( minus_minus_int @ B @ D ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_954_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_955_image__divide__atLeastAtMost,axiom,
! [D: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ( image_real_real
@ ^ [C2: real] : ( divide_divide_real @ C2 @ D )
@ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ ( divide_divide_real @ A @ D ) @ ( divide_divide_real @ B @ D ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_956_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_957_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_958_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_959_minus__int__code_I1_J,axiom,
! [K2: int] :
( ( minus_minus_int @ K2 @ zero_zero_int )
= K2 ) ).
% minus_int_code(1)
thf(fact_960_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_961_times__int__code_I1_J,axiom,
! [K2: int] :
( ( times_times_int @ K2 @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_962_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_963_zmult__zless__mono2,axiom,
! [I: int,J: int,K2: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ord_less_int @ ( times_times_int @ K2 @ I ) @ ( times_times_int @ K2 @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_964_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U2: nat] :
( ( set_or4665077453230672383an_nat @ L @ ( suc @ U2 ) )
= ( set_or1269000886237332187st_nat @ L @ U2 ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_965_Henstock__Kurzweil__Integration_Ointegral__cong,axiom,
! [S2: set_real,F: real > real,G: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( hensto2714581292692559302l_real @ S2 @ F )
= ( hensto2714581292692559302l_real @ S2 @ G ) ) ) ).
% Henstock_Kurzweil_Integration.integral_cong
thf(fact_966_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > real,M: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_967_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_968_sum_Onested__swap,axiom,
! [A: nat > nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( groups6591440286371151544t_real @ ( A @ I4 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I4 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( A @ I4 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% sum.nested_swap
thf(fact_969_sum_Onested__swap,axiom,
! [A: nat > nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( groups3542108847815614940at_nat @ ( A @ I4 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I4 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( A @ I4 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% sum.nested_swap
thf(fact_970_sum__shift__lb__Suc0__0,axiom,
! [F: nat > int,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_int )
=> ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_971_sum__shift__lb__Suc0__0,axiom,
! [F: nat > real,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_972_sum__shift__lb__Suc0__0,axiom,
! [F: nat > nat,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_973_sum__Suc__diff,axiom,
! [M: nat,N: nat,F: nat > int] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% sum_Suc_diff
thf(fact_974_sum__Suc__diff,axiom,
! [M: nat,N: nat,F: nat > real] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% sum_Suc_diff
thf(fact_975_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups6591440286371151544t_real
@ ^ [K: nat] : ( G @ ( suc @ K ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_976_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [K: nat] : ( G @ ( suc @ K ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_977_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_978_nat__mult__eq__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_979_nat__mult__less__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_980_nat__mult__div__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( K2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= zero_zero_nat ) )
& ( ( K2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_981_nat__mult__le__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_982_nat__mult__div__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_983_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A: nat > nat,B: nat > nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
=> ( ord_less_eq_nat
@ ( times_times_nat @ N
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( B @ I4 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
@ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_984_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_985_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K2: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_986_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_987_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
= ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_988_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_989_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_990_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_991_image__int__atLeastLessThan,axiom,
! [A: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
= ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastLessThan
thf(fact_992_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_993_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_994_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_995_verit__comp__simplify1_I2_J,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_996_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_997_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_998_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_999_minf_I7_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ~ ( ord_less_nat @ T2 @ X3 ) ) ).
% minf(7)
thf(fact_1000_minf_I7_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ~ ( ord_less_real @ T2 @ X3 ) ) ).
% minf(7)
thf(fact_1001_minf_I7_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z2 )
=> ~ ( ord_less_int @ T2 @ X3 ) ) ).
% minf(7)
thf(fact_1002_minf_I5_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ord_less_nat @ X3 @ T2 ) ) ).
% minf(5)
thf(fact_1003_minf_I5_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ( ord_less_real @ X3 @ T2 ) ) ).
% minf(5)
thf(fact_1004_minf_I5_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z2 )
=> ( ord_less_int @ X3 @ T2 ) ) ).
% minf(5)
thf(fact_1005_minf_I4_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( X3 != T2 ) ) ).
% minf(4)
thf(fact_1006_minf_I4_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ( X3 != T2 ) ) ).
% minf(4)
thf(fact_1007_minf_I4_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z2 )
=> ( X3 != T2 ) ) ).
% minf(4)
thf(fact_1008_minf_I3_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( X3 != T2 ) ) ).
% minf(3)
thf(fact_1009_minf_I3_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ( X3 != T2 ) ) ).
% minf(3)
thf(fact_1010_minf_I3_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z2 )
=> ( X3 != T2 ) ) ).
% minf(3)
thf(fact_1011_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ( ( P @ X3 )
| ( Q3 @ X3 ) )
= ( ( P4 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1012_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q3: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ( ( ( P @ X3 )
| ( Q3 @ X3 ) )
= ( ( P4 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1013_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q3: int > $o,Q4: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z2 )
=> ( ( ( P @ X3 )
| ( Q3 @ X3 ) )
= ( ( P4 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1014_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ( ( P @ X3 )
& ( Q3 @ X3 ) )
= ( ( P4 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1015_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q3: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ( ( ( P @ X3 )
& ( Q3 @ X3 ) )
= ( ( P4 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1016_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q3: int > $o,Q4: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z2 )
=> ( ( ( P @ X3 )
& ( Q3 @ X3 ) )
= ( ( P4 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1017_pinf_I7_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( ord_less_nat @ T2 @ X3 ) ) ).
% pinf(7)
thf(fact_1018_pinf_I7_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ord_less_real @ T2 @ X3 ) ) ).
% pinf(7)
thf(fact_1019_pinf_I7_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ Z2 @ X3 )
=> ( ord_less_int @ T2 @ X3 ) ) ).
% pinf(7)
thf(fact_1020_pinf_I5_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ~ ( ord_less_nat @ X3 @ T2 ) ) ).
% pinf(5)
thf(fact_1021_pinf_I5_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ~ ( ord_less_real @ X3 @ T2 ) ) ).
% pinf(5)
thf(fact_1022_pinf_I5_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ Z2 @ X3 )
=> ~ ( ord_less_int @ X3 @ T2 ) ) ).
% pinf(5)
thf(fact_1023_pinf_I4_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( X3 != T2 ) ) ).
% pinf(4)
thf(fact_1024_pinf_I4_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( X3 != T2 ) ) ).
% pinf(4)
thf(fact_1025_pinf_I4_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ Z2 @ X3 )
=> ( X3 != T2 ) ) ).
% pinf(4)
thf(fact_1026_pinf_I3_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( X3 != T2 ) ) ).
% pinf(3)
thf(fact_1027_pinf_I3_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( X3 != T2 ) ) ).
% pinf(3)
thf(fact_1028_pinf_I3_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ Z2 @ X3 )
=> ( X3 != T2 ) ) ).
% pinf(3)
thf(fact_1029_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( ( ( P @ X3 )
| ( Q3 @ X3 ) )
= ( ( P4 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1030_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q3: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ( ( P @ X3 )
| ( Q3 @ X3 ) )
= ( ( P4 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1031_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q3: int > $o,Q4: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ Z2 @ X3 )
=> ( ( ( P @ X3 )
| ( Q3 @ X3 ) )
= ( ( P4 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1032_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( ( ( P @ X3 )
& ( Q3 @ X3 ) )
= ( ( P4 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1033_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q3: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ( ( P @ X3 )
& ( Q3 @ X3 ) )
= ( ( P4 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1034_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q3: int > $o,Q4: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( Q3 @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ Z2 @ X3 )
=> ( ( ( P @ X3 )
& ( Q3 @ X3 ) )
= ( ( P4 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1035_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1036_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1037_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1038_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_1039_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_1040_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1041_minf_I8_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ~ ( ord_less_eq_real @ T2 @ X3 ) ) ).
% minf(8)
thf(fact_1042_minf_I8_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ T2 @ X3 ) ) ).
% minf(8)
thf(fact_1043_minf_I8_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z2 )
=> ~ ( ord_less_eq_int @ T2 @ X3 ) ) ).
% minf(8)
thf(fact_1044_minf_I6_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ( ord_less_eq_real @ X3 @ T2 ) ) ).
% minf(6)
thf(fact_1045_minf_I6_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ T2 ) ) ).
% minf(6)
thf(fact_1046_minf_I6_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z2 )
=> ( ord_less_eq_int @ X3 @ T2 ) ) ).
% minf(6)
thf(fact_1047_pinf_I8_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ord_less_eq_real @ T2 @ X3 ) ) ).
% pinf(8)
thf(fact_1048_pinf_I8_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ( ord_less_eq_nat @ T2 @ X3 ) ) ).
% pinf(8)
thf(fact_1049_pinf_I8_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ Z2 @ X3 )
=> ( ord_less_eq_int @ T2 @ X3 ) ) ).
% pinf(8)
thf(fact_1050_pinf_I6_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ T2 ) ) ).
% pinf(6)
thf(fact_1051_pinf_I6_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T2 ) ) ).
% pinf(6)
thf(fact_1052_pinf_I6_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ Z2 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ T2 ) ) ).
% pinf(6)
thf(fact_1053_verit__comp__simplify1_I3_J,axiom,
! [B3: real,A4: real] :
( ( ~ ( ord_less_eq_real @ B3 @ A4 ) )
= ( ord_less_real @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1054_verit__comp__simplify1_I3_J,axiom,
! [B3: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B3 @ A4 ) )
= ( ord_less_nat @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1055_verit__comp__simplify1_I3_J,axiom,
! [B3: int,A4: int] :
( ( ~ ( ord_less_eq_int @ B3 @ A4 ) )
= ( ord_less_int @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1056_inf__period_I1_J,axiom,
! [P: real > $o,D3: real,Q3: real > $o] :
( ! [X4: real,K3: real] :
( ( P @ X4 )
= ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D3 ) ) ) )
=> ( ! [X4: real,K3: real] :
( ( Q3 @ X4 )
= ( Q3 @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D3 ) ) ) )
=> ! [X3: real,K5: real] :
( ( ( P @ X3 )
& ( Q3 @ X3 ) )
= ( ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K5 @ D3 ) ) )
& ( Q3 @ ( minus_minus_real @ X3 @ ( times_times_real @ K5 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1057_inf__period_I1_J,axiom,
! [P: int > $o,D3: int,Q3: int > $o] :
( ! [X4: int,K3: int] :
( ( P @ X4 )
= ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ( ! [X4: int,K3: int] :
( ( Q3 @ X4 )
= ( Q3 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ! [X3: int,K5: int] :
( ( ( P @ X3 )
& ( Q3 @ X3 ) )
= ( ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K5 @ D3 ) ) )
& ( Q3 @ ( minus_minus_int @ X3 @ ( times_times_int @ K5 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1058_inf__period_I2_J,axiom,
! [P: real > $o,D3: real,Q3: real > $o] :
( ! [X4: real,K3: real] :
( ( P @ X4 )
= ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D3 ) ) ) )
=> ( ! [X4: real,K3: real] :
( ( Q3 @ X4 )
= ( Q3 @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D3 ) ) ) )
=> ! [X3: real,K5: real] :
( ( ( P @ X3 )
| ( Q3 @ X3 ) )
= ( ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K5 @ D3 ) ) )
| ( Q3 @ ( minus_minus_real @ X3 @ ( times_times_real @ K5 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1059_inf__period_I2_J,axiom,
! [P: int > $o,D3: int,Q3: int > $o] :
( ! [X4: int,K3: int] :
( ( P @ X4 )
= ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ( ! [X4: int,K3: int] :
( ( Q3 @ X4 )
= ( Q3 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ! [X3: int,K5: int] :
( ( ( P @ X3 )
| ( Q3 @ X3 ) )
= ( ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K5 @ D3 ) ) )
| ( Q3 @ ( minus_minus_int @ X3 @ ( times_times_int @ K5 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1060_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_1061_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_1062_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1063_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1064_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1065_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_1066_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_1067_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1068_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_less_as_int
thf(fact_1069_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K3: int] :
( ( P1 @ X4 )
= ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P1 @ X4 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1070_plusinfinity,axiom,
! [D: int,P4: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K3: int] :
( ( P4 @ X4 )
= ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1071_zero__to__b__eq,axiom,
( ( set_or1222579329274155063t_real @ zero_zero_real @ b )
= ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [K: nat] : ( set_or1222579329274155063t_real @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K ) ) ) @ ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) )
@ ( set_ord_lessThan_nat @ n ) ) ) ) ).
% zero_to_b_eq
thf(fact_1072_psubsetI,axiom,
! [A2: set_real,B5: set_real] :
( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ( A2 != B5 )
=> ( ord_less_set_real @ A2 @ B5 ) ) ) ).
% psubsetI
thf(fact_1073_square__continuous,axiom,
! [E: real,X: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ Y5 @ X ) ) @ D4 )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( times_times_real @ Y5 @ Y5 ) @ ( times_times_real @ X @ X ) ) ) @ E ) ) ) ) ).
% square_continuous
thf(fact_1074_image__eqI,axiom,
! [B: real,F: real > real,X: real,A2: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ B @ ( image_real_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1075_image__eqI,axiom,
! [B: nat,F: real > nat,X: real,A2: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A2 )
=> ( member_nat @ B @ ( image_real_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1076_image__eqI,axiom,
! [B: set_real,F: real > set_real,X: real,A2: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A2 )
=> ( member_set_real @ B @ ( image_real_set_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1077_image__eqI,axiom,
! [B: int,F: nat > int,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_int @ B @ ( image_nat_int @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1078_image__eqI,axiom,
! [B: real,F: nat > real,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_real @ B @ ( image_nat_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1079_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1080_image__eqI,axiom,
! [B: set_real,F: nat > set_real,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_set_real @ B @ ( image_nat_set_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1081_image__eqI,axiom,
! [B: real,F: set_real > real,X: set_real,A2: set_set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_set_real @ X @ A2 )
=> ( member_real @ B @ ( image_set_real_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1082_image__eqI,axiom,
! [B: nat,F: set_real > nat,X: set_real,A2: set_set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_set_real @ X @ A2 )
=> ( member_nat @ B @ ( image_set_real_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1083_image__eqI,axiom,
! [B: set_real,F: set_real > set_real,X: set_real,A2: set_set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_set_real @ X @ A2 )
=> ( member_set_real @ B @ ( image_2436557299294012491t_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1084_subset__antisym,axiom,
! [A2: set_real,B5: set_real] :
( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ( ord_less_eq_set_real @ B5 @ A2 )
=> ( A2 = B5 ) ) ) ).
% subset_antisym
thf(fact_1085_subsetI,axiom,
! [A2: set_nat,B5: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B5 ) )
=> ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).
% subsetI
thf(fact_1086_subsetI,axiom,
! [A2: set_set_real,B5: set_set_real] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
=> ( member_set_real @ X4 @ B5 ) )
=> ( ord_le3558479182127378552t_real @ A2 @ B5 ) ) ).
% subsetI
thf(fact_1087_subsetI,axiom,
! [A2: set_real,B5: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( member_real @ X4 @ B5 ) )
=> ( ord_less_eq_set_real @ A2 @ B5 ) ) ).
% subsetI
thf(fact_1088_Diff__iff,axiom,
! [C: real,A2: set_real,B5: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
= ( ( member_real @ C @ A2 )
& ~ ( member_real @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_1089_Diff__iff,axiom,
! [C: nat,A2: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_1090_Diff__iff,axiom,
! [C: set_real,A2: set_set_real,B5: set_set_real] :
( ( member_set_real @ C @ ( minus_5467046032205032049t_real @ A2 @ B5 ) )
= ( ( member_set_real @ C @ A2 )
& ~ ( member_set_real @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_1091_DiffI,axiom,
! [C: real,A2: set_real,B5: set_real] :
( ( member_real @ C @ A2 )
=> ( ~ ( member_real @ C @ B5 )
=> ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) ) ) ) ).
% DiffI
thf(fact_1092_DiffI,axiom,
! [C: nat,A2: set_nat,B5: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B5 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ) ).
% DiffI
thf(fact_1093_DiffI,axiom,
! [C: set_real,A2: set_set_real,B5: set_set_real] :
( ( member_set_real @ C @ A2 )
=> ( ~ ( member_set_real @ C @ B5 )
=> ( member_set_real @ C @ ( minus_5467046032205032049t_real @ A2 @ B5 ) ) ) ) ).
% DiffI
thf(fact_1094_image__ident,axiom,
! [Y6: set_real] :
( ( image_real_real
@ ^ [X2: real] : X2
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_1095_image__ident,axiom,
! [Y6: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : X2
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_1096_Sup__atLeastAtMost,axiom,
! [X: set_real,Y: set_real] :
( ( ord_less_eq_set_real @ X @ Y )
=> ( ( comple3096694443085538997t_real @ ( set_or7743017856606604397t_real @ X @ Y ) )
= Y ) ) ).
% Sup_atLeastAtMost
thf(fact_1097_rev__image__eqI,axiom,
! [X: real,A2: set_real,B: real,F: real > real] :
( ( member_real @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_real_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1098_rev__image__eqI,axiom,
! [X: real,A2: set_real,B: nat,F: real > nat] :
( ( member_real @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_real_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1099_rev__image__eqI,axiom,
! [X: real,A2: set_real,B: set_real,F: real > set_real] :
( ( member_real @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_real @ B @ ( image_real_set_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1100_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: int,F: nat > int] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_nat_int @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1101_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: real,F: nat > real] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_nat_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1102_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1103_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: set_real,F: nat > set_real] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_real @ B @ ( image_nat_set_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1104_rev__image__eqI,axiom,
! [X: set_real,A2: set_set_real,B: real,F: set_real > real] :
( ( member_set_real @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_set_real_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1105_rev__image__eqI,axiom,
! [X: set_real,A2: set_set_real,B: nat,F: set_real > nat] :
( ( member_set_real @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_set_real_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1106_rev__image__eqI,axiom,
! [X: set_real,A2: set_set_real,B: set_real,F: set_real > set_real] :
( ( member_set_real @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_real @ B @ ( image_2436557299294012491t_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1107_ball__imageD,axiom,
! [F: real > real,A2: set_real,P: real > $o] :
( ! [X4: real] :
( ( member_real @ X4 @ ( image_real_real @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_1108_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_1109_ball__imageD,axiom,
! [F: nat > int,A2: set_nat,P: int > $o] :
( ! [X4: int] :
( ( member_int @ X4 @ ( image_nat_int @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_1110_ball__imageD,axiom,
! [F: nat > set_real,A2: set_nat,P: set_real > $o] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ ( image_nat_set_real @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_1111_image__cong,axiom,
! [M7: set_real,N5: set_real,F: real > real,G: real > real] :
( ( M7 = N5 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ N5 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_real_real @ F @ M7 )
= ( image_real_real @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_1112_image__cong,axiom,
! [M7: set_nat,N5: set_nat,F: nat > nat,G: nat > nat] :
( ( M7 = N5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_nat @ F @ M7 )
= ( image_nat_nat @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_1113_image__cong,axiom,
! [M7: set_nat,N5: set_nat,F: nat > int,G: nat > int] :
( ( M7 = N5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_int @ F @ M7 )
= ( image_nat_int @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_1114_image__cong,axiom,
! [M7: set_nat,N5: set_nat,F: nat > set_real,G: nat > set_real] :
( ( M7 = N5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_set_real @ F @ M7 )
= ( image_nat_set_real @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_1115_bex__imageD,axiom,
! [F: real > real,A2: set_real,P: real > $o] :
( ? [X3: real] :
( ( member_real @ X3 @ ( image_real_real @ F @ A2 ) )
& ( P @ X3 ) )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_1116_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X3 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_1117_bex__imageD,axiom,
! [F: nat > int,A2: set_nat,P: int > $o] :
( ? [X3: int] :
( ( member_int @ X3 @ ( image_nat_int @ F @ A2 ) )
& ( P @ X3 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_1118_bex__imageD,axiom,
! [F: nat > set_real,A2: set_nat,P: set_real > $o] :
( ? [X3: set_real] :
( ( member_set_real @ X3 @ ( image_nat_set_real @ F @ A2 ) )
& ( P @ X3 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_1119_image__iff,axiom,
! [Z: int,F: nat > int,A2: set_nat] :
( ( member_int @ Z @ ( image_nat_int @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1120_image__iff,axiom,
! [Z: real,F: real > real,A2: set_real] :
( ( member_real @ Z @ ( image_real_real @ F @ A2 ) )
= ( ? [X2: real] :
( ( member_real @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1121_image__iff,axiom,
! [Z: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1122_image__iff,axiom,
! [Z: set_real,F: nat > set_real,A2: set_nat] :
( ( member_set_real @ Z @ ( image_nat_set_real @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1123_imageI,axiom,
! [X: real,A2: set_real,F: real > real] :
( ( member_real @ X @ A2 )
=> ( member_real @ ( F @ X ) @ ( image_real_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_1124_imageI,axiom,
! [X: real,A2: set_real,F: real > nat] :
( ( member_real @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_real_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1125_imageI,axiom,
! [X: real,A2: set_real,F: real > set_real] :
( ( member_real @ X @ A2 )
=> ( member_set_real @ ( F @ X ) @ ( image_real_set_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_1126_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > int] :
( ( member_nat @ X @ A2 )
=> ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ A2 ) ) ) ).
% imageI
thf(fact_1127_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > real] :
( ( member_nat @ X @ A2 )
=> ( member_real @ ( F @ X ) @ ( image_nat_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_1128_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1129_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > set_real] :
( ( member_nat @ X @ A2 )
=> ( member_set_real @ ( F @ X ) @ ( image_nat_set_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_1130_imageI,axiom,
! [X: set_real,A2: set_set_real,F: set_real > real] :
( ( member_set_real @ X @ A2 )
=> ( member_real @ ( F @ X ) @ ( image_set_real_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_1131_imageI,axiom,
! [X: set_real,A2: set_set_real,F: set_real > nat] :
( ( member_set_real @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_set_real_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1132_imageI,axiom,
! [X: set_real,A2: set_set_real,F: set_real > set_real] :
( ( member_set_real @ X @ A2 )
=> ( member_set_real @ ( F @ X ) @ ( image_2436557299294012491t_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_1133_Collect__mono__iff,axiom,
! [P: real > $o,Q3: real > $o] :
( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q3 ) )
= ( ! [X2: real] :
( ( P @ X2 )
=> ( Q3 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1134_set__eq__subset,axiom,
( ( ^ [Y4: set_real,Z3: set_real] : ( Y4 = Z3 ) )
= ( ^ [A6: set_real,B6: set_real] :
( ( ord_less_eq_set_real @ A6 @ B6 )
& ( ord_less_eq_set_real @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_1135_subset__trans,axiom,
! [A2: set_real,B5: set_real,C3: set_real] :
( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ( ord_less_eq_set_real @ B5 @ C3 )
=> ( ord_less_eq_set_real @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_1136_Collect__mono,axiom,
! [P: real > $o,Q3: real > $o] :
( ! [X4: real] :
( ( P @ X4 )
=> ( Q3 @ X4 ) )
=> ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q3 ) ) ) ).
% Collect_mono
thf(fact_1137_subset__refl,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ A2 ) ).
% subset_refl
thf(fact_1138_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A6 )
=> ( member_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_1139_subset__iff,axiom,
( ord_le3558479182127378552t_real
= ( ^ [A6: set_set_real,B6: set_set_real] :
! [T3: set_real] :
( ( member_set_real @ T3 @ A6 )
=> ( member_set_real @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_1140_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [T3: real] :
( ( member_real @ T3 @ A6 )
=> ( member_real @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_1141_equalityD2,axiom,
! [A2: set_real,B5: set_real] :
( ( A2 = B5 )
=> ( ord_less_eq_set_real @ B5 @ A2 ) ) ).
% equalityD2
thf(fact_1142_equalityD1,axiom,
! [A2: set_real,B5: set_real] :
( ( A2 = B5 )
=> ( ord_less_eq_set_real @ A2 @ B5 ) ) ).
% equalityD1
thf(fact_1143_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( member_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_1144_subset__eq,axiom,
( ord_le3558479182127378552t_real
= ( ^ [A6: set_set_real,B6: set_set_real] :
! [X2: set_real] :
( ( member_set_real @ X2 @ A6 )
=> ( member_set_real @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_1145_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [X2: real] :
( ( member_real @ X2 @ A6 )
=> ( member_real @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_1146_equalityE,axiom,
! [A2: set_real,B5: set_real] :
( ( A2 = B5 )
=> ~ ( ( ord_less_eq_set_real @ A2 @ B5 )
=> ~ ( ord_less_eq_set_real @ B5 @ A2 ) ) ) ).
% equalityE
thf(fact_1147_subsetD,axiom,
! [A2: set_nat,B5: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_1148_subsetD,axiom,
! [A2: set_set_real,B5: set_set_real,C: set_real] :
( ( ord_le3558479182127378552t_real @ A2 @ B5 )
=> ( ( member_set_real @ C @ A2 )
=> ( member_set_real @ C @ B5 ) ) ) ).
% subsetD
thf(fact_1149_subsetD,axiom,
! [A2: set_real,B5: set_real,C: real] :
( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B5 ) ) ) ).
% subsetD
thf(fact_1150_in__mono,axiom,
! [A2: set_nat,B5: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_1151_in__mono,axiom,
! [A2: set_set_real,B5: set_set_real,X: set_real] :
( ( ord_le3558479182127378552t_real @ A2 @ B5 )
=> ( ( member_set_real @ X @ A2 )
=> ( member_set_real @ X @ B5 ) ) ) ).
% in_mono
thf(fact_1152_in__mono,axiom,
! [A2: set_real,B5: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B5 ) ) ) ).
% in_mono
thf(fact_1153_DiffD2,axiom,
! [C: real,A2: set_real,B5: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
=> ~ ( member_real @ C @ B5 ) ) ).
% DiffD2
thf(fact_1154_DiffD2,axiom,
! [C: nat,A2: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
=> ~ ( member_nat @ C @ B5 ) ) ).
% DiffD2
thf(fact_1155_DiffD2,axiom,
! [C: set_real,A2: set_set_real,B5: set_set_real] :
( ( member_set_real @ C @ ( minus_5467046032205032049t_real @ A2 @ B5 ) )
=> ~ ( member_set_real @ C @ B5 ) ) ).
% DiffD2
thf(fact_1156_DiffD1,axiom,
! [C: real,A2: set_real,B5: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
=> ( member_real @ C @ A2 ) ) ).
% DiffD1
thf(fact_1157_DiffD1,axiom,
! [C: nat,A2: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_1158_DiffD1,axiom,
! [C: set_real,A2: set_set_real,B5: set_set_real] :
( ( member_set_real @ C @ ( minus_5467046032205032049t_real @ A2 @ B5 ) )
=> ( member_set_real @ C @ A2 ) ) ).
% DiffD1
thf(fact_1159_DiffE,axiom,
! [C: real,A2: set_real,B5: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
=> ~ ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B5 ) ) ) ).
% DiffE
thf(fact_1160_DiffE,axiom,
! [C: nat,A2: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B5 ) ) ) ).
% DiffE
thf(fact_1161_DiffE,axiom,
! [C: set_real,A2: set_set_real,B5: set_set_real] :
( ( member_set_real @ C @ ( minus_5467046032205032049t_real @ A2 @ B5 ) )
=> ~ ( ( member_set_real @ C @ A2 )
=> ( member_set_real @ C @ B5 ) ) ) ).
% DiffE
thf(fact_1162_psubsetD,axiom,
! [A2: set_real,B5: set_real,C: real] :
( ( ord_less_set_real @ A2 @ B5 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_1163_psubsetD,axiom,
! [A2: set_nat,B5: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B5 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_1164_psubsetD,axiom,
! [A2: set_set_real,B5: set_set_real,C: set_real] :
( ( ord_le7926960851185191020t_real @ A2 @ B5 )
=> ( ( member_set_real @ C @ A2 )
=> ( member_set_real @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_1165_Compr__image__eq,axiom,
! [F: nat > int,A2: set_nat,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_nat_int @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_int @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1166_Compr__image__eq,axiom,
! [F: real > real,A2: set_real,P: real > $o] :
( ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ ( image_real_real @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_real_real @ F
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1167_Compr__image__eq,axiom,
! [F: nat > real,A2: set_nat,P: real > $o] :
( ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ ( image_nat_real @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_real @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1168_Compr__image__eq,axiom,
! [F: set_real > real,A2: set_set_real,P: real > $o] :
( ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ ( image_set_real_real @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_set_real_real @ F
@ ( collect_set_real
@ ^ [X2: set_real] :
( ( member_set_real @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1169_Compr__image__eq,axiom,
! [F: real > nat,A2: set_real,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_real_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_real_nat @ F
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1170_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1171_Compr__image__eq,axiom,
! [F: set_real > nat,A2: set_set_real,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_set_real_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_set_real_nat @ F
@ ( collect_set_real
@ ^ [X2: set_real] :
( ( member_set_real @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1172_Compr__image__eq,axiom,
! [F: real > set_real,A2: set_real,P: set_real > $o] :
( ( collect_set_real
@ ^ [X2: set_real] :
( ( member_set_real @ X2 @ ( image_real_set_real @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_real_set_real @ F
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1173_Compr__image__eq,axiom,
! [F: nat > set_real,A2: set_nat,P: set_real > $o] :
( ( collect_set_real
@ ^ [X2: set_real] :
( ( member_set_real @ X2 @ ( image_nat_set_real @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_set_real @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1174_Compr__image__eq,axiom,
! [F: set_real > set_real,A2: set_set_real,P: set_real > $o] :
( ( collect_set_real
@ ^ [X2: set_real] :
( ( member_set_real @ X2 @ ( image_2436557299294012491t_real @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_2436557299294012491t_real @ F
@ ( collect_set_real
@ ^ [X2: set_real] :
( ( member_set_real @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1175_image__image,axiom,
! [F: int > nat,G: nat > int,A2: set_nat] :
( ( image_int_nat @ F @ ( image_nat_int @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1176_image__image,axiom,
! [F: int > int,G: nat > int,A2: set_nat] :
( ( image_int_int @ F @ ( image_nat_int @ G @ A2 ) )
= ( image_nat_int
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1177_image__image,axiom,
! [F: int > set_real,G: nat > int,A2: set_nat] :
( ( image_int_set_real @ F @ ( image_nat_int @ G @ A2 ) )
= ( image_nat_set_real
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1178_image__image,axiom,
! [F: set_real > nat,G: nat > set_real,A2: set_nat] :
( ( image_set_real_nat @ F @ ( image_nat_set_real @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1179_image__image,axiom,
! [F: set_real > int,G: nat > set_real,A2: set_nat] :
( ( image_set_real_int @ F @ ( image_nat_set_real @ G @ A2 ) )
= ( image_nat_int
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1180_image__image,axiom,
! [F: set_real > set_real,G: nat > set_real,A2: set_nat] :
( ( image_2436557299294012491t_real @ F @ ( image_nat_set_real @ G @ A2 ) )
= ( image_nat_set_real
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1181_image__image,axiom,
! [F: real > real,G: real > real,A2: set_real] :
( ( image_real_real @ F @ ( image_real_real @ G @ A2 ) )
= ( image_real_real
@ ^ [X2: real] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1182_image__image,axiom,
! [F: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1183_image__image,axiom,
! [F: nat > int,G: nat > nat,A2: set_nat] :
( ( image_nat_int @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_int
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1184_image__image,axiom,
! [F: nat > set_real,G: nat > nat,A2: set_nat] :
( ( image_nat_set_real @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_set_real
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1185_imageE,axiom,
! [B: int,F: nat > int,A2: set_nat] :
( ( member_int @ B @ ( image_nat_int @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_1186_imageE,axiom,
! [B: real,F: real > real,A2: set_real] :
( ( member_real @ B @ ( image_real_real @ F @ A2 ) )
=> ~ ! [X4: real] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_real @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_1187_imageE,axiom,
! [B: real,F: nat > real,A2: set_nat] :
( ( member_real @ B @ ( image_nat_real @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_1188_imageE,axiom,
! [B: real,F: set_real > real,A2: set_set_real] :
( ( member_real @ B @ ( image_set_real_real @ F @ A2 ) )
=> ~ ! [X4: set_real] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_set_real @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_1189_imageE,axiom,
! [B: nat,F: real > nat,A2: set_real] :
( ( member_nat @ B @ ( image_real_nat @ F @ A2 ) )
=> ~ ! [X4: real] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_real @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_1190_imageE,axiom,
! [B: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_1191_imageE,axiom,
! [B: nat,F: set_real > nat,A2: set_set_real] :
( ( member_nat @ B @ ( image_set_real_nat @ F @ A2 ) )
=> ~ ! [X4: set_real] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_set_real @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_1192_imageE,axiom,
! [B: set_real,F: real > set_real,A2: set_real] :
( ( member_set_real @ B @ ( image_real_set_real @ F @ A2 ) )
=> ~ ! [X4: real] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_real @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_1193_imageE,axiom,
! [B: set_real,F: nat > set_real,A2: set_nat] :
( ( member_set_real @ B @ ( image_nat_set_real @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_1194_imageE,axiom,
! [B: set_real,F: set_real > set_real,A2: set_set_real] :
( ( member_set_real @ B @ ( image_2436557299294012491t_real @ F @ A2 ) )
=> ~ ! [X4: set_real] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_set_real @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_1195_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_1196_less__eq__set__def,axiom,
( ord_le3558479182127378552t_real
= ( ^ [A6: set_set_real,B6: set_set_real] :
( ord_le2392157289819280397real_o
@ ^ [X2: set_real] : ( member_set_real @ X2 @ A6 )
@ ^ [X2: set_real] : ( member_set_real @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_1197_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
( ord_less_eq_real_o
@ ^ [X2: real] : ( member_real @ X2 @ A6 )
@ ^ [X2: real] : ( member_real @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_1198_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1199_Collect__subset,axiom,
! [A2: set_set_real,P: set_real > $o] :
( ord_le3558479182127378552t_real
@ ( collect_set_real
@ ^ [X2: set_real] :
( ( member_set_real @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1200_Collect__subset,axiom,
! [A2: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1201_minus__set__def,axiom,
( minus_minus_set_real
= ( ^ [A6: set_real,B6: set_real] :
( collect_real
@ ( minus_minus_real_o
@ ^ [X2: real] : ( member_real @ X2 @ A6 )
@ ^ [X2: real] : ( member_real @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_1202_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_1203_minus__set__def,axiom,
( minus_5467046032205032049t_real
= ( ^ [A6: set_set_real,B6: set_set_real] :
( collect_set_real
@ ( minus_2011193488284532564real_o
@ ^ [X2: set_real] : ( member_set_real @ X2 @ A6 )
@ ^ [X2: set_real] : ( member_set_real @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_1204_set__diff__eq,axiom,
( minus_minus_set_real
= ( ^ [A6: set_real,B6: set_real] :
( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A6 )
& ~ ( member_real @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1205_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A6 )
& ~ ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1206_set__diff__eq,axiom,
( minus_5467046032205032049t_real
= ( ^ [A6: set_set_real,B6: set_set_real] :
( collect_set_real
@ ^ [X2: set_real] :
( ( member_set_real @ X2 @ A6 )
& ~ ( member_set_real @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1207_less__set__def,axiom,
( ord_less_set_real
= ( ^ [A6: set_real,B6: set_real] :
( ord_less_real_o
@ ^ [X2: real] : ( member_real @ X2 @ A6 )
@ ^ [X2: real] : ( member_real @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_1208_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_1209_less__set__def,axiom,
( ord_le7926960851185191020t_real
= ( ^ [A6: set_set_real,B6: set_set_real] :
( ord_less_set_real_o
@ ^ [X2: set_real] : ( member_set_real @ X2 @ A6 )
@ ^ [X2: set_real] : ( member_set_real @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_1210_image__mono,axiom,
! [A2: set_nat,B5: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B5 ) ) ) ).
% image_mono
thf(fact_1211_image__mono,axiom,
! [A2: set_nat,B5: set_nat,F: nat > int] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ ( image_nat_int @ F @ B5 ) ) ) ).
% image_mono
thf(fact_1212_image__mono,axiom,
! [A2: set_nat,B5: set_nat,F: nat > set_real] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ord_le3558479182127378552t_real @ ( image_nat_set_real @ F @ A2 ) @ ( image_nat_set_real @ F @ B5 ) ) ) ).
% image_mono
thf(fact_1213_image__mono,axiom,
! [A2: set_real,B5: set_real,F: real > real] :
( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A2 ) @ ( image_real_real @ F @ B5 ) ) ) ).
% image_mono
thf(fact_1214_image__subsetI,axiom,
! [A2: set_real,F: real > nat,B5: set_nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_real_nat @ F @ A2 ) @ B5 ) ) ).
% image_subsetI
thf(fact_1215_image__subsetI,axiom,
! [A2: set_real,F: real > set_real,B5: set_set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( member_set_real @ ( F @ X4 ) @ B5 ) )
=> ( ord_le3558479182127378552t_real @ ( image_real_set_real @ F @ A2 ) @ B5 ) ) ).
% image_subsetI
thf(fact_1216_image__subsetI,axiom,
! [A2: set_nat,F: nat > int,B5: set_int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_int @ ( F @ X4 ) @ B5 ) )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ B5 ) ) ).
% image_subsetI
thf(fact_1217_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B5: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B5 ) ) ).
% image_subsetI
thf(fact_1218_image__subsetI,axiom,
! [A2: set_nat,F: nat > set_real,B5: set_set_real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_set_real @ ( F @ X4 ) @ B5 ) )
=> ( ord_le3558479182127378552t_real @ ( image_nat_set_real @ F @ A2 ) @ B5 ) ) ).
% image_subsetI
thf(fact_1219_image__subsetI,axiom,
! [A2: set_set_real,F: set_real > nat,B5: set_nat] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_set_real_nat @ F @ A2 ) @ B5 ) ) ).
% image_subsetI
thf(fact_1220_image__subsetI,axiom,
! [A2: set_set_real,F: set_real > set_real,B5: set_set_real] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
=> ( member_set_real @ ( F @ X4 ) @ B5 ) )
=> ( ord_le3558479182127378552t_real @ ( image_2436557299294012491t_real @ F @ A2 ) @ B5 ) ) ).
% image_subsetI
thf(fact_1221_image__subsetI,axiom,
! [A2: set_real,F: real > real,B5: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( member_real @ ( F @ X4 ) @ B5 ) )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A2 ) @ B5 ) ) ).
% image_subsetI
thf(fact_1222_image__subsetI,axiom,
! [A2: set_nat,F: nat > real,B5: set_real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_real @ ( F @ X4 ) @ B5 ) )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ A2 ) @ B5 ) ) ).
% image_subsetI
thf(fact_1223_image__subsetI,axiom,
! [A2: set_set_real,F: set_real > real,B5: set_real] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
=> ( member_real @ ( F @ X4 ) @ B5 ) )
=> ( ord_less_eq_set_real @ ( image_set_real_real @ F @ A2 ) @ B5 ) ) ).
% image_subsetI
thf(fact_1224_subset__imageE,axiom,
! [B5: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( B5
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1225_subset__imageE,axiom,
! [B5: set_int,F: nat > int,A2: set_nat] :
( ( ord_less_eq_set_int @ B5 @ ( image_nat_int @ F @ A2 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( B5
!= ( image_nat_int @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1226_subset__imageE,axiom,
! [B5: set_set_real,F: nat > set_real,A2: set_nat] :
( ( ord_le3558479182127378552t_real @ B5 @ ( image_nat_set_real @ F @ A2 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( B5
!= ( image_nat_set_real @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1227_subset__imageE,axiom,
! [B5: set_real,F: real > real,A2: set_real] :
( ( ord_less_eq_set_real @ B5 @ ( image_real_real @ F @ A2 ) )
=> ~ ! [C4: set_real] :
( ( ord_less_eq_set_real @ C4 @ A2 )
=> ( B5
!= ( image_real_real @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1228_image__subset__iff,axiom,
! [F: nat > int,A2: set_nat,B5: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ B5 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_int @ ( F @ X2 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_1229_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B5 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_1230_image__subset__iff,axiom,
! [F: nat > set_real,A2: set_nat,B5: set_set_real] :
( ( ord_le3558479182127378552t_real @ ( image_nat_set_real @ F @ A2 ) @ B5 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_real @ ( F @ X2 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_1231_image__subset__iff,axiom,
! [F: real > real,A2: set_real,B5: set_real] :
( ( ord_less_eq_set_real @ ( image_real_real @ F @ A2 ) @ B5 )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( member_real @ ( F @ X2 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_1232_subset__image__iff,axiom,
! [B5: set_real,F: real > real,A2: set_real] :
( ( ord_less_eq_set_real @ B5 @ ( image_real_real @ F @ A2 ) )
= ( ? [AA: set_real] :
( ( ord_less_eq_set_real @ AA @ A2 )
& ( B5
= ( image_real_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1233_interval__bounds__real_I1_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( comple1385675409528146559p_real @ ( set_or1222579329274155063t_real @ A @ B ) )
= B ) ) ).
% interval_bounds_real(1)
thf(fact_1234_int__f2__D,axiom,
! [K4: set_real] :
( ( member_set_real @ K4 @ ( regular_division @ zero_zero_real @ a @ n ) )
=> ( hensto240673015341029504l_real @ f2 @ ( times_times_real @ ( f @ ( comple1385675409528146559p_real @ K4 ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) ) @ K4 ) ) ).
% int_f2_D
thf(fact_1235_lemma__interval,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D4 )
=> ( ( ord_less_eq_real @ A @ Y5 )
& ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).
% lemma_interval
thf(fact_1236_lemma__interval__lt,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D4 )
=> ( ( ord_less_real @ A @ Y5 )
& ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_1237_f12,axiom,
( hensto240673015341029504l_real
@ ^ [X2: real] : ( minus_minus_real @ ( f2 @ X2 ) @ ( f1 @ X2 ) )
@ ( groups8702937949983641418l_real
@ ^ [K6: set_real] : ( times_times_real @ ( minus_minus_real @ ( f @ ( comple1385675409528146559p_real @ K6 ) ) @ ( f @ ( comple4887499456419720421f_real @ K6 ) ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) )
@ ( regular_division @ zero_zero_real @ a @ n ) )
@ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ) ).
% f12
thf(fact_1238_int__21__D,axiom,
! [K4: set_real] :
( ( member_set_real @ K4 @ ( regular_division @ zero_zero_real @ a @ n ) )
=> ( hensto240673015341029504l_real
@ ^ [X2: real] : ( minus_minus_real @ ( f2 @ X2 ) @ ( f1 @ X2 ) )
@ ( times_times_real @ ( minus_minus_real @ ( f @ ( comple1385675409528146559p_real @ K4 ) ) @ ( f @ ( comple4887499456419720421f_real @ K4 ) ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) )
@ K4 ) ) ).
% int_21_D
thf(fact_1239_int__f1__D,axiom,
! [K4: set_real] :
( ( member_set_real @ K4 @ ( regular_division @ zero_zero_real @ a @ n ) )
=> ( hensto240673015341029504l_real @ f1 @ ( times_times_real @ ( f @ ( comple4887499456419720421f_real @ K4 ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) ) @ K4 ) ) ).
% int_f1_D
thf(fact_1240_less,axiom,
! [K4: set_real] :
( ( member_set_real @ K4 @ ( regular_division @ zero_zero_real @ a @ n ) )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( f @ ( comple1385675409528146559p_real @ K4 ) ) @ ( f @ ( comple4887499456419720421f_real @ K4 ) ) ) ) @ ( divide_divide_real @ epsilon @ a ) ) ) ).
% less
thf(fact_1241__092_060open_062_I_092_060Sum_062K_092_060in_062regular__division_A0_Aa_An_O_A_If_A_ISup_AK_J_A_N_Af_A_IInf_AK_J_J_A_K_A_Ia_A_P_Areal_An_J_J_A_060_A_092_060epsilon_062_092_060close_062,axiom,
( ord_less_real
@ ( groups8702937949983641418l_real
@ ^ [K6: set_real] : ( times_times_real @ ( minus_minus_real @ ( f @ ( comple1385675409528146559p_real @ K6 ) ) @ ( f @ ( comple4887499456419720421f_real @ K6 ) ) ) @ ( divide_divide_real @ a @ ( semiri5074537144036343181t_real @ n ) ) )
@ ( regular_division @ zero_zero_real @ a @ n ) )
@ epsilon ) ).
% \<open>(\<Sum>K\<in>regular_division 0 a n. (f (Sup K) - f (Inf K)) * (a / real n)) < \<epsilon>\<close>
thf(fact_1242_interval__bounds__real_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( comple4887499456419720421f_real @ ( set_or1222579329274155063t_real @ A @ B ) )
= A ) ) ).
% interval_bounds_real(2)
thf(fact_1243_D__ne,axiom,
( ( regular_division @ zero_zero_real @ a @ n )
!= bot_bot_set_set_real ) ).
% D_ne
thf(fact_1244_Inf__nat__def,axiom,
( complete_Inf_Inf_nat
= ( ^ [X7: set_nat] :
( ord_Least_nat
@ ^ [N4: nat] : ( member_nat @ N4 @ X7 ) ) ) ) ).
% Inf_nat_def
thf(fact_1245_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1246_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1247_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1248_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_1249_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1250_Union__regular__division,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( N = zero_zero_nat )
=> ( ( comple3096694443085538997t_real @ ( regular_division @ A @ B @ N ) )
= bot_bot_set_real ) )
& ( ( N != zero_zero_nat )
=> ( ( comple3096694443085538997t_real @ ( regular_division @ A @ B @ N ) )
= ( set_or1222579329274155063t_real @ A @ B ) ) ) ) ) ).
% Union_regular_division
thf(fact_1251_Union__segment__image,axiom,
! [K2: nat,N: nat] :
( ( ( K2 = zero_zero_nat )
=> ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ ( segment @ N ) @ ( set_ord_lessThan_nat @ K2 ) ) )
= bot_bot_set_real ) )
& ( ( K2 != zero_zero_nat )
=> ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ ( segment @ N ) @ ( set_ord_lessThan_nat @ K2 ) ) )
= ( set_or1222579329274155063t_real @ zero_zero_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% Union_segment_image
thf(fact_1252_segment__nonempty,axiom,
! [N: nat,K2: nat] :
( ( segment @ N @ K2 )
!= bot_bot_set_real ) ).
% segment_nonempty
thf(fact_1253_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1254_atLeastAtMost__subset__contains__Inf,axiom,
! [A2: set_real,A: real,B: real] :
( ( A2 != bot_bot_set_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_set_real @ A2 @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( member_real @ ( comple4887499456419720421f_real @ A2 ) @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ) ) ).
% atLeastAtMost_subset_contains_Inf
thf(fact_1255_f2__def,axiom,
( f2
= ( comp_real_real_real @ f @ upper ) ) ).
% f2_def
thf(fact_1256__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_1257_f1__def,axiom,
( f1
= ( comp_real_real_real @ f @ lower ) ) ).
% f1_def
thf(fact_1258_cont__0a,axiom,
topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) @ f ).
% cont_0a
thf(fact_1259_cont,axiom,
topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ f ).
% cont
thf(fact_1260_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 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( ( hensto2714581292692559302l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
= zero_zero_real )
= ( ! [X2: real] :
( ( member_real @ X2 @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ( F @ X2 )
= zero_zero_real ) ) ) ) ) ) ) ).
% integral_eq_0_iff
thf(fact_1261_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 )
=> ? [C5: real,D4: real] :
( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ C5 @ D4 ) )
& ( ord_less_eq_real @ C5 @ D4 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_1262_sm,axiom,
monoto4017252874604999745l_real @ ( set_ord_atLeast_real @ zero_zero_real ) @ ord_less_real @ ord_less_real @ f ).
% sm
thf(fact_1263_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A5: real,B4: real,C5: real] :
( ( P @ A5 @ B4 )
=> ( ( P @ B4 @ C5 )
=> ( ( ord_less_eq_real @ A5 @ B4 )
=> ( ( ord_less_eq_real @ B4 @ C5 )
=> ( P @ A5 @ C5 ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [A5: real,B4: real] :
( ( ( ord_less_eq_real @ A5 @ X4 )
& ( ord_less_eq_real @ X4 @ B4 )
& ( ord_less_real @ ( minus_minus_real @ B4 @ A5 ) @ D5 ) )
=> ( P @ A5 @ B4 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_1264_div,axiom,
tagged6100619406677346166f_real @ ( regular_division @ zero_zero_real @ a @ n ) @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) ).
% div
thf(fact_1265__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_1266_sm__0a,axiom,
monoto4017252874604999745l_real @ ( set_or1222579329274155063t_real @ zero_zero_real @ a ) @ ord_less_real @ ord_less_real @ f ).
% sm_0a
thf(fact_1267_regular__division__division__of,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( tagged6100619406677346166f_real @ ( regular_division @ A @ B @ N ) @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ) ).
% regular_division_division_of
thf(fact_1268_strict__mono__continuous__invD,axiom,
! [A: real,F: real > real,G: real > real] :
( ( monoto4017252874604999745l_real @ ( set_ord_atLeast_real @ A ) @ ord_less_real @ ord_less_real @ F )
=> ( ( topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ A ) @ F )
=> ( ( ( image_real_real @ F @ ( set_ord_atLeast_real @ A ) )
= ( set_ord_atLeast_real @ ( F @ A ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( G @ ( F @ X4 ) )
= X4 ) )
=> ( topolo5044208981011980120l_real @ ( set_ord_atLeast_real @ ( F @ A ) ) @ G ) ) ) ) ) ).
% strict_mono_continuous_invD
thf(fact_1269__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>
% Helper facts (5)
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 ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( hensto240673015341029504l_real @ g1
@ ( minus_minus_real @ ( times_times_real @ a @ b )
@ ( divide_divide_real
@ ( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [K: nat] : ( f @ ( a_seg @ ( semiri5074537144036343181t_real @ K ) ) )
@ ( set_ord_lessThan_nat @ n ) )
@ a )
@ ( semiri5074537144036343181t_real @ n ) ) )
@ ( set_or1222579329274155063t_real @ zero_zero_real @ b ) ) ).
%------------------------------------------------------------------------------