%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP035^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n022.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:01 AM UTC 2026
% Result : Theorem 2.67s 1.64s
% Output : Refutation 2.67s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 2
% Syntax : Number of formulae : 12 ( 12 unt; 0 typ; 0 def)
% Number of atoms : 12 ( 11 equ; 0 cnn)
% Maximal formula atoms : 1 ( 1 avg)
% Number of connectives : 21 ( 4 ~; 0 |; 0 &; 17 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 2 ( 2 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 70 ( 68 usr; 10 con; 0-4 aty)
% Number of variables : 7 ( 5 ^; 2 !; 0 ?; 7 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
complex: $tType ).
thf(type_def_7,type,
set_complex: $tType ).
thf(type_def_8,type,
real: $tType ).
thf(type_def_9,type,
set_complex_complex: $tType ).
thf(type_def_10,type,
set_real: $tType ).
thf(type_def_11,type,
nat: $tType ).
thf(type_def_12,type,
set_nat: $tType ).
thf(func_def_0,type,
set_complex_complex: $tType ).
thf(func_def_1,type,
set_complex: $tType ).
thf(func_def_2,type,
set_real: $tType ).
thf(func_def_3,type,
set_nat: $tType ).
thf(func_def_4,type,
complex: $tType ).
thf(func_def_5,type,
real: $tType ).
thf(func_def_6,type,
nat: $tType ).
thf(func_def_7,type,
comple372758642hic_on: ( complex > complex ) > set_complex > $o ).
thf(func_def_8,type,
deriv_complex: ( complex > complex ) > complex > complex ).
thf(func_def_9,type,
deriv_real: ( real > real ) > real > real ).
thf(func_def_10,type,
comp_c1610621014omplex: ( ( complex > complex ) > complex > complex ) > ( ( complex > complex ) > complex > complex ) > ( complex > complex ) > complex > complex ).
thf(func_def_11,type,
comp_c881053372omplex: ( ( complex > complex ) > complex ) > ( complex > complex > complex ) > complex > complex ).
thf(func_def_12,type,
comp_c1884694328l_real: ( ( complex > complex ) > real ) > ( real > complex > complex ) > real > real ).
thf(func_def_13,type,
comp_c606622857omplex: ( complex > complex > complex ) > ( ( complex > complex ) > complex ) > ( complex > complex ) > complex > complex ).
thf(func_def_14,type,
comp_c130555887omplex: ( complex > complex ) > ( complex > complex ) > complex > complex ).
thf(func_def_15,type,
comp_c819638635l_real: ( complex > real ) > ( real > complex ) > real > real ).
thf(func_def_16,type,
comp_r2009261319omplex: ( real > complex > complex ) > ( ( complex > complex ) > real ) > ( complex > complex ) > complex > complex ).
thf(func_def_17,type,
comp_r667767405omplex: ( real > complex ) > ( complex > real ) > complex > complex ).
thf(func_def_18,type,
comp_real_real_real: ( real > real ) > ( real > real ) > real > real ).
thf(func_def_19,type,
id_complex_complex: ( complex > complex ) > complex > complex ).
thf(func_def_20,type,
id_real_real: ( real > real ) > real > real ).
thf(func_def_21,type,
id_complex: complex > complex ).
thf(func_def_22,type,
id_real: real > real ).
thf(func_def_23,type,
id_set1618538368omplex: set_complex_complex > set_complex_complex ).
thf(func_def_24,type,
id_set_complex: set_complex > set_complex ).
thf(func_def_25,type,
id_set_real: set_real > set_real ).
thf(func_def_26,type,
one_one_complex: complex ).
thf(func_def_27,type,
one_one_nat: nat ).
thf(func_def_28,type,
one_one_real: real ).
thf(func_def_29,type,
times_times_complex: complex > complex > complex ).
thf(func_def_30,type,
times_times_nat: nat > nat > nat ).
thf(func_def_31,type,
times_times_real: real > real > real ).
thf(func_def_32,type,
times_1316095593omplex: set_complex > set_complex > set_complex ).
thf(func_def_33,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_34,type,
times_times_set_real: set_real > set_real > set_real ).
thf(func_def_35,type,
zero_zero_complex: complex ).
thf(func_def_36,type,
zero_zero_nat: nat ).
thf(func_def_37,type,
zero_zero_real: real ).
thf(func_def_38,type,
if_complex: $o > complex > complex > complex ).
thf(func_def_39,type,
if_nat: $o > nat > nat > nat ).
thf(func_def_40,type,
if_real: $o > real > real > real ).
thf(func_def_41,type,
compow1098280738omplex: nat > ( ( complex > complex ) > complex > complex ) > ( complex > complex ) > complex > complex ).
thf(func_def_42,type,
compow1723822618l_real: nat > ( ( real > real ) > real > real ) > ( real > real ) > real > real ).
thf(func_def_43,type,
compow1667379464omplex: nat > ( complex > complex ) > complex > complex ).
thf(func_def_44,type,
compow_real_real: nat > ( real > real ) > real > real ).
thf(func_def_45,type,
funpow1854104714omplex: nat > ( ( complex > complex ) > complex > complex ) > ( complex > complex ) > complex > complex ).
thf(func_def_46,type,
funpow_real_real: nat > ( ( real > real ) > real > real ) > ( real > real ) > real > real ).
thf(func_def_47,type,
funpow_complex: nat > ( complex > complex ) > complex > complex ).
thf(func_def_48,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_49,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_50,type,
ord_le701908932omplex: set_complex > set_complex > $o ).
thf(func_def_51,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(func_def_52,type,
image_944012797omplex: ( ( complex > complex ) > complex > complex ) > set_complex_complex > set_complex_complex ).
thf(func_def_53,type,
image_58037603omplex: ( complex > complex ) > set_complex > set_complex ).
thf(func_def_54,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(func_def_55,type,
topolo935673511omplex: set_complex > $o ).
thf(func_def_56,type,
topolo2127351575omplex: set_complex > $o ).
thf(func_def_57,type,
arcosh_complex: complex > complex ).
thf(func_def_58,type,
arcosh_real: real > real ).
thf(func_def_59,type,
arsinh_complex: complex > complex ).
thf(func_def_60,type,
arsinh_real: real > real ).
thf(func_def_61,type,
artanh_complex: complex > complex ).
thf(func_def_62,type,
artanh_real: real > real ).
thf(func_def_63,type,
ln_ln_complex: complex > complex ).
thf(func_def_64,type,
ln_ln_real: real > real ).
thf(func_def_65,type,
member_complex: complex > set_complex > $o ).
thf(func_def_66,type,
member_nat: nat > set_nat > $o ).
thf(func_def_67,type,
member_real: real > set_real > $o ).
thf(func_def_68,type,
s: set_complex ).
thf(func_def_69,type,
t: set_complex ).
thf(func_def_70,type,
f: complex > complex ).
thf(func_def_71,type,
g: complex > complex ).
thf(func_def_72,type,
w: complex ).
thf(func_def_84,type,
sK0: ( complex > complex ) > complex ).
thf(func_def_86,type,
ph2:
!>[X0: $tType] : X0 ).
thf(f896,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f895]) ).
thf(f895,plain,
one_one_complex != one_one_complex,
inference(superposition,[status(thm)],[f872,f887]) ).
thf(f887,plain,
! [X1: complex] :
( one_one_complex
= ( deriv_complex @ id_complex @ X1 ) ),
inference(beta_eta_normalization,[status(thm)],[f886]) ).
thf(f886,plain,
! [X1: complex] :
( ( deriv_complex @ id_complex @ X1 )
= ( ^ [Y0: complex] : one_one_complex
@ X1 ) ),
inference(argument_congruence,[status(thm)],[f871]) ).
thf(f871,plain,
( ( deriv_complex @ id_complex )
= ( ^ [Y0: complex] : one_one_complex ) ),
inference(cnf_transformation,[status(thm)],[f540]) ).
thf(f540,plain,
( ( deriv_complex @ id_complex )
= ( ^ [Y0: complex] : one_one_complex ) ),
inference(fool_elimination,[status(thm)],[f539]) ).
thf(f539,plain,
( ( deriv_complex @ id_complex )
= ( ^ [X0: complex] : one_one_complex ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
( ( deriv_complex @ id_complex )
= ( ^ [X1: complex] : one_one_complex ) ),
file('/export/starexec/sandbox/tmp/tmp.gvY0VWrXGK/DTF2THF_15200.p',fact_3_deriv__id) ).
thf(f872,plain,
( one_one_complex
!= ( deriv_complex @ id_complex @ w ) ),
inference(cnf_transformation,[status(thm)],[f825]) ).
thf(f825,plain,
( one_one_complex
!= ( deriv_complex @ id_complex @ w ) ),
inference(flattening,[status(thm)],[f361]) ).
thf(f361,negated_conjecture,
( one_one_complex
!= ( deriv_complex @ id_complex @ w ) ),
inference(negated_conjecture,[status(cth)],[f360]) ).
thf(f360,conjecture,
( one_one_complex
= ( deriv_complex @ id_complex @ w ) ),
file('/export/starexec/sandbox/tmp/tmp.gvY0VWrXGK/DTF2THF_15200.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP035^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n022.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:24:21 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.27/0.49 ---- Original DTF file ---
% 0.27/0.49 thf(spec,logic,$$dhol).
% 0.27/0.49 %------------------------------------------------------------------------------
% 0.27/0.49 % File : ITP035^1 : TPTP v9.2.1. Released v7.5.0.
% 0.27/0.49 % Domain : Interactive Theorem Proving
% 0.27/0.49 % Problem : Sledgehammer Cartan problem prob_32__6530878_1
% 0.27/0.49 % Version : Especial.
% 0.27/0.49 % English :
% 0.27/0.49
% 0.27/0.49 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.27/0.49 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.27/0.49 % Source : [Des21]
% 0.27/0.49 % Names : Cartan/prob_32__6530878_1 [Des21]
% 0.27/0.49
% 0.27/0.49 % Status : Theorem
% 0.27/0.49 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0
% 0.27/0.49 % Syntax : Number of formulae : 433 ( 205 unt; 73 typ; 0 def)
% 0.27/0.49 % Number of atoms : 909 ( 532 equ; 0 cnn)
% 0.27/0.49 % Maximal formula atoms : 11 ( 2 avg)
% 0.27/0.49 % Number of connectives : 2406 ( 60 ~; 38 |; 72 &;1899 @)
% 0.27/0.49 % ( 0 <=>; 337 =>; 0 <=; 0 <~>)
% 0.27/0.49 % Maximal formula depth : 18 ( 5 avg)
% 0.27/0.49 % Number of types : 8 ( 7 usr)
% 0.27/0.49 % Number of type conns : 426 ( 426 >; 0 *; 0 +; 0 <<)
% 0.27/0.49 % Number of symbols : 69 ( 66 usr; 11 con; 0-4 aty)
% 0.27/0.49 % Number of variables : 832 ( 41 ^; 785 !; 6 ?; 832 :)
% 0.27/0.49 % SPC : TH0_THM_EQU_NAR
% 0.27/0.49
% 0.27/0.49 % Comments : This file was generated by Sledgehammer 2021-02-23 15:38:30.825
% 0.27/0.49 %------------------------------------------------------------------------------
% 0.27/0.49 % Could-be-implicit typings (7)
% 0.27/0.49 thf(ty_n_t__Set__Oset_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 0.27/0.49 set_complex_complex: $tType ).
% 0.27/0.49
% 0.27/0.49 thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.27/0.49 set_complex: $tType ).
% 0.27/0.49
% 0.27/0.49 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.27/0.49 set_real: $tType ).
% 0.27/0.49
% 0.27/0.49 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.27/0.49 set_nat: $tType ).
% 0.27/0.49
% 0.27/0.49 thf(ty_n_t__Complex__Ocomplex,type,
% 0.27/0.49 complex: $tType ).
% 0.27/0.49
% 0.27/0.49 thf(ty_n_t__Real__Oreal,type,
% 0.27/0.49 real: $tType ).
% 0.27/0.49
% 0.27/0.49 thf(ty_n_t__Nat__Onat,type,
% 0.27/0.49 nat: $tType ).
% 0.27/0.49
% 0.27/0.49 % Explicit typings (66)
% 0.27/0.49 thf(sy_c_Complex__Analysis__Basics_Oholomorphic__on,type,
% 0.27/0.49 comple372758642hic_on: ( complex > complex ) > set_complex > $o ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Derivative_Oderiv_001t__Complex__Ocomplex,type,
% 0.27/0.49 deriv_complex: ( complex > complex ) > complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Derivative_Oderiv_001t__Real__Oreal,type,
% 0.27/0.49 deriv_real: ( real > real ) > real > real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 0.27/0.49 comp_c1610621014omplex: ( ( complex > complex ) > complex > complex ) > ( ( complex > complex ) > complex > complex ) > ( complex > complex ) > complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 0.27/0.49 comp_c881053372omplex: ( ( complex > complex ) > complex ) > ( complex > complex > complex ) > complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.27/0.49 comp_c1884694328l_real: ( ( complex > complex ) > real ) > ( real > complex > complex ) > real > real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 0.27/0.49 comp_c606622857omplex: ( complex > complex > complex ) > ( ( complex > complex ) > complex ) > ( complex > complex ) > complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 0.27/0.49 comp_c130555887omplex: ( complex > complex ) > ( complex > complex ) > complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.27/0.49 comp_c819638635l_real: ( complex > real ) > ( real > complex ) > real > real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 0.27/0.49 comp_r2009261319omplex: ( real > complex > complex ) > ( ( complex > complex ) > real ) > ( complex > complex ) > complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 0.27/0.49 comp_r667767405omplex: ( real > complex ) > ( complex > real ) > complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.27/0.49 comp_real_real_real: ( real > real ) > ( real > real ) > real > real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Fun_Oid_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 0.27/0.49 id_complex_complex: ( complex > complex ) > complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Fun_Oid_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.27/0.49 id_real_real: ( real > real ) > real > real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Fun_Oid_001t__Complex__Ocomplex,type,
% 0.27/0.49 id_complex: complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Fun_Oid_001t__Real__Oreal,type,
% 0.27/0.49 id_real: real > real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Fun_Oid_001t__Set__Oset_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 0.27/0.49 id_set1618538368omplex: set_complex_complex > set_complex_complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Fun_Oid_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.27/0.49 id_set_complex: set_complex > set_complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Fun_Oid_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.27/0.49 id_set_real: set_real > set_real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
% 0.27/0.49 one_one_complex: complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 0.27/0.49 one_one_nat: nat ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 0.27/0.49 one_one_real: real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
% 0.27/0.49 times_times_complex: complex > complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 0.27/0.49 times_times_nat: nat > nat > nat ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
% 0.27/0.49 times_times_real: real > real > real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.27/0.49 times_1316095593omplex: set_complex > set_complex > set_complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.27/0.49 times_times_set_nat: set_nat > set_nat > set_nat ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.27/0.49 times_times_set_real: set_real > set_real > set_real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
% 0.27/0.49 zero_zero_complex: complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.27/0.49 zero_zero_nat: nat ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.27/0.49 zero_zero_real: real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_If_001t__Complex__Ocomplex,type,
% 0.27/0.49 if_complex: $o > complex > complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_If_001t__Nat__Onat,type,
% 0.27/0.49 if_nat: $o > nat > nat > nat ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_If_001t__Real__Oreal,type,
% 0.27/0.49 if_real: $o > real > real > real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Nat_Ocompow_001_062_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_M_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 0.27/0.49 compow1098280738omplex: nat > ( ( complex > complex ) > complex > complex ) > ( complex > complex ) > complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Nat_Ocompow_001_062_I_062_It__Real__Oreal_Mt__Real__Oreal_J_M_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 0.27/0.49 compow1723822618l_real: nat > ( ( real > real ) > real > real ) > ( real > real ) > real > real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Nat_Ocompow_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 0.27/0.49 compow1667379464omplex: nat > ( complex > complex ) > complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Nat_Ocompow_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.27/0.49 compow_real_real: nat > ( real > real ) > real > real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Nat_Ofunpow_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 0.27/0.49 funpow1854104714omplex: nat > ( ( complex > complex ) > complex > complex ) > ( complex > complex ) > complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Nat_Ofunpow_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.27/0.49 funpow_real_real: nat > ( ( real > real ) > real > real ) > ( real > real ) > real > real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Nat_Ofunpow_001t__Complex__Ocomplex,type,
% 0.27/0.49 funpow_complex: nat > ( complex > complex ) > complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.27/0.49 ord_less_eq_nat: nat > nat > $o ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.27/0.49 ord_less_eq_real: real > real > $o ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.27/0.49 ord_le701908932omplex: set_complex > set_complex > $o ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
% 0.27/0.49 collect_complex: ( complex > $o ) > set_complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Set_Oimage_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 0.27/0.49 image_944012797omplex: ( ( complex > complex ) > complex > complex ) > set_complex_complex > set_complex_complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 0.27/0.49 image_58037603omplex: ( complex > complex ) > set_complex > set_complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.27/0.49 image_real_real: ( real > real ) > set_real > set_real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Complex__Ocomplex,type,
% 0.27/0.49 topolo935673511omplex: set_complex > $o ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Topological__Spaces_Otopological__space__class_Oconnected_001t__Complex__Ocomplex,type,
% 0.27/0.49 topolo2127351575omplex: set_complex > $o ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Transcendental_Oarcosh_001t__Complex__Ocomplex,type,
% 0.27/0.49 arcosh_complex: complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 0.27/0.49 arcosh_real: real > real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Transcendental_Oarsinh_001t__Complex__Ocomplex,type,
% 0.27/0.49 arsinh_complex: complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 0.27/0.49 arsinh_real: real > real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Transcendental_Oartanh_001t__Complex__Ocomplex,type,
% 0.27/0.49 artanh_complex: complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 0.27/0.49 artanh_real: real > real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Transcendental_Oln__class_Oln_001t__Complex__Ocomplex,type,
% 0.27/0.49 ln_ln_complex: complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 0.27/0.49 ln_ln_real: real > real ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_member_001t__Complex__Ocomplex,type,
% 0.27/0.49 member_complex: complex > set_complex > $o ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_member_001t__Nat__Onat,type,
% 0.27/0.49 member_nat: nat > set_nat > $o ).
% 0.27/0.49
% 0.27/0.49 thf(sy_c_member_001t__Real__Oreal,type,
% 0.27/0.49 member_real: real > set_real > $o ).
% 0.27/0.49
% 0.27/0.49 thf(sy_v_S,type,
% 0.27/0.49 s: set_complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_v_T,type,
% 0.27/0.49 t: set_complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_v_f,type,
% 0.27/0.49 f: complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_v_g,type,
% 0.27/0.49 g: complex > complex ).
% 0.27/0.49
% 0.27/0.49 thf(sy_v_w,type,
% 0.27/0.49 w: complex ).
% 0.27/0.49
% 0.27/0.49 % Relevant facts (352)
% 0.27/0.49 thf(fact_0__092_060open_062_092_060And_062z_O_A_Ideriv_A_094_094_A1_J_Aid_Az_A_061_A_Iif_A1_A_061_A0_Athen_Az_Aelse_Aif_A1_A_061_A1_Athen_A1_058_058_063_Ha_Aelse_A_I0_058_058_063_Ha_J_J_092_060close_062,axiom,
% 0.27/0.49 ! [Z: real] :
% 0.27/0.49 ( ( ( one_one_nat = zero_zero_nat )
% 0.27/0.49 => ( ( compow1723822618l_real @ one_one_nat @ deriv_real @ id_real @ Z )
% 0.27/0.49 = Z ) )
% 0.27/0.49 & ( ( one_one_nat != zero_zero_nat )
% 0.27/0.49 => ( ( compow1723822618l_real @ one_one_nat @ deriv_real @ id_real @ Z )
% 0.27/0.49 = one_one_real ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % \<open>\<And>z. (deriv ^^ 1) id z = (if 1 = 0 then z else if 1 = 1 then 1::?'a else (0::?'a))\<close>
% 0.27/0.49 thf(fact_1__092_060open_062_092_060And_062z_O_A_Ideriv_A_094_094_A1_J_Aid_Az_A_061_A_Iif_A1_A_061_A0_Athen_Az_Aelse_Aif_A1_A_061_A1_Athen_A1_058_058_063_Ha_Aelse_A_I0_058_058_063_Ha_J_J_092_060close_062,axiom,
% 0.27/0.49 ! [Z: complex] :
% 0.27/0.49 ( ( ( one_one_nat = zero_zero_nat )
% 0.27/0.49 => ( ( compow1098280738omplex @ one_one_nat @ deriv_complex @ id_complex @ Z )
% 0.27/0.49 = Z ) )
% 0.27/0.49 & ( ( one_one_nat != zero_zero_nat )
% 0.27/0.49 => ( ( compow1098280738omplex @ one_one_nat @ deriv_complex @ id_complex @ Z )
% 0.27/0.49 = one_one_complex ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % \<open>\<And>z. (deriv ^^ 1) id z = (if 1 = 0 then z else if 1 = 1 then 1::?'a else (0::?'a))\<close>
% 0.27/0.49 thf(fact_2_assms_I7_J,axiom,
% 0.27/0.49 member_complex @ w @ s ).
% 0.27/0.49
% 0.27/0.49 % assms(7)
% 0.27/0.49 thf(fact_3_deriv__id,axiom,
% 0.27/0.49 ( ( deriv_complex @ id_complex )
% 0.27/0.49 = ( ^ [Z2: complex] : one_one_complex ) ) ).
% 0.27/0.49
% 0.27/0.49 % deriv_id
% 0.27/0.49 thf(fact_4_deriv__id,axiom,
% 0.27/0.49 ( ( deriv_real @ id_real )
% 0.27/0.49 = ( ^ [Z2: real] : one_one_real ) ) ).
% 0.27/0.49
% 0.27/0.49 % deriv_id
% 0.27/0.49 thf(fact_5_id__apply,axiom,
% 0.27/0.49 ( id_real
% 0.27/0.49 = ( ^ [X: real] : X ) ) ).
% 0.27/0.49
% 0.27/0.49 % id_apply
% 0.27/0.49 thf(fact_6_id__apply,axiom,
% 0.27/0.49 ( id_complex_complex
% 0.27/0.49 = ( ^ [X: complex > complex] : X ) ) ).
% 0.27/0.49
% 0.27/0.49 % id_apply
% 0.27/0.49 thf(fact_7_id__apply,axiom,
% 0.27/0.49 ( id_complex
% 0.27/0.49 = ( ^ [X: complex] : X ) ) ).
% 0.27/0.49
% 0.27/0.49 % id_apply
% 0.27/0.49 thf(fact_8_id__def,axiom,
% 0.27/0.49 ( id_real
% 0.27/0.49 = ( ^ [X: real] : X ) ) ).
% 0.27/0.49
% 0.27/0.49 % id_def
% 0.27/0.49 thf(fact_9_id__def,axiom,
% 0.27/0.49 ( id_complex_complex
% 0.27/0.49 = ( ^ [X: complex > complex] : X ) ) ).
% 0.27/0.49
% 0.27/0.49 % id_def
% 0.27/0.49 thf(fact_10_id__def,axiom,
% 0.27/0.49 ( id_complex
% 0.27/0.49 = ( ^ [X: complex] : X ) ) ).
% 0.27/0.49
% 0.27/0.49 % id_def
% 0.27/0.49 thf(fact_11_eq__id__iff,axiom,
% 0.27/0.49 ! [F: real > real] :
% 0.27/0.49 ( ( ! [X: real] :
% 0.27/0.49 ( ( F @ X )
% 0.27/0.49 = X ) )
% 0.27/0.49 = ( F = id_real ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_id_iff
% 0.27/0.49 thf(fact_12_eq__id__iff,axiom,
% 0.27/0.49 ! [F: ( complex > complex ) > complex > complex] :
% 0.27/0.49 ( ( ! [X: complex > complex] :
% 0.27/0.49 ( ( F @ X )
% 0.27/0.49 = X ) )
% 0.27/0.49 = ( F = id_complex_complex ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_id_iff
% 0.27/0.49 thf(fact_13_eq__id__iff,axiom,
% 0.27/0.49 ! [F: complex > complex] :
% 0.27/0.49 ( ( ! [X: complex] :
% 0.27/0.49 ( ( F @ X )
% 0.27/0.49 = X ) )
% 0.27/0.49 = ( F = id_complex ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_id_iff
% 0.27/0.49 thf(fact_14_higher__deriv__id,axiom,
% 0.27/0.49 ! [N: nat,Z: real] :
% 0.27/0.49 ( ( ( N = zero_zero_nat )
% 0.27/0.49 => ( ( compow1723822618l_real @ N @ deriv_real @ id_real @ Z )
% 0.27/0.49 = Z ) )
% 0.27/0.49 & ( ( N != zero_zero_nat )
% 0.27/0.49 => ( ( ( N = one_one_nat )
% 0.27/0.49 => ( ( compow1723822618l_real @ N @ deriv_real @ id_real @ Z )
% 0.27/0.49 = one_one_real ) )
% 0.27/0.49 & ( ( N != one_one_nat )
% 0.27/0.49 => ( ( compow1723822618l_real @ N @ deriv_real @ id_real @ Z )
% 0.27/0.49 = zero_zero_real ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % higher_deriv_id
% 0.27/0.49 thf(fact_15_higher__deriv__id,axiom,
% 0.27/0.49 ! [N: nat,Z: complex] :
% 0.27/0.49 ( ( ( N = zero_zero_nat )
% 0.27/0.49 => ( ( compow1098280738omplex @ N @ deriv_complex @ id_complex @ Z )
% 0.27/0.49 = Z ) )
% 0.27/0.49 & ( ( N != zero_zero_nat )
% 0.27/0.49 => ( ( ( N = one_one_nat )
% 0.27/0.49 => ( ( compow1098280738omplex @ N @ deriv_complex @ id_complex @ Z )
% 0.27/0.49 = one_one_complex ) )
% 0.27/0.49 & ( ( N != one_one_nat )
% 0.27/0.49 => ( ( compow1098280738omplex @ N @ deriv_complex @ id_complex @ Z )
% 0.27/0.49 = zero_zero_complex ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % higher_deriv_id
% 0.27/0.49 thf(fact_16_one__reorient,axiom,
% 0.27/0.49 ! [X2: complex] :
% 0.27/0.49 ( ( one_one_complex = X2 )
% 0.27/0.49 = ( X2 = one_one_complex ) ) ).
% 0.27/0.49
% 0.27/0.49 % one_reorient
% 0.27/0.49 thf(fact_17_one__reorient,axiom,
% 0.27/0.49 ! [X2: nat] :
% 0.27/0.49 ( ( one_one_nat = X2 )
% 0.27/0.49 = ( X2 = one_one_nat ) ) ).
% 0.27/0.49
% 0.27/0.49 % one_reorient
% 0.27/0.49 thf(fact_18_one__reorient,axiom,
% 0.27/0.49 ! [X2: real] :
% 0.27/0.49 ( ( one_one_real = X2 )
% 0.27/0.49 = ( X2 = one_one_real ) ) ).
% 0.27/0.49
% 0.27/0.49 % one_reorient
% 0.27/0.49 thf(fact_19_id__funpow,axiom,
% 0.27/0.49 ! [N: nat] :
% 0.27/0.49 ( ( compow_real_real @ N @ id_real )
% 0.27/0.49 = id_real ) ).
% 0.27/0.49
% 0.27/0.49 % id_funpow
% 0.27/0.49 thf(fact_20_id__funpow,axiom,
% 0.27/0.49 ! [N: nat] :
% 0.27/0.49 ( ( compow1723822618l_real @ N @ id_real_real )
% 0.27/0.49 = id_real_real ) ).
% 0.27/0.49
% 0.27/0.49 % id_funpow
% 0.27/0.49 thf(fact_21_id__funpow,axiom,
% 0.27/0.49 ! [N: nat] :
% 0.27/0.49 ( ( compow1667379464omplex @ N @ id_complex )
% 0.27/0.49 = id_complex ) ).
% 0.27/0.49
% 0.27/0.49 % id_funpow
% 0.27/0.49 thf(fact_22_id__funpow,axiom,
% 0.27/0.49 ! [N: nat] :
% 0.27/0.49 ( ( compow1098280738omplex @ N @ id_complex_complex )
% 0.27/0.49 = id_complex_complex ) ).
% 0.27/0.49
% 0.27/0.49 % id_funpow
% 0.27/0.49 thf(fact_23_calculation,axiom,
% 0.27/0.49 ( ( times_times_complex @ ( deriv_complex @ f @ w ) @ ( deriv_complex @ g @ ( f @ w ) ) )
% 0.27/0.49 = ( deriv_complex @ id_complex @ w ) ) ).
% 0.27/0.49
% 0.27/0.49 % calculation
% 0.27/0.49 thf(fact_24__092_060open_062deriv_A_Ig_A_092_060circ_062_Af_J_Aw_A_061_Aderiv_Aid_Aw_092_060close_062,axiom,
% 0.27/0.49 ( ( deriv_complex @ ( comp_c130555887omplex @ g @ f ) @ w )
% 0.27/0.49 = ( deriv_complex @ id_complex @ w ) ) ).
% 0.27/0.49
% 0.27/0.49 % \<open>deriv (g \<circ> f) w = deriv id w\<close>
% 0.27/0.49 thf(fact_25_zero__neq__one,axiom,
% 0.27/0.49 zero_zero_nat != one_one_nat ).
% 0.27/0.49
% 0.27/0.49 % zero_neq_one
% 0.27/0.49 thf(fact_26_zero__neq__one,axiom,
% 0.27/0.49 zero_zero_real != one_one_real ).
% 0.27/0.49
% 0.27/0.49 % zero_neq_one
% 0.27/0.49 thf(fact_27_zero__neq__one,axiom,
% 0.27/0.49 zero_zero_complex != one_one_complex ).
% 0.27/0.49
% 0.27/0.49 % zero_neq_one
% 0.27/0.49 thf(fact_28__092_060open_062deriv_Af_Aw_A_K_Aderiv_Ag_A_If_Aw_J_A_061_Aderiv_Ag_A_If_Aw_J_A_K_Aderiv_Af_Aw_092_060close_062,axiom,
% 0.27/0.49 ( ( times_times_complex @ ( deriv_complex @ f @ w ) @ ( deriv_complex @ g @ ( f @ w ) ) )
% 0.27/0.49 = ( times_times_complex @ ( deriv_complex @ g @ ( f @ w ) ) @ ( deriv_complex @ f @ w ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % \<open>deriv f w * deriv g (f w) = deriv g (f w) * deriv f w\<close>
% 0.27/0.49 thf(fact_29_comp__apply,axiom,
% 0.27/0.49 ( comp_c130555887omplex
% 0.27/0.49 = ( ^ [F2: complex > complex,G: complex > complex,X: complex] : ( F2 @ ( G @ X ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % comp_apply
% 0.27/0.49 thf(fact_30_mult__zero__left,axiom,
% 0.27/0.49 ! [A: complex] :
% 0.27/0.49 ( ( times_times_complex @ zero_zero_complex @ A )
% 0.27/0.49 = zero_zero_complex ) ).
% 0.27/0.49
% 0.27/0.49 % mult_zero_left
% 0.27/0.49 thf(fact_31_mult__zero__left,axiom,
% 0.27/0.49 ! [A: nat] :
% 0.27/0.49 ( ( times_times_nat @ zero_zero_nat @ A )
% 0.27/0.49 = zero_zero_nat ) ).
% 0.27/0.49
% 0.27/0.49 % mult_zero_left
% 0.27/0.49 thf(fact_32_mult__zero__left,axiom,
% 0.27/0.49 ! [A: real] :
% 0.27/0.49 ( ( times_times_real @ zero_zero_real @ A )
% 0.27/0.49 = zero_zero_real ) ).
% 0.27/0.49
% 0.27/0.49 % mult_zero_left
% 0.27/0.49 thf(fact_33_mult__zero__right,axiom,
% 0.27/0.49 ! [A: complex] :
% 0.27/0.49 ( ( times_times_complex @ A @ zero_zero_complex )
% 0.27/0.49 = zero_zero_complex ) ).
% 0.27/0.49
% 0.27/0.49 % mult_zero_right
% 0.27/0.49 thf(fact_34_mult__zero__right,axiom,
% 0.27/0.49 ! [A: nat] :
% 0.27/0.49 ( ( times_times_nat @ A @ zero_zero_nat )
% 0.27/0.49 = zero_zero_nat ) ).
% 0.27/0.49
% 0.27/0.49 % mult_zero_right
% 0.27/0.49 thf(fact_35_mult__zero__right,axiom,
% 0.27/0.49 ! [A: real] :
% 0.27/0.49 ( ( times_times_real @ A @ zero_zero_real )
% 0.27/0.49 = zero_zero_real ) ).
% 0.27/0.49
% 0.27/0.49 % mult_zero_right
% 0.27/0.49 thf(fact_36_mult__eq__0__iff,axiom,
% 0.27/0.49 ! [A: complex,B: complex] :
% 0.27/0.49 ( ( ( times_times_complex @ A @ B )
% 0.27/0.49 = zero_zero_complex )
% 0.27/0.49 = ( ( A = zero_zero_complex )
% 0.27/0.49 | ( B = zero_zero_complex ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_eq_0_iff
% 0.27/0.49 thf(fact_37_mult__eq__0__iff,axiom,
% 0.27/0.49 ! [A: nat,B: nat] :
% 0.27/0.49 ( ( ( times_times_nat @ A @ B )
% 0.27/0.49 = zero_zero_nat )
% 0.27/0.49 = ( ( A = zero_zero_nat )
% 0.27/0.49 | ( B = zero_zero_nat ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_eq_0_iff
% 0.27/0.49 thf(fact_38_mult__eq__0__iff,axiom,
% 0.27/0.49 ! [A: real,B: real] :
% 0.27/0.49 ( ( ( times_times_real @ A @ B )
% 0.27/0.49 = zero_zero_real )
% 0.27/0.49 = ( ( A = zero_zero_real )
% 0.27/0.49 | ( B = zero_zero_real ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_eq_0_iff
% 0.27/0.49 thf(fact_39_mult__cancel__left,axiom,
% 0.27/0.49 ! [C: complex,A: complex,B: complex] :
% 0.27/0.49 ( ( ( times_times_complex @ C @ A )
% 0.27/0.49 = ( times_times_complex @ C @ B ) )
% 0.27/0.49 = ( ( C = zero_zero_complex )
% 0.27/0.49 | ( A = B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_cancel_left
% 0.27/0.49 thf(fact_40_mult__cancel__left,axiom,
% 0.27/0.49 ! [C: nat,A: nat,B: nat] :
% 0.27/0.49 ( ( ( times_times_nat @ C @ A )
% 0.27/0.49 = ( times_times_nat @ C @ B ) )
% 0.27/0.49 = ( ( C = zero_zero_nat )
% 0.27/0.49 | ( A = B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_cancel_left
% 0.27/0.49 thf(fact_41_mult__cancel__left,axiom,
% 0.27/0.49 ! [C: real,A: real,B: real] :
% 0.27/0.49 ( ( ( times_times_real @ C @ A )
% 0.27/0.49 = ( times_times_real @ C @ B ) )
% 0.27/0.49 = ( ( C = zero_zero_real )
% 0.27/0.49 | ( A = B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_cancel_left
% 0.27/0.49 thf(fact_42_mult__cancel__right,axiom,
% 0.27/0.49 ! [A: complex,C: complex,B: complex] :
% 0.27/0.49 ( ( ( times_times_complex @ A @ C )
% 0.27/0.49 = ( times_times_complex @ B @ C ) )
% 0.27/0.49 = ( ( C = zero_zero_complex )
% 0.27/0.49 | ( A = B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_cancel_right
% 0.27/0.49 thf(fact_43_mult__cancel__right,axiom,
% 0.27/0.49 ! [A: nat,C: nat,B: nat] :
% 0.27/0.49 ( ( ( times_times_nat @ A @ C )
% 0.27/0.49 = ( times_times_nat @ B @ C ) )
% 0.27/0.49 = ( ( C = zero_zero_nat )
% 0.27/0.49 | ( A = B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_cancel_right
% 0.27/0.49 thf(fact_44_mult__cancel__right,axiom,
% 0.27/0.49 ! [A: real,C: real,B: real] :
% 0.27/0.49 ( ( ( times_times_real @ A @ C )
% 0.27/0.49 = ( times_times_real @ B @ C ) )
% 0.27/0.49 = ( ( C = zero_zero_real )
% 0.27/0.49 | ( A = B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_cancel_right
% 0.27/0.49 thf(fact_45_mult_Oright__neutral,axiom,
% 0.27/0.49 ! [A: complex] :
% 0.27/0.49 ( ( times_times_complex @ A @ one_one_complex )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % mult.right_neutral
% 0.27/0.49 thf(fact_46_mult_Oright__neutral,axiom,
% 0.27/0.49 ! [A: nat] :
% 0.27/0.49 ( ( times_times_nat @ A @ one_one_nat )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % mult.right_neutral
% 0.27/0.49 thf(fact_47_mult_Oright__neutral,axiom,
% 0.27/0.49 ! [A: real] :
% 0.27/0.49 ( ( times_times_real @ A @ one_one_real )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % mult.right_neutral
% 0.27/0.49 thf(fact_48_mult_Oleft__neutral,axiom,
% 0.27/0.49 ! [A: complex] :
% 0.27/0.49 ( ( times_times_complex @ one_one_complex @ A )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % mult.left_neutral
% 0.27/0.49 thf(fact_49_mult_Oleft__neutral,axiom,
% 0.27/0.49 ! [A: nat] :
% 0.27/0.49 ( ( times_times_nat @ one_one_nat @ A )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % mult.left_neutral
% 0.27/0.49 thf(fact_50_mult_Oleft__neutral,axiom,
% 0.27/0.49 ! [A: real] :
% 0.27/0.49 ( ( times_times_real @ one_one_real @ A )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % mult.left_neutral
% 0.27/0.49 thf(fact_51_id__comp,axiom,
% 0.27/0.49 ! [G2: complex > complex] :
% 0.27/0.49 ( ( comp_c130555887omplex @ id_complex @ G2 )
% 0.27/0.49 = G2 ) ).
% 0.27/0.49
% 0.27/0.49 % id_comp
% 0.27/0.49 thf(fact_52_comp__id,axiom,
% 0.27/0.49 ! [F: complex > complex] :
% 0.27/0.49 ( ( comp_c130555887omplex @ F @ id_complex )
% 0.27/0.49 = F ) ).
% 0.27/0.49
% 0.27/0.49 % comp_id
% 0.27/0.49 thf(fact_53_funpow__0,axiom,
% 0.27/0.49 ! [F: ( complex > complex ) > complex > complex,X2: complex > complex] :
% 0.27/0.49 ( ( compow1098280738omplex @ zero_zero_nat @ F @ X2 )
% 0.27/0.49 = X2 ) ).
% 0.27/0.49
% 0.27/0.49 % funpow_0
% 0.27/0.49 thf(fact_54_funpow__0,axiom,
% 0.27/0.49 ! [F: ( real > real ) > real > real,X2: real > real] :
% 0.27/0.49 ( ( compow1723822618l_real @ zero_zero_nat @ F @ X2 )
% 0.27/0.49 = X2 ) ).
% 0.27/0.49
% 0.27/0.49 % funpow_0
% 0.27/0.49 thf(fact_55_funpow__0,axiom,
% 0.27/0.49 ! [F: complex > complex,X2: complex] :
% 0.27/0.49 ( ( compow1667379464omplex @ zero_zero_nat @ F @ X2 )
% 0.27/0.49 = X2 ) ).
% 0.27/0.49
% 0.27/0.49 % funpow_0
% 0.27/0.49 thf(fact_56_assms_I3_J,axiom,
% 0.27/0.49 topolo935673511omplex @ s ).
% 0.27/0.49
% 0.27/0.49 % assms(3)
% 0.27/0.49 thf(fact_57__092_060open_062deriv_Ag_A_If_Aw_J_A_K_Aderiv_Af_Aw_A_061_Aderiv_A_Ig_A_092_060circ_062_Af_J_Aw_092_060close_062,axiom,
% 0.27/0.49 ( ( times_times_complex @ ( deriv_complex @ g @ ( f @ w ) ) @ ( deriv_complex @ f @ w ) )
% 0.27/0.49 = ( deriv_complex @ ( comp_c130555887omplex @ g @ f ) @ w ) ) ).
% 0.27/0.49
% 0.27/0.49 % \<open>deriv g (f w) * deriv f w = deriv (g \<circ> f) w\<close>
% 0.27/0.49 thf(fact_58_mult__cancel__right2,axiom,
% 0.27/0.49 ! [A: complex,C: complex] :
% 0.27/0.49 ( ( ( times_times_complex @ A @ C )
% 0.27/0.49 = C )
% 0.27/0.49 = ( ( C = zero_zero_complex )
% 0.27/0.49 | ( A = one_one_complex ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_cancel_right2
% 0.27/0.49 thf(fact_59_mult__cancel__right2,axiom,
% 0.27/0.49 ! [A: real,C: real] :
% 0.27/0.49 ( ( ( times_times_real @ A @ C )
% 0.27/0.49 = C )
% 0.27/0.49 = ( ( C = zero_zero_real )
% 0.27/0.49 | ( A = one_one_real ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_cancel_right2
% 0.27/0.49 thf(fact_60_mult__cancel__right1,axiom,
% 0.27/0.49 ! [C: complex,B: complex] :
% 0.27/0.49 ( ( C
% 0.27/0.49 = ( times_times_complex @ B @ C ) )
% 0.27/0.49 = ( ( C = zero_zero_complex )
% 0.27/0.49 | ( B = one_one_complex ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_cancel_right1
% 0.27/0.49 thf(fact_61_mult__cancel__right1,axiom,
% 0.27/0.49 ! [C: real,B: real] :
% 0.27/0.49 ( ( C
% 0.27/0.49 = ( times_times_real @ B @ C ) )
% 0.27/0.49 = ( ( C = zero_zero_real )
% 0.27/0.49 | ( B = one_one_real ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_cancel_right1
% 0.27/0.49 thf(fact_62_mult__cancel__left2,axiom,
% 0.27/0.49 ! [C: complex,A: complex] :
% 0.27/0.49 ( ( ( times_times_complex @ C @ A )
% 0.27/0.49 = C )
% 0.27/0.49 = ( ( C = zero_zero_complex )
% 0.27/0.49 | ( A = one_one_complex ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_cancel_left2
% 0.27/0.49 thf(fact_63_mult__cancel__left2,axiom,
% 0.27/0.49 ! [C: real,A: real] :
% 0.27/0.49 ( ( ( times_times_real @ C @ A )
% 0.27/0.49 = C )
% 0.27/0.49 = ( ( C = zero_zero_real )
% 0.27/0.49 | ( A = one_one_real ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_cancel_left2
% 0.27/0.49 thf(fact_64_mult__cancel__left1,axiom,
% 0.27/0.49 ! [C: complex,B: complex] :
% 0.27/0.49 ( ( C
% 0.27/0.49 = ( times_times_complex @ C @ B ) )
% 0.27/0.49 = ( ( C = zero_zero_complex )
% 0.27/0.49 | ( B = one_one_complex ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_cancel_left1
% 0.27/0.49 thf(fact_65_mult__cancel__left1,axiom,
% 0.27/0.49 ! [C: real,B: real] :
% 0.27/0.49 ( ( C
% 0.27/0.49 = ( times_times_real @ C @ B ) )
% 0.27/0.49 = ( ( C = zero_zero_real )
% 0.27/0.49 | ( B = one_one_real ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_cancel_left1
% 0.27/0.49 thf(fact_66_assms_I6_J,axiom,
% 0.27/0.49 ! [Z: complex] :
% 0.27/0.49 ( ( member_complex @ Z @ s )
% 0.27/0.49 => ( ( g @ ( f @ Z ) )
% 0.27/0.49 = Z ) ) ).
% 0.27/0.49
% 0.27/0.49 % assms(6)
% 0.27/0.49 thf(fact_67_assms_I1_J,axiom,
% 0.27/0.49 comple372758642hic_on @ f @ s ).
% 0.27/0.49
% 0.27/0.49 % assms(1)
% 0.27/0.49 thf(fact_68_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.27/0.49 ! [A: complex,B: complex,C: complex] :
% 0.27/0.49 ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
% 0.27/0.49 = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % ab_semigroup_mult_class.mult_ac(1)
% 0.27/0.49 thf(fact_69_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.27/0.49 ! [A: nat,B: nat,C: nat] :
% 0.27/0.49 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.27/0.49 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % ab_semigroup_mult_class.mult_ac(1)
% 0.27/0.49 thf(fact_70_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.27/0.49 ! [A: real,B: real,C: real] :
% 0.27/0.49 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.27/0.49 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % ab_semigroup_mult_class.mult_ac(1)
% 0.27/0.49 thf(fact_71_comp__def,axiom,
% 0.27/0.49 ( comp_c130555887omplex
% 0.27/0.49 = ( ^ [F2: complex > complex,G: complex > complex,X: complex] : ( F2 @ ( G @ X ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % comp_def
% 0.27/0.49 thf(fact_72_comp__assoc,axiom,
% 0.27/0.49 ! [F: complex > complex,G2: complex > complex,H: complex > complex] :
% 0.27/0.49 ( ( comp_c130555887omplex @ ( comp_c130555887omplex @ F @ G2 ) @ H )
% 0.27/0.49 = ( comp_c130555887omplex @ F @ ( comp_c130555887omplex @ G2 @ H ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % comp_assoc
% 0.27/0.49 thf(fact_73_comp__funpow,axiom,
% 0.27/0.49 ! [N: nat,F: complex > complex] :
% 0.27/0.49 ( ( compow1098280738omplex @ N @ ( comp_c130555887omplex @ F ) )
% 0.27/0.49 = ( comp_c130555887omplex @ ( compow1667379464omplex @ N @ F ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % comp_funpow
% 0.27/0.49 thf(fact_74_comp__funpow,axiom,
% 0.27/0.49 ! [N: nat,F: real > real] :
% 0.27/0.49 ( ( compow1723822618l_real @ N @ ( comp_real_real_real @ F ) )
% 0.27/0.49 = ( comp_real_real_real @ ( compow_real_real @ N @ F ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % comp_funpow
% 0.27/0.49 thf(fact_75_comp__eq__dest,axiom,
% 0.27/0.49 ! [A: complex > complex,B: complex > complex,C: complex > complex,D: complex > complex,V: complex] :
% 0.27/0.49 ( ( ( comp_c130555887omplex @ A @ B )
% 0.27/0.49 = ( comp_c130555887omplex @ C @ D ) )
% 0.27/0.49 => ( ( A @ ( B @ V ) )
% 0.27/0.49 = ( C @ ( D @ V ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % comp_eq_dest
% 0.27/0.49 thf(fact_76_comp__eq__elim,axiom,
% 0.27/0.49 ! [A: complex > complex,B: complex > complex,C: complex > complex,D: complex > complex] :
% 0.27/0.49 ( ( ( comp_c130555887omplex @ A @ B )
% 0.27/0.49 = ( comp_c130555887omplex @ C @ D ) )
% 0.27/0.49 => ! [V2: complex] :
% 0.27/0.49 ( ( A @ ( B @ V2 ) )
% 0.27/0.49 = ( C @ ( D @ V2 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % comp_eq_elim
% 0.27/0.49 thf(fact_77_funpow__swap1,axiom,
% 0.27/0.49 ! [F: ( complex > complex ) > complex > complex,N: nat,X2: complex > complex] :
% 0.27/0.49 ( ( F @ ( compow1098280738omplex @ N @ F @ X2 ) )
% 0.27/0.49 = ( compow1098280738omplex @ N @ F @ ( F @ X2 ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funpow_swap1
% 0.27/0.49 thf(fact_78_funpow__swap1,axiom,
% 0.27/0.49 ! [F: ( real > real ) > real > real,N: nat,X2: real > real] :
% 0.27/0.49 ( ( F @ ( compow1723822618l_real @ N @ F @ X2 ) )
% 0.27/0.49 = ( compow1723822618l_real @ N @ F @ ( F @ X2 ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funpow_swap1
% 0.27/0.49 thf(fact_79_funpow__swap1,axiom,
% 0.27/0.49 ! [F: complex > complex,N: nat,X2: complex] :
% 0.27/0.49 ( ( F @ ( compow1667379464omplex @ N @ F @ X2 ) )
% 0.27/0.49 = ( compow1667379464omplex @ N @ F @ ( F @ X2 ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funpow_swap1
% 0.27/0.49 thf(fact_80_comp__eq__dest__lhs,axiom,
% 0.27/0.49 ! [A: complex > complex,B: complex > complex,C: complex > complex,V: complex] :
% 0.27/0.49 ( ( ( comp_c130555887omplex @ A @ B )
% 0.27/0.49 = C )
% 0.27/0.49 => ( ( A @ ( B @ V ) )
% 0.27/0.49 = ( C @ V ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % comp_eq_dest_lhs
% 0.27/0.49 thf(fact_81_zero__reorient,axiom,
% 0.27/0.49 ! [X2: nat] :
% 0.27/0.49 ( ( zero_zero_nat = X2 )
% 0.27/0.49 = ( X2 = zero_zero_nat ) ) ).
% 0.27/0.49
% 0.27/0.49 % zero_reorient
% 0.27/0.49 thf(fact_82_zero__reorient,axiom,
% 0.27/0.49 ! [X2: real] :
% 0.27/0.49 ( ( zero_zero_real = X2 )
% 0.27/0.49 = ( X2 = zero_zero_real ) ) ).
% 0.27/0.49
% 0.27/0.49 % zero_reorient
% 0.27/0.49 thf(fact_83_zero__reorient,axiom,
% 0.27/0.49 ! [X2: complex] :
% 0.27/0.49 ( ( zero_zero_complex = X2 )
% 0.27/0.49 = ( X2 = zero_zero_complex ) ) ).
% 0.27/0.49
% 0.27/0.49 % zero_reorient
% 0.27/0.49 thf(fact_84_mult__not__zero,axiom,
% 0.27/0.49 ! [A: complex,B: complex] :
% 0.27/0.49 ( ( ( times_times_complex @ A @ B )
% 0.27/0.49 != zero_zero_complex )
% 0.27/0.49 => ( ( A != zero_zero_complex )
% 0.27/0.49 & ( B != zero_zero_complex ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_not_zero
% 0.27/0.49 thf(fact_85_mult__not__zero,axiom,
% 0.27/0.49 ! [A: nat,B: nat] :
% 0.27/0.49 ( ( ( times_times_nat @ A @ B )
% 0.27/0.49 != zero_zero_nat )
% 0.27/0.49 => ( ( A != zero_zero_nat )
% 0.27/0.49 & ( B != zero_zero_nat ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_not_zero
% 0.27/0.49 thf(fact_86_mult__not__zero,axiom,
% 0.27/0.49 ! [A: real,B: real] :
% 0.27/0.49 ( ( ( times_times_real @ A @ B )
% 0.27/0.49 != zero_zero_real )
% 0.27/0.49 => ( ( A != zero_zero_real )
% 0.27/0.49 & ( B != zero_zero_real ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_not_zero
% 0.27/0.49 thf(fact_87_mem__Collect__eq,axiom,
% 0.27/0.49 ! [A: complex,P: complex > $o] :
% 0.27/0.49 ( ( member_complex @ A @ ( collect_complex @ P ) )
% 0.27/0.49 = ( P @ A ) ) ).
% 0.27/0.49
% 0.27/0.49 % mem_Collect_eq
% 0.27/0.49 thf(fact_88_Collect__mem__eq,axiom,
% 0.27/0.49 ! [A2: set_complex] :
% 0.27/0.49 ( ( collect_complex
% 0.27/0.49 @ ^ [X: complex] : ( member_complex @ X @ A2 ) )
% 0.27/0.49 = A2 ) ).
% 0.27/0.49
% 0.27/0.49 % Collect_mem_eq
% 0.27/0.49 thf(fact_89_mult_Oassoc,axiom,
% 0.27/0.49 ! [A: complex,B: complex,C: complex] :
% 0.27/0.49 ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
% 0.27/0.49 = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult.assoc
% 0.27/0.49 thf(fact_90_mult_Oassoc,axiom,
% 0.27/0.49 ! [A: nat,B: nat,C: nat] :
% 0.27/0.49 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.27/0.49 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult.assoc
% 0.27/0.49 thf(fact_91_mult_Oassoc,axiom,
% 0.27/0.49 ! [A: real,B: real,C: real] :
% 0.27/0.49 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.27/0.49 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult.assoc
% 0.27/0.49 thf(fact_92_mult_Ocommute,axiom,
% 0.27/0.49 ( times_times_complex
% 0.27/0.49 = ( ^ [A3: complex,B2: complex] : ( times_times_complex @ B2 @ A3 ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult.commute
% 0.27/0.49 thf(fact_93_mult_Ocommute,axiom,
% 0.27/0.49 ( times_times_nat
% 0.27/0.49 = ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult.commute
% 0.27/0.49 thf(fact_94_mult_Ocommute,axiom,
% 0.27/0.49 ( times_times_real
% 0.27/0.49 = ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult.commute
% 0.27/0.49 thf(fact_95_mult_Oleft__commute,axiom,
% 0.27/0.49 ! [B: complex,A: complex,C: complex] :
% 0.27/0.49 ( ( times_times_complex @ B @ ( times_times_complex @ A @ C ) )
% 0.27/0.49 = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult.left_commute
% 0.27/0.49 thf(fact_96_mult_Oleft__commute,axiom,
% 0.27/0.49 ! [B: nat,A: nat,C: nat] :
% 0.27/0.49 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 0.27/0.49 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult.left_commute
% 0.27/0.49 thf(fact_97_mult_Oleft__commute,axiom,
% 0.27/0.49 ! [B: real,A: real,C: real] :
% 0.27/0.49 ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 0.27/0.49 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult.left_commute
% 0.27/0.49 thf(fact_98_divisors__zero,axiom,
% 0.27/0.49 ! [A: complex,B: complex] :
% 0.27/0.49 ( ( ( times_times_complex @ A @ B )
% 0.27/0.49 = zero_zero_complex )
% 0.27/0.49 => ( ( A = zero_zero_complex )
% 0.27/0.49 | ( B = zero_zero_complex ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % divisors_zero
% 0.27/0.49 thf(fact_99_divisors__zero,axiom,
% 0.27/0.49 ! [A: nat,B: nat] :
% 0.27/0.49 ( ( ( times_times_nat @ A @ B )
% 0.27/0.49 = zero_zero_nat )
% 0.27/0.49 => ( ( A = zero_zero_nat )
% 0.27/0.49 | ( B = zero_zero_nat ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % divisors_zero
% 0.27/0.49 thf(fact_100_divisors__zero,axiom,
% 0.27/0.49 ! [A: real,B: real] :
% 0.27/0.49 ( ( ( times_times_real @ A @ B )
% 0.27/0.49 = zero_zero_real )
% 0.27/0.49 => ( ( A = zero_zero_real )
% 0.27/0.49 | ( B = zero_zero_real ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % divisors_zero
% 0.27/0.49 thf(fact_101_no__zero__divisors,axiom,
% 0.27/0.49 ! [A: complex,B: complex] :
% 0.27/0.49 ( ( A != zero_zero_complex )
% 0.27/0.49 => ( ( B != zero_zero_complex )
% 0.27/0.49 => ( ( times_times_complex @ A @ B )
% 0.27/0.49 != zero_zero_complex ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % no_zero_divisors
% 0.27/0.49 thf(fact_102_no__zero__divisors,axiom,
% 0.27/0.49 ! [A: nat,B: nat] :
% 0.27/0.49 ( ( A != zero_zero_nat )
% 0.27/0.49 => ( ( B != zero_zero_nat )
% 0.27/0.49 => ( ( times_times_nat @ A @ B )
% 0.27/0.49 != zero_zero_nat ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % no_zero_divisors
% 0.27/0.49 thf(fact_103_no__zero__divisors,axiom,
% 0.27/0.49 ! [A: real,B: real] :
% 0.27/0.49 ( ( A != zero_zero_real )
% 0.27/0.49 => ( ( B != zero_zero_real )
% 0.27/0.49 => ( ( times_times_real @ A @ B )
% 0.27/0.49 != zero_zero_real ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % no_zero_divisors
% 0.27/0.49 thf(fact_104_mult__left__cancel,axiom,
% 0.27/0.49 ! [C: complex,A: complex,B: complex] :
% 0.27/0.49 ( ( C != zero_zero_complex )
% 0.27/0.49 => ( ( ( times_times_complex @ C @ A )
% 0.27/0.49 = ( times_times_complex @ C @ B ) )
% 0.27/0.49 = ( A = B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_left_cancel
% 0.27/0.49 thf(fact_105_mult__left__cancel,axiom,
% 0.27/0.49 ! [C: nat,A: nat,B: nat] :
% 0.27/0.49 ( ( C != zero_zero_nat )
% 0.27/0.49 => ( ( ( times_times_nat @ C @ A )
% 0.27/0.49 = ( times_times_nat @ C @ B ) )
% 0.27/0.49 = ( A = B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_left_cancel
% 0.27/0.49 thf(fact_106_mult__left__cancel,axiom,
% 0.27/0.49 ! [C: real,A: real,B: real] :
% 0.27/0.49 ( ( C != zero_zero_real )
% 0.27/0.49 => ( ( ( times_times_real @ C @ A )
% 0.27/0.49 = ( times_times_real @ C @ B ) )
% 0.27/0.49 = ( A = B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_left_cancel
% 0.27/0.49 thf(fact_107_mult__right__cancel,axiom,
% 0.27/0.49 ! [C: complex,A: complex,B: complex] :
% 0.27/0.49 ( ( C != zero_zero_complex )
% 0.27/0.49 => ( ( ( times_times_complex @ A @ C )
% 0.27/0.49 = ( times_times_complex @ B @ C ) )
% 0.27/0.49 = ( A = B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_right_cancel
% 0.27/0.49 thf(fact_108_mult__right__cancel,axiom,
% 0.27/0.49 ! [C: nat,A: nat,B: nat] :
% 0.27/0.49 ( ( C != zero_zero_nat )
% 0.27/0.49 => ( ( ( times_times_nat @ A @ C )
% 0.27/0.49 = ( times_times_nat @ B @ C ) )
% 0.27/0.49 = ( A = B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_right_cancel
% 0.27/0.49 thf(fact_109_mult__right__cancel,axiom,
% 0.27/0.49 ! [C: real,A: real,B: real] :
% 0.27/0.49 ( ( C != zero_zero_real )
% 0.27/0.49 => ( ( ( times_times_real @ A @ C )
% 0.27/0.49 = ( times_times_real @ B @ C ) )
% 0.27/0.49 = ( A = B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_right_cancel
% 0.27/0.49 thf(fact_110_comp__eq__id__dest,axiom,
% 0.27/0.49 ! [A: complex > complex,B: complex > complex,C: complex > complex,V: complex] :
% 0.27/0.49 ( ( ( comp_c130555887omplex @ A @ B )
% 0.27/0.49 = ( comp_c130555887omplex @ id_complex @ C ) )
% 0.27/0.49 => ( ( A @ ( B @ V ) )
% 0.27/0.49 = ( C @ V ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % comp_eq_id_dest
% 0.27/0.49 thf(fact_111_mult_Ocomm__neutral,axiom,
% 0.27/0.49 ! [A: complex] :
% 0.27/0.49 ( ( times_times_complex @ A @ one_one_complex )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % mult.comm_neutral
% 0.27/0.49 thf(fact_112_mult_Ocomm__neutral,axiom,
% 0.27/0.49 ! [A: nat] :
% 0.27/0.49 ( ( times_times_nat @ A @ one_one_nat )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % mult.comm_neutral
% 0.27/0.49 thf(fact_113_mult_Ocomm__neutral,axiom,
% 0.27/0.49 ! [A: real] :
% 0.27/0.49 ( ( times_times_real @ A @ one_one_real )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % mult.comm_neutral
% 0.27/0.49 thf(fact_114_comm__monoid__mult__class_Omult__1,axiom,
% 0.27/0.49 ! [A: complex] :
% 0.27/0.49 ( ( times_times_complex @ one_one_complex @ A )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % comm_monoid_mult_class.mult_1
% 0.27/0.49 thf(fact_115_comm__monoid__mult__class_Omult__1,axiom,
% 0.27/0.49 ! [A: nat] :
% 0.27/0.49 ( ( times_times_nat @ one_one_nat @ A )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % comm_monoid_mult_class.mult_1
% 0.27/0.49 thf(fact_116_comm__monoid__mult__class_Omult__1,axiom,
% 0.27/0.49 ! [A: real] :
% 0.27/0.49 ( ( times_times_real @ one_one_real @ A )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % comm_monoid_mult_class.mult_1
% 0.27/0.49 thf(fact_117_funpow__simps__right_I1_J,axiom,
% 0.27/0.49 ! [F: real > real] :
% 0.27/0.49 ( ( compow_real_real @ zero_zero_nat @ F )
% 0.27/0.49 = id_real ) ).
% 0.27/0.49
% 0.27/0.49 % funpow_simps_right(1)
% 0.27/0.49 thf(fact_118_funpow__simps__right_I1_J,axiom,
% 0.27/0.49 ! [F: ( complex > complex ) > complex > complex] :
% 0.27/0.49 ( ( compow1098280738omplex @ zero_zero_nat @ F )
% 0.27/0.49 = id_complex_complex ) ).
% 0.27/0.49
% 0.27/0.49 % funpow_simps_right(1)
% 0.27/0.49 thf(fact_119_funpow__simps__right_I1_J,axiom,
% 0.27/0.49 ! [F: ( real > real ) > real > real] :
% 0.27/0.49 ( ( compow1723822618l_real @ zero_zero_nat @ F )
% 0.27/0.49 = id_real_real ) ).
% 0.27/0.49
% 0.27/0.49 % funpow_simps_right(1)
% 0.27/0.49 thf(fact_120_funpow__simps__right_I1_J,axiom,
% 0.27/0.49 ! [F: complex > complex] :
% 0.27/0.49 ( ( compow1667379464omplex @ zero_zero_nat @ F )
% 0.27/0.49 = id_complex ) ).
% 0.27/0.49
% 0.27/0.49 % funpow_simps_right(1)
% 0.27/0.49 thf(fact_121_fun_Omap__id,axiom,
% 0.27/0.49 ! [T: complex > complex] :
% 0.27/0.49 ( ( comp_c130555887omplex @ id_complex @ T )
% 0.27/0.49 = T ) ).
% 0.27/0.49
% 0.27/0.49 % fun.map_id
% 0.27/0.49 thf(fact_122_vector__space__over__itself_Oscale__one,axiom,
% 0.27/0.49 ! [X2: complex] :
% 0.27/0.49 ( ( times_times_complex @ one_one_complex @ X2 )
% 0.27/0.49 = X2 ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_one
% 0.27/0.49 thf(fact_123_vector__space__over__itself_Oscale__one,axiom,
% 0.27/0.49 ! [X2: real] :
% 0.27/0.49 ( ( times_times_real @ one_one_real @ X2 )
% 0.27/0.49 = X2 ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_one
% 0.27/0.49 thf(fact_124_vector__space__over__itself_Oscale__eq__0__iff,axiom,
% 0.27/0.49 ! [A: complex,X2: complex] :
% 0.27/0.49 ( ( ( times_times_complex @ A @ X2 )
% 0.27/0.49 = zero_zero_complex )
% 0.27/0.49 = ( ( A = zero_zero_complex )
% 0.27/0.49 | ( X2 = zero_zero_complex ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_eq_0_iff
% 0.27/0.49 thf(fact_125_vector__space__over__itself_Oscale__eq__0__iff,axiom,
% 0.27/0.49 ! [A: real,X2: real] :
% 0.27/0.49 ( ( ( times_times_real @ A @ X2 )
% 0.27/0.49 = zero_zero_real )
% 0.27/0.49 = ( ( A = zero_zero_real )
% 0.27/0.49 | ( X2 = zero_zero_real ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_eq_0_iff
% 0.27/0.49 thf(fact_126_vector__space__over__itself_Oscale__zero__left,axiom,
% 0.27/0.49 ! [X2: complex] :
% 0.27/0.49 ( ( times_times_complex @ zero_zero_complex @ X2 )
% 0.27/0.49 = zero_zero_complex ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_zero_left
% 0.27/0.49 thf(fact_127_vector__space__over__itself_Oscale__zero__left,axiom,
% 0.27/0.49 ! [X2: real] :
% 0.27/0.49 ( ( times_times_real @ zero_zero_real @ X2 )
% 0.27/0.49 = zero_zero_real ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_zero_left
% 0.27/0.49 thf(fact_128_vector__space__over__itself_Oscale__zero__right,axiom,
% 0.27/0.49 ! [A: complex] :
% 0.27/0.49 ( ( times_times_complex @ A @ zero_zero_complex )
% 0.27/0.49 = zero_zero_complex ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_zero_right
% 0.27/0.49 thf(fact_129_vector__space__over__itself_Oscale__zero__right,axiom,
% 0.27/0.49 ! [A: real] :
% 0.27/0.49 ( ( times_times_real @ A @ zero_zero_real )
% 0.27/0.49 = zero_zero_real ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_zero_right
% 0.27/0.49 thf(fact_130_vector__space__over__itself_Oscale__cancel__left,axiom,
% 0.27/0.49 ! [A: complex,X2: complex,Y: complex] :
% 0.27/0.49 ( ( ( times_times_complex @ A @ X2 )
% 0.27/0.49 = ( times_times_complex @ A @ Y ) )
% 0.27/0.49 = ( ( X2 = Y )
% 0.27/0.49 | ( A = zero_zero_complex ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_cancel_left
% 0.27/0.49 thf(fact_131_vector__space__over__itself_Oscale__cancel__left,axiom,
% 0.27/0.49 ! [A: real,X2: real,Y: real] :
% 0.27/0.49 ( ( ( times_times_real @ A @ X2 )
% 0.27/0.49 = ( times_times_real @ A @ Y ) )
% 0.27/0.49 = ( ( X2 = Y )
% 0.27/0.49 | ( A = zero_zero_real ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_cancel_left
% 0.27/0.49 thf(fact_132_vector__space__over__itself_Oscale__cancel__right,axiom,
% 0.27/0.49 ! [A: complex,X2: complex,B: complex] :
% 0.27/0.49 ( ( ( times_times_complex @ A @ X2 )
% 0.27/0.49 = ( times_times_complex @ B @ X2 ) )
% 0.27/0.49 = ( ( A = B )
% 0.27/0.49 | ( X2 = zero_zero_complex ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_cancel_right
% 0.27/0.49 thf(fact_133_vector__space__over__itself_Oscale__cancel__right,axiom,
% 0.27/0.49 ! [A: real,X2: real,B: real] :
% 0.27/0.49 ( ( ( times_times_real @ A @ X2 )
% 0.27/0.49 = ( times_times_real @ B @ X2 ) )
% 0.27/0.49 = ( ( A = B )
% 0.27/0.49 | ( X2 = zero_zero_real ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_cancel_right
% 0.27/0.49 thf(fact_134_mult__if__delta,axiom,
% 0.27/0.49 ! [P: $o,Q: complex] :
% 0.27/0.49 ( ( P
% 0.27/0.49 => ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q )
% 0.27/0.49 = Q ) )
% 0.27/0.49 & ( ~ P
% 0.27/0.49 => ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q )
% 0.27/0.49 = zero_zero_complex ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_if_delta
% 0.27/0.49 thf(fact_135_mult__if__delta,axiom,
% 0.27/0.49 ! [P: $o,Q: nat] :
% 0.27/0.49 ( ( P
% 0.27/0.49 => ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
% 0.27/0.49 = Q ) )
% 0.27/0.49 & ( ~ P
% 0.27/0.49 => ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
% 0.27/0.49 = zero_zero_nat ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_if_delta
% 0.27/0.49 thf(fact_136_mult__if__delta,axiom,
% 0.27/0.49 ! [P: $o,Q: real] :
% 0.27/0.49 ( ( P
% 0.27/0.49 => ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
% 0.27/0.49 = Q ) )
% 0.27/0.49 & ( ~ P
% 0.27/0.49 => ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
% 0.27/0.49 = zero_zero_real ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_if_delta
% 0.27/0.49 thf(fact_137_set__times__intro,axiom,
% 0.27/0.49 ! [A: complex,C2: set_complex,B: complex,D2: set_complex] :
% 0.27/0.49 ( ( member_complex @ A @ C2 )
% 0.27/0.49 => ( ( member_complex @ B @ D2 )
% 0.27/0.49 => ( member_complex @ ( times_times_complex @ A @ B ) @ ( times_1316095593omplex @ C2 @ D2 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % set_times_intro
% 0.27/0.49 thf(fact_138_set__times__intro,axiom,
% 0.27/0.49 ! [A: nat,C2: set_nat,B: nat,D2: set_nat] :
% 0.27/0.49 ( ( member_nat @ A @ C2 )
% 0.27/0.49 => ( ( member_nat @ B @ D2 )
% 0.27/0.49 => ( member_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C2 @ D2 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % set_times_intro
% 0.27/0.49 thf(fact_139_set__times__intro,axiom,
% 0.27/0.49 ! [A: real,C2: set_real,B: real,D2: set_real] :
% 0.27/0.49 ( ( member_real @ A @ C2 )
% 0.27/0.49 => ( ( member_real @ B @ D2 )
% 0.27/0.49 => ( member_real @ ( times_times_real @ A @ B ) @ ( times_times_set_real @ C2 @ D2 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % set_times_intro
% 0.27/0.49 thf(fact_140_funpow__code__def,axiom,
% 0.27/0.49 funpow1854104714omplex = compow1098280738omplex ).
% 0.27/0.49
% 0.27/0.49 % funpow_code_def
% 0.27/0.49 thf(fact_141_funpow__code__def,axiom,
% 0.27/0.49 funpow_real_real = compow1723822618l_real ).
% 0.27/0.49
% 0.27/0.49 % funpow_code_def
% 0.27/0.49 thf(fact_142_funpow__code__def,axiom,
% 0.27/0.49 funpow_complex = compow1667379464omplex ).
% 0.27/0.49
% 0.27/0.49 % funpow_code_def
% 0.27/0.49 thf(fact_143_assms_I4_J,axiom,
% 0.27/0.49 topolo935673511omplex @ t ).
% 0.27/0.49
% 0.27/0.49 % assms(4)
% 0.27/0.49 thf(fact_144_assms_I2_J,axiom,
% 0.27/0.49 comple372758642hic_on @ g @ t ).
% 0.27/0.49
% 0.27/0.49 % assms(2)
% 0.27/0.49 thf(fact_145_mult__cancel2,axiom,
% 0.27/0.49 ! [M: nat,K: nat,N: nat] :
% 0.27/0.49 ( ( ( times_times_nat @ M @ K )
% 0.27/0.49 = ( times_times_nat @ N @ K ) )
% 0.27/0.49 = ( ( M = N )
% 0.27/0.49 | ( K = zero_zero_nat ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_cancel2
% 0.27/0.49 thf(fact_146_mult__cancel1,axiom,
% 0.27/0.49 ! [K: nat,M: nat,N: nat] :
% 0.27/0.49 ( ( ( times_times_nat @ K @ M )
% 0.27/0.49 = ( times_times_nat @ K @ N ) )
% 0.27/0.49 = ( ( M = N )
% 0.27/0.49 | ( K = zero_zero_nat ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_cancel1
% 0.27/0.49 thf(fact_147_mult__0__right,axiom,
% 0.27/0.49 ! [M: nat] :
% 0.27/0.49 ( ( times_times_nat @ M @ zero_zero_nat )
% 0.27/0.49 = zero_zero_nat ) ).
% 0.27/0.49
% 0.27/0.49 % mult_0_right
% 0.27/0.49 thf(fact_148_mult__is__0,axiom,
% 0.27/0.49 ! [M: nat,N: nat] :
% 0.27/0.49 ( ( ( times_times_nat @ M @ N )
% 0.27/0.49 = zero_zero_nat )
% 0.27/0.49 = ( ( M = zero_zero_nat )
% 0.27/0.49 | ( N = zero_zero_nat ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_is_0
% 0.27/0.49 thf(fact_149_nat__mult__eq__1__iff,axiom,
% 0.27/0.49 ! [M: nat,N: nat] :
% 0.27/0.49 ( ( ( times_times_nat @ M @ N )
% 0.27/0.49 = one_one_nat )
% 0.27/0.49 = ( ( M = one_one_nat )
% 0.27/0.49 & ( N = one_one_nat ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % nat_mult_eq_1_iff
% 0.27/0.49 thf(fact_150_nat__1__eq__mult__iff,axiom,
% 0.27/0.49 ! [M: nat,N: nat] :
% 0.27/0.49 ( ( one_one_nat
% 0.27/0.49 = ( times_times_nat @ M @ N ) )
% 0.27/0.49 = ( ( M = one_one_nat )
% 0.27/0.49 & ( N = one_one_nat ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % nat_1_eq_mult_iff
% 0.27/0.49 thf(fact_151_higher__deriv__transform__within__open,axiom,
% 0.27/0.49 ! [F: complex > complex,S: set_complex,G2: complex > complex,Z: complex,I: nat] :
% 0.27/0.49 ( ( comple372758642hic_on @ F @ S )
% 0.27/0.49 => ( ( comple372758642hic_on @ G2 @ S )
% 0.27/0.49 => ( ( topolo935673511omplex @ S )
% 0.27/0.49 => ( ( member_complex @ Z @ S )
% 0.27/0.49 => ( ! [W: complex] :
% 0.27/0.49 ( ( member_complex @ W @ S )
% 0.27/0.49 => ( ( F @ W )
% 0.27/0.49 = ( G2 @ W ) ) )
% 0.27/0.49 => ( ( compow1098280738omplex @ I @ deriv_complex @ F @ Z )
% 0.27/0.49 = ( compow1098280738omplex @ I @ deriv_complex @ G2 @ Z ) ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % higher_deriv_transform_within_open
% 0.27/0.49 thf(fact_152_holomorphic__higher__deriv,axiom,
% 0.27/0.49 ! [F: complex > complex,S: set_complex,N: nat] :
% 0.27/0.49 ( ( comple372758642hic_on @ F @ S )
% 0.27/0.49 => ( ( topolo935673511omplex @ S )
% 0.27/0.49 => ( comple372758642hic_on @ ( compow1098280738omplex @ N @ deriv_complex @ F ) @ S ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % holomorphic_higher_deriv
% 0.27/0.49 thf(fact_153_holomorphic__deriv,axiom,
% 0.27/0.49 ! [F: complex > complex,S: set_complex] :
% 0.27/0.49 ( ( comple372758642hic_on @ F @ S )
% 0.27/0.49 => ( ( topolo935673511omplex @ S )
% 0.27/0.49 => ( comple372758642hic_on @ ( deriv_complex @ F ) @ S ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % holomorphic_deriv
% 0.27/0.49 thf(fact_154_mult__0,axiom,
% 0.27/0.49 ! [N: nat] :
% 0.27/0.49 ( ( times_times_nat @ zero_zero_nat @ N )
% 0.27/0.49 = zero_zero_nat ) ).
% 0.27/0.49
% 0.27/0.49 % mult_0
% 0.27/0.49 thf(fact_155_nat__mult__1__right,axiom,
% 0.27/0.49 ! [N: nat] :
% 0.27/0.49 ( ( times_times_nat @ N @ one_one_nat )
% 0.27/0.49 = N ) ).
% 0.27/0.49
% 0.27/0.49 % nat_mult_1_right
% 0.27/0.49 thf(fact_156_nat__mult__1,axiom,
% 0.27/0.49 ! [N: nat] :
% 0.27/0.49 ( ( times_times_nat @ one_one_nat @ N )
% 0.27/0.49 = N ) ).
% 0.27/0.49
% 0.27/0.49 % nat_mult_1
% 0.27/0.49 thf(fact_157_funpow__mult,axiom,
% 0.27/0.49 ! [N: nat,M: nat,F: ( complex > complex ) > complex > complex] :
% 0.27/0.49 ( ( compow1098280738omplex @ N @ ( compow1098280738omplex @ M @ F ) )
% 0.27/0.49 = ( compow1098280738omplex @ ( times_times_nat @ M @ N ) @ F ) ) ).
% 0.27/0.49
% 0.27/0.49 % funpow_mult
% 0.27/0.49 thf(fact_158_funpow__mult,axiom,
% 0.27/0.49 ! [N: nat,M: nat,F: ( real > real ) > real > real] :
% 0.27/0.49 ( ( compow1723822618l_real @ N @ ( compow1723822618l_real @ M @ F ) )
% 0.27/0.49 = ( compow1723822618l_real @ ( times_times_nat @ M @ N ) @ F ) ) ).
% 0.27/0.49
% 0.27/0.49 % funpow_mult
% 0.27/0.49 thf(fact_159_funpow__mult,axiom,
% 0.27/0.49 ! [N: nat,M: nat,F: complex > complex] :
% 0.27/0.49 ( ( compow1667379464omplex @ N @ ( compow1667379464omplex @ M @ F ) )
% 0.27/0.49 = ( compow1667379464omplex @ ( times_times_nat @ M @ N ) @ F ) ) ).
% 0.27/0.49
% 0.27/0.49 % funpow_mult
% 0.27/0.49 thf(fact_160_mult__eq__self__implies__10,axiom,
% 0.27/0.49 ! [M: nat,N: nat] :
% 0.27/0.49 ( ( M
% 0.27/0.49 = ( times_times_nat @ M @ N ) )
% 0.27/0.49 => ( ( N = one_one_nat )
% 0.27/0.49 | ( M = zero_zero_nat ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mult_eq_self_implies_10
% 0.27/0.49 thf(fact_161_vector__space__over__itself_Oscale__left__commute,axiom,
% 0.27/0.49 ! [A: complex,B: complex,X2: complex] :
% 0.27/0.49 ( ( times_times_complex @ A @ ( times_times_complex @ B @ X2 ) )
% 0.27/0.49 = ( times_times_complex @ B @ ( times_times_complex @ A @ X2 ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_left_commute
% 0.27/0.49 thf(fact_162_vector__space__over__itself_Oscale__left__commute,axiom,
% 0.27/0.49 ! [A: real,B: real,X2: real] :
% 0.27/0.49 ( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
% 0.27/0.49 = ( times_times_real @ B @ ( times_times_real @ A @ X2 ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_left_commute
% 0.27/0.49 thf(fact_163_vector__space__over__itself_Oscale__scale,axiom,
% 0.27/0.49 ! [A: complex,B: complex,X2: complex] :
% 0.27/0.49 ( ( times_times_complex @ A @ ( times_times_complex @ B @ X2 ) )
% 0.27/0.49 = ( times_times_complex @ ( times_times_complex @ A @ B ) @ X2 ) ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_scale
% 0.27/0.49 thf(fact_164_vector__space__over__itself_Oscale__scale,axiom,
% 0.27/0.49 ! [A: real,B: real,X2: real] :
% 0.27/0.49 ( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
% 0.27/0.49 = ( times_times_real @ ( times_times_real @ A @ B ) @ X2 ) ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_scale
% 0.27/0.49 thf(fact_165_set__times__elim,axiom,
% 0.27/0.49 ! [X2: complex,A2: set_complex,B3: set_complex] :
% 0.27/0.49 ( ( member_complex @ X2 @ ( times_1316095593omplex @ A2 @ B3 ) )
% 0.27/0.49 => ~ ! [A4: complex,B4: complex] :
% 0.27/0.49 ( ( X2
% 0.27/0.49 = ( times_times_complex @ A4 @ B4 ) )
% 0.27/0.49 => ( ( member_complex @ A4 @ A2 )
% 0.27/0.49 => ~ ( member_complex @ B4 @ B3 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % set_times_elim
% 0.27/0.49 thf(fact_166_set__times__elim,axiom,
% 0.27/0.49 ! [X2: nat,A2: set_nat,B3: set_nat] :
% 0.27/0.49 ( ( member_nat @ X2 @ ( times_times_set_nat @ A2 @ B3 ) )
% 0.27/0.49 => ~ ! [A4: nat,B4: nat] :
% 0.27/0.49 ( ( X2
% 0.27/0.49 = ( times_times_nat @ A4 @ B4 ) )
% 0.27/0.49 => ( ( member_nat @ A4 @ A2 )
% 0.27/0.49 => ~ ( member_nat @ B4 @ B3 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % set_times_elim
% 0.27/0.49 thf(fact_167_set__times__elim,axiom,
% 0.27/0.49 ! [X2: real,A2: set_real,B3: set_real] :
% 0.27/0.49 ( ( member_real @ X2 @ ( times_times_set_real @ A2 @ B3 ) )
% 0.27/0.49 => ~ ! [A4: real,B4: real] :
% 0.27/0.49 ( ( X2
% 0.27/0.49 = ( times_times_real @ A4 @ B4 ) )
% 0.27/0.49 => ( ( member_real @ A4 @ A2 )
% 0.27/0.49 => ~ ( member_real @ B4 @ B3 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % set_times_elim
% 0.27/0.49 thf(fact_168_fun_Omap__comp,axiom,
% 0.27/0.49 ! [G2: complex > complex,F: complex > complex,V: complex > complex] :
% 0.27/0.49 ( ( comp_c130555887omplex @ G2 @ ( comp_c130555887omplex @ F @ V ) )
% 0.27/0.49 = ( comp_c130555887omplex @ ( comp_c130555887omplex @ G2 @ F ) @ V ) ) ).
% 0.27/0.49
% 0.27/0.49 % fun.map_comp
% 0.27/0.49 thf(fact_169_vector__space__over__itself_Oscale__right__imp__eq,axiom,
% 0.27/0.49 ! [X2: complex,A: complex,B: complex] :
% 0.27/0.49 ( ( X2 != zero_zero_complex )
% 0.27/0.49 => ( ( ( times_times_complex @ A @ X2 )
% 0.27/0.49 = ( times_times_complex @ B @ X2 ) )
% 0.27/0.49 => ( A = B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_right_imp_eq
% 0.27/0.49 thf(fact_170_vector__space__over__itself_Oscale__right__imp__eq,axiom,
% 0.27/0.49 ! [X2: real,A: real,B: real] :
% 0.27/0.49 ( ( X2 != zero_zero_real )
% 0.27/0.49 => ( ( ( times_times_real @ A @ X2 )
% 0.27/0.49 = ( times_times_real @ B @ X2 ) )
% 0.27/0.49 => ( A = B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_right_imp_eq
% 0.27/0.49 thf(fact_171_vector__space__over__itself_Oscale__left__imp__eq,axiom,
% 0.27/0.49 ! [A: complex,X2: complex,Y: complex] :
% 0.27/0.49 ( ( A != zero_zero_complex )
% 0.27/0.49 => ( ( ( times_times_complex @ A @ X2 )
% 0.27/0.49 = ( times_times_complex @ A @ Y ) )
% 0.27/0.49 => ( X2 = Y ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % vector_space_over_itself.scale_left_imp_eq
% 0.27/0.49 thf(fact_172_vector__space__over__itself_Oscale__left__imp__eq,axiom,
% 0.27/0.49 ! [A: real,X2: real,Y: real] :
% 0.27/0.49 ( ( A != zero_zero_real )
% 0.27/0.49 => ( ( ( times_times_real @ A @ X2 )
% 0.27/0.49 = ( times_times_real @ A @ Y ) )
% 0.27/0.49 => ( X2 = Y ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % vector_space_over_itself.scale_left_imp_eq
% 0.27/0.50 thf(fact_173_fun_Omap__id0,axiom,
% 0.27/0.50 ( ( comp_c130555887omplex @ id_complex )
% 0.27/0.50 = id_complex_complex ) ).
% 0.27/0.50
% 0.27/0.50 % fun.map_id0
% 0.27/0.50 thf(fact_174_complex__derivative__transform__within__open,axiom,
% 0.27/0.50 ! [F: complex > complex,S2: set_complex,G2: complex > complex,Z: complex] :
% 0.27/0.50 ( ( comple372758642hic_on @ F @ S2 )
% 0.27/0.50 => ( ( comple372758642hic_on @ G2 @ S2 )
% 0.27/0.50 => ( ( topolo935673511omplex @ S2 )
% 0.27/0.50 => ( ( member_complex @ Z @ S2 )
% 0.27/0.50 => ( ! [W: complex] :
% 0.27/0.50 ( ( member_complex @ W @ S2 )
% 0.27/0.50 => ( ( F @ W )
% 0.27/0.50 = ( G2 @ W ) ) )
% 0.27/0.50 => ( ( deriv_complex @ F @ Z )
% 0.27/0.50 = ( deriv_complex @ G2 @ Z ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % complex_derivative_transform_within_open
% 0.27/0.50 thf(fact_175_holomorphic__on__id,axiom,
% 0.27/0.50 ! [S2: set_complex] : ( comple372758642hic_on @ id_complex @ S2 ) ).
% 0.27/0.50
% 0.27/0.50 % holomorphic_on_id
% 0.27/0.50 thf(fact_176_holomorphic__on__linear,axiom,
% 0.27/0.50 ! [C: complex,S2: set_complex] : ( comple372758642hic_on @ ( times_times_complex @ C ) @ S2 ) ).
% 0.27/0.50
% 0.27/0.50 % holomorphic_on_linear
% 0.27/0.50 thf(fact_177_pointfree__idE,axiom,
% 0.27/0.50 ! [F: complex > complex,G2: complex > complex,X2: complex] :
% 0.27/0.50 ( ( ( comp_c130555887omplex @ F @ G2 )
% 0.27/0.50 = id_complex )
% 0.27/0.50 => ( ( F @ ( G2 @ X2 ) )
% 0.27/0.50 = X2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % pointfree_idE
% 0.27/0.50 thf(fact_178_isomorphism__expand,axiom,
% 0.27/0.50 ! [F: real > complex,G2: complex > real] :
% 0.27/0.50 ( ( ( ( comp_r667767405omplex @ F @ G2 )
% 0.27/0.50 = id_complex )
% 0.27/0.50 & ( ( comp_c819638635l_real @ G2 @ F )
% 0.27/0.50 = id_real ) )
% 0.27/0.50 = ( ! [X: complex] :
% 0.27/0.50 ( ( F @ ( G2 @ X ) )
% 0.27/0.50 = X )
% 0.27/0.50 & ! [X: real] :
% 0.27/0.50 ( ( G2 @ ( F @ X ) )
% 0.27/0.50 = X ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % isomorphism_expand
% 0.27/0.50 thf(fact_179_isomorphism__expand,axiom,
% 0.27/0.50 ! [F: ( complex > complex ) > complex,G2: complex > complex > complex] :
% 0.27/0.50 ( ( ( ( comp_c881053372omplex @ F @ G2 )
% 0.27/0.50 = id_complex )
% 0.27/0.50 & ( ( comp_c606622857omplex @ G2 @ F )
% 0.27/0.50 = id_complex_complex ) )
% 0.27/0.50 = ( ! [X: complex] :
% 0.27/0.50 ( ( F @ ( G2 @ X ) )
% 0.27/0.50 = X )
% 0.27/0.50 & ! [X: complex > complex] :
% 0.27/0.50 ( ( G2 @ ( F @ X ) )
% 0.27/0.50 = X ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % isomorphism_expand
% 0.27/0.50 thf(fact_180_isomorphism__expand,axiom,
% 0.27/0.50 ! [F: complex > real,G2: real > complex] :
% 0.27/0.50 ( ( ( ( comp_c819638635l_real @ F @ G2 )
% 0.27/0.50 = id_real )
% 0.27/0.50 & ( ( comp_r667767405omplex @ G2 @ F )
% 0.27/0.50 = id_complex ) )
% 0.27/0.50 = ( ! [X: real] :
% 0.27/0.50 ( ( F @ ( G2 @ X ) )
% 0.27/0.50 = X )
% 0.27/0.50 & ! [X: complex] :
% 0.27/0.50 ( ( G2 @ ( F @ X ) )
% 0.27/0.50 = X ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % isomorphism_expand
% 0.27/0.50 thf(fact_181_isomorphism__expand,axiom,
% 0.27/0.50 ! [F: real > real,G2: real > real] :
% 0.27/0.50 ( ( ( ( comp_real_real_real @ F @ G2 )
% 0.27/0.50 = id_real )
% 0.27/0.50 & ( ( comp_real_real_real @ G2 @ F )
% 0.27/0.50 = id_real ) )
% 0.27/0.50 = ( ! [X: real] :
% 0.27/0.50 ( ( F @ ( G2 @ X ) )
% 0.27/0.50 = X )
% 0.27/0.50 & ! [X: real] :
% 0.27/0.50 ( ( G2 @ ( F @ X ) )
% 0.27/0.50 = X ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % isomorphism_expand
% 0.27/0.50 thf(fact_182_isomorphism__expand,axiom,
% 0.27/0.50 ! [F: ( complex > complex ) > real,G2: real > complex > complex] :
% 0.27/0.50 ( ( ( ( comp_c1884694328l_real @ F @ G2 )
% 0.27/0.50 = id_real )
% 0.27/0.50 & ( ( comp_r2009261319omplex @ G2 @ F )
% 0.27/0.50 = id_complex_complex ) )
% 0.27/0.50 = ( ! [X: real] :
% 0.27/0.50 ( ( F @ ( G2 @ X ) )
% 0.27/0.50 = X )
% 0.27/0.50 & ! [X: complex > complex] :
% 0.27/0.50 ( ( G2 @ ( F @ X ) )
% 0.27/0.50 = X ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % isomorphism_expand
% 0.27/0.50 thf(fact_183_isomorphism__expand,axiom,
% 0.27/0.50 ! [F: complex > complex > complex,G2: ( complex > complex ) > complex] :
% 0.27/0.50 ( ( ( ( comp_c606622857omplex @ F @ G2 )
% 0.27/0.50 = id_complex_complex )
% 0.27/0.50 & ( ( comp_c881053372omplex @ G2 @ F )
% 0.27/0.50 = id_complex ) )
% 0.27/0.50 = ( ! [X: complex > complex] :
% 0.27/0.50 ( ( F @ ( G2 @ X ) )
% 0.27/0.50 = X )
% 0.27/0.50 & ! [X: complex] :
% 0.27/0.50 ( ( G2 @ ( F @ X ) )
% 0.27/0.50 = X ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % isomorphism_expand
% 0.27/0.50 thf(fact_184_isomorphism__expand,axiom,
% 0.27/0.50 ! [F: real > complex > complex,G2: ( complex > complex ) > real] :
% 0.27/0.50 ( ( ( ( comp_r2009261319omplex @ F @ G2 )
% 0.27/0.50 = id_complex_complex )
% 0.27/0.50 & ( ( comp_c1884694328l_real @ G2 @ F )
% 0.27/0.50 = id_real ) )
% 0.27/0.50 = ( ! [X: complex > complex] :
% 0.27/0.50 ( ( F @ ( G2 @ X ) )
% 0.27/0.50 = X )
% 0.27/0.50 & ! [X: real] :
% 0.27/0.50 ( ( G2 @ ( F @ X ) )
% 0.27/0.50 = X ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % isomorphism_expand
% 0.27/0.50 thf(fact_185_isomorphism__expand,axiom,
% 0.27/0.50 ! [F: ( complex > complex ) > complex > complex,G2: ( complex > complex ) > complex > complex] :
% 0.27/0.50 ( ( ( ( comp_c1610621014omplex @ F @ G2 )
% 0.27/0.50 = id_complex_complex )
% 0.27/0.50 & ( ( comp_c1610621014omplex @ G2 @ F )
% 0.27/0.50 = id_complex_complex ) )
% 0.27/0.50 = ( ! [X: complex > complex] :
% 0.27/0.50 ( ( F @ ( G2 @ X ) )
% 0.27/0.50 = X )
% 0.27/0.50 & ! [X: complex > complex] :
% 0.27/0.50 ( ( G2 @ ( F @ X ) )
% 0.27/0.50 = X ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % isomorphism_expand
% 0.27/0.50 thf(fact_186_isomorphism__expand,axiom,
% 0.27/0.50 ! [F: complex > complex,G2: complex > complex] :
% 0.27/0.50 ( ( ( ( comp_c130555887omplex @ F @ G2 )
% 0.27/0.50 = id_complex )
% 0.27/0.50 & ( ( comp_c130555887omplex @ G2 @ F )
% 0.27/0.50 = id_complex ) )
% 0.27/0.50 = ( ! [X: complex] :
% 0.27/0.50 ( ( F @ ( G2 @ X ) )
% 0.27/0.50 = X )
% 0.27/0.50 & ! [X: complex] :
% 0.27/0.50 ( ( G2 @ ( F @ X ) )
% 0.27/0.50 = X ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % isomorphism_expand
% 0.27/0.50 thf(fact_187_left__right__inverse__eq,axiom,
% 0.27/0.50 ! [F: real > complex,G2: complex > real,H: real > complex] :
% 0.27/0.50 ( ( ( comp_r667767405omplex @ F @ G2 )
% 0.27/0.50 = id_complex )
% 0.27/0.50 => ( ( ( comp_c819638635l_real @ G2 @ H )
% 0.27/0.50 = id_real )
% 0.27/0.50 => ( F = H ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % left_right_inverse_eq
% 0.27/0.50 thf(fact_188_left__right__inverse__eq,axiom,
% 0.27/0.50 ! [F: ( complex > complex ) > complex,G2: complex > complex > complex,H: ( complex > complex ) > complex] :
% 0.27/0.50 ( ( ( comp_c881053372omplex @ F @ G2 )
% 0.27/0.50 = id_complex )
% 0.27/0.50 => ( ( ( comp_c606622857omplex @ G2 @ H )
% 0.27/0.50 = id_complex_complex )
% 0.27/0.50 => ( F = H ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % left_right_inverse_eq
% 0.27/0.50 thf(fact_189_left__right__inverse__eq,axiom,
% 0.27/0.50 ! [F: complex > real,G2: real > complex,H: complex > real] :
% 0.27/0.50 ( ( ( comp_c819638635l_real @ F @ G2 )
% 0.27/0.50 = id_real )
% 0.27/0.50 => ( ( ( comp_r667767405omplex @ G2 @ H )
% 0.27/0.50 = id_complex )
% 0.27/0.50 => ( F = H ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % left_right_inverse_eq
% 0.27/0.50 thf(fact_190_left__right__inverse__eq,axiom,
% 0.27/0.50 ! [F: real > real,G2: real > real,H: real > real] :
% 0.27/0.50 ( ( ( comp_real_real_real @ F @ G2 )
% 0.27/0.50 = id_real )
% 0.27/0.50 => ( ( ( comp_real_real_real @ G2 @ H )
% 0.27/0.50 = id_real )
% 0.27/0.50 => ( F = H ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % left_right_inverse_eq
% 0.27/0.50 thf(fact_191_left__right__inverse__eq,axiom,
% 0.27/0.50 ! [F: ( complex > complex ) > real,G2: real > complex > complex,H: ( complex > complex ) > real] :
% 0.27/0.50 ( ( ( comp_c1884694328l_real @ F @ G2 )
% 0.27/0.50 = id_real )
% 0.27/0.50 => ( ( ( comp_r2009261319omplex @ G2 @ H )
% 0.27/0.50 = id_complex_complex )
% 0.27/0.50 => ( F = H ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % left_right_inverse_eq
% 0.27/0.50 thf(fact_192_left__right__inverse__eq,axiom,
% 0.27/0.50 ! [F: complex > complex > complex,G2: ( complex > complex ) > complex,H: complex > complex > complex] :
% 0.27/0.50 ( ( ( comp_c606622857omplex @ F @ G2 )
% 0.27/0.50 = id_complex_complex )
% 0.27/0.50 => ( ( ( comp_c881053372omplex @ G2 @ H )
% 0.27/0.50 = id_complex )
% 0.27/0.50 => ( F = H ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % left_right_inverse_eq
% 0.27/0.50 thf(fact_193_left__right__inverse__eq,axiom,
% 0.27/0.50 ! [F: real > complex > complex,G2: ( complex > complex ) > real,H: real > complex > complex] :
% 0.27/0.50 ( ( ( comp_r2009261319omplex @ F @ G2 )
% 0.27/0.50 = id_complex_complex )
% 0.27/0.50 => ( ( ( comp_c1884694328l_real @ G2 @ H )
% 0.27/0.50 = id_real )
% 0.27/0.50 => ( F = H ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % left_right_inverse_eq
% 0.27/0.50 thf(fact_194_left__right__inverse__eq,axiom,
% 0.27/0.50 ! [F: ( complex > complex ) > complex > complex,G2: ( complex > complex ) > complex > complex,H: ( complex > complex ) > complex > complex] :
% 0.27/0.50 ( ( ( comp_c1610621014omplex @ F @ G2 )
% 0.27/0.50 = id_complex_complex )
% 0.27/0.50 => ( ( ( comp_c1610621014omplex @ G2 @ H )
% 0.27/0.50 = id_complex_complex )
% 0.27/0.50 => ( F = H ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % left_right_inverse_eq
% 0.27/0.50 thf(fact_195_left__right__inverse__eq,axiom,
% 0.27/0.50 ! [F: complex > complex,G2: complex > complex,H: complex > complex] :
% 0.27/0.50 ( ( ( comp_c130555887omplex @ F @ G2 )
% 0.27/0.50 = id_complex )
% 0.27/0.50 => ( ( ( comp_c130555887omplex @ G2 @ H )
% 0.27/0.50 = id_complex )
% 0.27/0.50 => ( F = H ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % left_right_inverse_eq
% 0.27/0.50 thf(fact_196_rewriteL__comp__comp,axiom,
% 0.27/0.50 ! [F: complex > complex,G2: complex > complex,L: complex > complex,H: complex > complex] :
% 0.27/0.50 ( ( ( comp_c130555887omplex @ F @ G2 )
% 0.27/0.50 = L )
% 0.27/0.50 => ( ( comp_c130555887omplex @ F @ ( comp_c130555887omplex @ G2 @ H ) )
% 0.27/0.50 = ( comp_c130555887omplex @ L @ H ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % rewriteL_comp_comp
% 0.27/0.50 thf(fact_197_rewriteR__comp__comp,axiom,
% 0.27/0.50 ! [G2: complex > complex,H: complex > complex,R: complex > complex,F: complex > complex] :
% 0.27/0.50 ( ( ( comp_c130555887omplex @ G2 @ H )
% 0.27/0.50 = R )
% 0.27/0.50 => ( ( comp_c130555887omplex @ ( comp_c130555887omplex @ F @ G2 ) @ H )
% 0.27/0.50 = ( comp_c130555887omplex @ F @ R ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % rewriteR_comp_comp
% 0.27/0.50 thf(fact_198_rewriteL__comp__comp2,axiom,
% 0.27/0.50 ! [F: complex > complex,G2: complex > complex,L1: complex > complex,L2: complex > complex,H: complex > complex,R: complex > complex] :
% 0.27/0.50 ( ( ( comp_c130555887omplex @ F @ G2 )
% 0.27/0.50 = ( comp_c130555887omplex @ L1 @ L2 ) )
% 0.27/0.50 => ( ( ( comp_c130555887omplex @ L2 @ H )
% 0.27/0.50 = R )
% 0.27/0.50 => ( ( comp_c130555887omplex @ F @ ( comp_c130555887omplex @ G2 @ H ) )
% 0.27/0.50 = ( comp_c130555887omplex @ L1 @ R ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % rewriteL_comp_comp2
% 0.27/0.50 thf(fact_199_rewriteR__comp__comp2,axiom,
% 0.27/0.50 ! [G2: complex > complex,H: complex > complex,R1: complex > complex,R2: complex > complex,F: complex > complex,L: complex > complex] :
% 0.27/0.50 ( ( ( comp_c130555887omplex @ G2 @ H )
% 0.27/0.50 = ( comp_c130555887omplex @ R1 @ R2 ) )
% 0.27/0.50 => ( ( ( comp_c130555887omplex @ F @ R1 )
% 0.27/0.50 = L )
% 0.27/0.50 => ( ( comp_c130555887omplex @ ( comp_c130555887omplex @ F @ G2 ) @ H )
% 0.27/0.50 = ( comp_c130555887omplex @ L @ R2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % rewriteR_comp_comp2
% 0.27/0.50 thf(fact_200_holomorphic__cong,axiom,
% 0.27/0.50 ! [S2: set_complex,T: set_complex,F: complex > complex,G2: complex > complex] :
% 0.27/0.50 ( ( S2 = T )
% 0.27/0.50 => ( ! [X3: complex] :
% 0.27/0.50 ( ( member_complex @ X3 @ S2 )
% 0.27/0.50 => ( ( F @ X3 )
% 0.27/0.50 = ( G2 @ X3 ) ) )
% 0.27/0.50 => ( ( comple372758642hic_on @ F @ S2 )
% 0.27/0.50 = ( comple372758642hic_on @ G2 @ T ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % holomorphic_cong
% 0.27/0.50 thf(fact_201_holomorphic__transform,axiom,
% 0.27/0.50 ! [F: complex > complex,S2: set_complex,G2: complex > complex] :
% 0.27/0.50 ( ( comple372758642hic_on @ F @ S2 )
% 0.27/0.50 => ( ! [X3: complex] :
% 0.27/0.50 ( ( member_complex @ X3 @ S2 )
% 0.27/0.50 => ( ( F @ X3 )
% 0.27/0.50 = ( G2 @ X3 ) ) )
% 0.27/0.50 => ( comple372758642hic_on @ G2 @ S2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % holomorphic_transform
% 0.27/0.50 thf(fact_202_assms_I5_J,axiom,
% 0.27/0.50 ord_le701908932omplex @ ( image_58037603omplex @ f @ s ) @ t ).
% 0.27/0.50
% 0.27/0.50 % assms(5)
% 0.27/0.50 thf(fact_203_nat__mult__eq__cancel__disj,axiom,
% 0.27/0.50 ! [K: nat,M: nat,N: nat] :
% 0.27/0.50 ( ( ( times_times_nat @ K @ M )
% 0.27/0.50 = ( times_times_nat @ K @ N ) )
% 0.27/0.50 = ( ( K = zero_zero_nat )
% 0.27/0.50 | ( M = N ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % nat_mult_eq_cancel_disj
% 0.27/0.50 thf(fact_204_mult__delta__left,axiom,
% 0.27/0.50 ! [B: $o,X2: complex,Y: complex] :
% 0.27/0.50 ( ( B
% 0.27/0.50 => ( ( times_times_complex @ ( if_complex @ B @ X2 @ zero_zero_complex ) @ Y )
% 0.27/0.50 = ( times_times_complex @ X2 @ Y ) ) )
% 0.27/0.50 & ( ~ B
% 0.27/0.50 => ( ( times_times_complex @ ( if_complex @ B @ X2 @ zero_zero_complex ) @ Y )
% 0.27/0.50 = zero_zero_complex ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_delta_left
% 0.27/0.50 thf(fact_205_mult__delta__left,axiom,
% 0.27/0.50 ! [B: $o,X2: nat,Y: nat] :
% 0.27/0.50 ( ( B
% 0.27/0.50 => ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y )
% 0.27/0.50 = ( times_times_nat @ X2 @ Y ) ) )
% 0.27/0.50 & ( ~ B
% 0.27/0.50 => ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y )
% 0.27/0.50 = zero_zero_nat ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_delta_left
% 0.27/0.50 thf(fact_206_mult__delta__left,axiom,
% 0.27/0.50 ! [B: $o,X2: real,Y: real] :
% 0.27/0.50 ( ( B
% 0.27/0.50 => ( ( times_times_real @ ( if_real @ B @ X2 @ zero_zero_real ) @ Y )
% 0.27/0.50 = ( times_times_real @ X2 @ Y ) ) )
% 0.27/0.50 & ( ~ B
% 0.27/0.50 => ( ( times_times_real @ ( if_real @ B @ X2 @ zero_zero_real ) @ Y )
% 0.27/0.50 = zero_zero_real ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_delta_left
% 0.27/0.50 thf(fact_207_mult__delta__right,axiom,
% 0.27/0.50 ! [B: $o,X2: complex,Y: complex] :
% 0.27/0.50 ( ( B
% 0.27/0.50 => ( ( times_times_complex @ X2 @ ( if_complex @ B @ Y @ zero_zero_complex ) )
% 0.27/0.50 = ( times_times_complex @ X2 @ Y ) ) )
% 0.27/0.50 & ( ~ B
% 0.27/0.50 => ( ( times_times_complex @ X2 @ ( if_complex @ B @ Y @ zero_zero_complex ) )
% 0.27/0.50 = zero_zero_complex ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_delta_right
% 0.27/0.50 thf(fact_208_mult__delta__right,axiom,
% 0.27/0.50 ! [B: $o,X2: nat,Y: nat] :
% 0.27/0.50 ( ( B
% 0.27/0.50 => ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y @ zero_zero_nat ) )
% 0.27/0.50 = ( times_times_nat @ X2 @ Y ) ) )
% 0.27/0.50 & ( ~ B
% 0.27/0.50 => ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y @ zero_zero_nat ) )
% 0.27/0.50 = zero_zero_nat ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_delta_right
% 0.27/0.50 thf(fact_209_mult__delta__right,axiom,
% 0.27/0.50 ! [B: $o,X2: real,Y: real] :
% 0.27/0.50 ( ( B
% 0.27/0.50 => ( ( times_times_real @ X2 @ ( if_real @ B @ Y @ zero_zero_real ) )
% 0.27/0.50 = ( times_times_real @ X2 @ Y ) ) )
% 0.27/0.50 & ( ~ B
% 0.27/0.50 => ( ( times_times_real @ X2 @ ( if_real @ B @ Y @ zero_zero_real ) )
% 0.27/0.50 = zero_zero_real ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_delta_right
% 0.27/0.50 thf(fact_210_arcosh__1,axiom,
% 0.27/0.50 ( ( arcosh_real @ one_one_real )
% 0.27/0.50 = zero_zero_real ) ).
% 0.27/0.50
% 0.27/0.50 % arcosh_1
% 0.27/0.50 thf(fact_211_arcosh__1,axiom,
% 0.27/0.50 ( ( arcosh_complex @ one_one_complex )
% 0.27/0.50 = zero_zero_complex ) ).
% 0.27/0.50
% 0.27/0.50 % arcosh_1
% 0.27/0.50 thf(fact_212_le__zero__eq,axiom,
% 0.27/0.50 ! [N: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.27/0.50 = ( N = zero_zero_nat ) ) ).
% 0.27/0.50
% 0.27/0.50 % le_zero_eq
% 0.27/0.50 thf(fact_213_image__id,axiom,
% 0.27/0.50 ( ( image_real_real @ id_real )
% 0.27/0.50 = id_set_real ) ).
% 0.27/0.50
% 0.27/0.50 % image_id
% 0.27/0.50 thf(fact_214_image__id,axiom,
% 0.27/0.50 ( ( image_944012797omplex @ id_complex_complex )
% 0.27/0.50 = id_set1618538368omplex ) ).
% 0.27/0.50
% 0.27/0.50 % image_id
% 0.27/0.50 thf(fact_215_image__id,axiom,
% 0.27/0.50 ( ( image_58037603omplex @ id_complex )
% 0.27/0.50 = id_set_complex ) ).
% 0.27/0.50
% 0.27/0.50 % image_id
% 0.27/0.50 thf(fact_216_set__times__mono2,axiom,
% 0.27/0.50 ! [C2: set_complex,D2: set_complex,E: set_complex,F3: set_complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ C2 @ D2 )
% 0.27/0.50 => ( ( ord_le701908932omplex @ E @ F3 )
% 0.27/0.50 => ( ord_le701908932omplex @ ( times_1316095593omplex @ C2 @ E ) @ ( times_1316095593omplex @ D2 @ F3 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % set_times_mono2
% 0.27/0.50 thf(fact_217_holomorphic__on__compose__gen,axiom,
% 0.27/0.50 ! [F: complex > complex,S2: set_complex,G2: complex > complex,T: set_complex] :
% 0.27/0.50 ( ( comple372758642hic_on @ F @ S2 )
% 0.27/0.50 => ( ( comple372758642hic_on @ G2 @ T )
% 0.27/0.50 => ( ( ord_le701908932omplex @ ( image_58037603omplex @ F @ S2 ) @ T )
% 0.27/0.50 => ( comple372758642hic_on @ ( comp_c130555887omplex @ G2 @ F ) @ S2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % holomorphic_on_compose_gen
% 0.27/0.50 thf(fact_218_image__comp,axiom,
% 0.27/0.50 ! [F: complex > complex,G2: complex > complex,R: set_complex] :
% 0.27/0.50 ( ( image_58037603omplex @ F @ ( image_58037603omplex @ G2 @ R ) )
% 0.27/0.50 = ( image_58037603omplex @ ( comp_c130555887omplex @ F @ G2 ) @ R ) ) ).
% 0.27/0.50
% 0.27/0.50 % image_comp
% 0.27/0.50 thf(fact_219_image__eq__imp__comp,axiom,
% 0.27/0.50 ! [F: complex > complex,A2: set_complex,G2: complex > complex,B3: set_complex,H: complex > complex] :
% 0.27/0.50 ( ( ( image_58037603omplex @ F @ A2 )
% 0.27/0.50 = ( image_58037603omplex @ G2 @ B3 ) )
% 0.27/0.50 => ( ( image_58037603omplex @ ( comp_c130555887omplex @ H @ F ) @ A2 )
% 0.27/0.50 = ( image_58037603omplex @ ( comp_c130555887omplex @ H @ G2 ) @ B3 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % image_eq_imp_comp
% 0.27/0.50 thf(fact_220_zero__le,axiom,
% 0.27/0.50 ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% 0.27/0.50
% 0.27/0.50 % zero_le
% 0.27/0.50 thf(fact_221_holomorphic__on__subset,axiom,
% 0.27/0.50 ! [F: complex > complex,S2: set_complex,T: set_complex] :
% 0.27/0.50 ( ( comple372758642hic_on @ F @ S2 )
% 0.27/0.50 => ( ( ord_le701908932omplex @ T @ S2 )
% 0.27/0.50 => ( comple372758642hic_on @ F @ T ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % holomorphic_on_subset
% 0.27/0.50 thf(fact_222_set__times__mono2__b,axiom,
% 0.27/0.50 ! [C2: set_complex,D2: set_complex,E: set_complex,F3: set_complex,X2: complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ C2 @ D2 )
% 0.27/0.50 => ( ( ord_le701908932omplex @ E @ F3 )
% 0.27/0.50 => ( ( member_complex @ X2 @ ( times_1316095593omplex @ C2 @ E ) )
% 0.27/0.50 => ( member_complex @ X2 @ ( times_1316095593omplex @ D2 @ F3 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % set_times_mono2_b
% 0.27/0.50 thf(fact_223_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.27/0.50 ! [A: nat,B: nat,C: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.27/0.50 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.27/0.50 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.27/0.50 thf(fact_224_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.27/0.50 ! [A: real,B: real,C: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.27/0.50 thf(fact_225_zero__le__mult__iff,axiom,
% 0.27/0.50 ! [A: real,B: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.27/0.50 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.27/0.50 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.27/0.50 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.27/0.50 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % zero_le_mult_iff
% 0.27/0.50 thf(fact_226_mult__nonneg__nonpos2,axiom,
% 0.27/0.50 ! [A: nat,B: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.27/0.50 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.27/0.50 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_nonneg_nonpos2
% 0.27/0.50 thf(fact_227_mult__nonneg__nonpos2,axiom,
% 0.27/0.50 ! [A: real,B: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.27/0.50 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_nonneg_nonpos2
% 0.27/0.50 thf(fact_228_mult__nonpos__nonneg,axiom,
% 0.27/0.50 ! [A: nat,B: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.27/0.50 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.27/0.50 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_nonpos_nonneg
% 0.27/0.50 thf(fact_229_mult__nonpos__nonneg,axiom,
% 0.27/0.50 ! [A: real,B: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_nonpos_nonneg
% 0.27/0.50 thf(fact_230_mult__nonneg__nonpos,axiom,
% 0.27/0.50 ! [A: nat,B: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.27/0.50 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.27/0.50 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_nonneg_nonpos
% 0.27/0.50 thf(fact_231_mult__nonneg__nonpos,axiom,
% 0.27/0.50 ! [A: real,B: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.27/0.50 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_nonneg_nonpos
% 0.27/0.50 thf(fact_232_mult__nonneg__nonneg,axiom,
% 0.27/0.50 ! [A: nat,B: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.27/0.50 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.27/0.50 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_nonneg_nonneg
% 0.27/0.50 thf(fact_233_mult__nonneg__nonneg,axiom,
% 0.27/0.50 ! [A: real,B: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.27/0.50 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_nonneg_nonneg
% 0.27/0.50 thf(fact_234_split__mult__neg__le,axiom,
% 0.27/0.50 ! [A: nat,B: nat] :
% 0.27/0.50 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.27/0.50 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 0.27/0.50 | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.27/0.50 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 0.27/0.50 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 0.27/0.50
% 0.27/0.50 % split_mult_neg_le
% 0.27/0.50 thf(fact_235_split__mult__neg__le,axiom,
% 0.27/0.50 ! [A: real,B: real] :
% 0.27/0.50 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.27/0.50 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.27/0.50 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.27/0.50 & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 0.27/0.50
% 0.27/0.50 % split_mult_neg_le
% 0.27/0.50 thf(fact_236_mult__le__0__iff,axiom,
% 0.27/0.50 ! [A: real,B: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.27/0.50 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.27/0.50 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.27/0.50 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.27/0.50 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_le_0_iff
% 0.27/0.50 thf(fact_237_mult__right__mono,axiom,
% 0.27/0.50 ! [A: nat,B: nat,C: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.27/0.50 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.27/0.50 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_right_mono
% 0.27/0.50 thf(fact_238_mult__right__mono,axiom,
% 0.27/0.50 ! [A: real,B: real,C: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_right_mono
% 0.27/0.50 thf(fact_239_mult__right__mono__neg,axiom,
% 0.27/0.50 ! [B: real,A: real,C: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ B @ A )
% 0.27/0.50 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_right_mono_neg
% 0.27/0.50 thf(fact_240_mult__left__mono,axiom,
% 0.27/0.50 ! [A: nat,B: nat,C: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.27/0.50 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.27/0.50 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_left_mono
% 0.27/0.50 thf(fact_241_mult__left__mono,axiom,
% 0.27/0.50 ! [A: real,B: real,C: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_left_mono
% 0.27/0.50 thf(fact_242_mult__nonpos__nonpos,axiom,
% 0.27/0.50 ! [A: real,B: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.27/0.50 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.27/0.50 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_nonpos_nonpos
% 0.27/0.50 thf(fact_243_mult__left__mono__neg,axiom,
% 0.27/0.50 ! [B: real,A: real,C: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ B @ A )
% 0.27/0.50 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_left_mono_neg
% 0.27/0.50 thf(fact_244_split__mult__pos__le,axiom,
% 0.27/0.50 ! [A: real,B: real] :
% 0.27/0.50 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.27/0.50 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.27/0.50 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.27/0.50 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.27/0.50 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % split_mult_pos_le
% 0.27/0.50 thf(fact_245_zero__le__square,axiom,
% 0.27/0.50 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 0.27/0.50
% 0.27/0.50 % zero_le_square
% 0.27/0.50 thf(fact_246_mult__mono_H,axiom,
% 0.27/0.50 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.27/0.50 => ( ( ord_less_eq_nat @ C @ D )
% 0.27/0.50 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.27/0.50 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.27/0.50 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_mono'
% 0.27/0.50 thf(fact_247_mult__mono_H,axiom,
% 0.27/0.50 ! [A: real,B: real,C: real,D: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.27/0.50 => ( ( ord_less_eq_real @ C @ D )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_mono'
% 0.27/0.50 thf(fact_248_mult__mono,axiom,
% 0.27/0.50 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.27/0.50 => ( ( ord_less_eq_nat @ C @ D )
% 0.27/0.50 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.27/0.50 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.27/0.50 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_mono
% 0.27/0.50 thf(fact_249_mult__mono,axiom,
% 0.27/0.50 ! [A: real,B: real,C: real,D: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.27/0.50 => ( ( ord_less_eq_real @ C @ D )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_mono
% 0.27/0.50 thf(fact_250_zero__le__one,axiom,
% 0.27/0.50 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.27/0.50
% 0.27/0.50 % zero_le_one
% 0.27/0.50 thf(fact_251_zero__le__one,axiom,
% 0.27/0.50 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.27/0.50
% 0.27/0.50 % zero_le_one
% 0.27/0.50 thf(fact_252_not__one__le__zero,axiom,
% 0.27/0.50 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.27/0.50
% 0.27/0.50 % not_one_le_zero
% 0.27/0.50 thf(fact_253_not__one__le__zero,axiom,
% 0.27/0.50 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 0.27/0.50
% 0.27/0.50 % not_one_le_zero
% 0.27/0.50 thf(fact_254_holomorphic__on__compose,axiom,
% 0.27/0.50 ! [F: complex > complex,S2: set_complex,G2: complex > complex] :
% 0.27/0.50 ( ( comple372758642hic_on @ F @ S2 )
% 0.27/0.50 => ( ( comple372758642hic_on @ G2 @ ( image_58037603omplex @ F @ S2 ) )
% 0.27/0.50 => ( comple372758642hic_on @ ( comp_c130555887omplex @ G2 @ F ) @ S2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % holomorphic_on_compose
% 0.27/0.50 thf(fact_255_deriv__left__inverse,axiom,
% 0.27/0.50 ! [F: complex > complex,S: set_complex,G2: complex > complex,T2: set_complex,W2: complex] :
% 0.27/0.50 ( ( comple372758642hic_on @ F @ S )
% 0.27/0.50 => ( ( comple372758642hic_on @ G2 @ T2 )
% 0.27/0.50 => ( ( topolo935673511omplex @ S )
% 0.27/0.50 => ( ( topolo935673511omplex @ T2 )
% 0.27/0.50 => ( ( ord_le701908932omplex @ ( image_58037603omplex @ F @ S ) @ T2 )
% 0.27/0.50 => ( ! [Z3: complex] :
% 0.27/0.50 ( ( member_complex @ Z3 @ S )
% 0.27/0.50 => ( ( G2 @ ( F @ Z3 ) )
% 0.27/0.50 = Z3 ) )
% 0.27/0.50 => ( ( member_complex @ W2 @ S )
% 0.27/0.50 => ( ( times_times_complex @ ( deriv_complex @ F @ W2 ) @ ( deriv_complex @ G2 @ ( F @ W2 ) ) )
% 0.27/0.50 = one_one_complex ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % deriv_left_inverse
% 0.27/0.50 thf(fact_256_mult__left__le,axiom,
% 0.27/0.50 ! [C: nat,A: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 0.27/0.50 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.27/0.50 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_left_le
% 0.27/0.50 thf(fact_257_mult__left__le,axiom,
% 0.27/0.50 ! [C: real,A: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ C @ one_one_real )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_left_le
% 0.27/0.50 thf(fact_258_mult__le__one,axiom,
% 0.27/0.50 ! [A: nat,B: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.27/0.50 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.27/0.50 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 0.27/0.50 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_le_one
% 0.27/0.50 thf(fact_259_mult__le__one,axiom,
% 0.27/0.50 ! [A: real,B: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ A @ one_one_real )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.27/0.50 => ( ( ord_less_eq_real @ B @ one_one_real )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_le_one
% 0.27/0.50 thf(fact_260_mult__right__le__one__le,axiom,
% 0.27/0.50 ! [X2: real,Y: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.27/0.50 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ X2 @ Y ) @ X2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_right_le_one_le
% 0.27/0.50 thf(fact_261_mult__left__le__one__le,axiom,
% 0.27/0.50 ! [X2: real,Y: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.27/0.50 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ Y @ X2 ) @ X2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_left_le_one_le
% 0.27/0.50 thf(fact_262_mult__eq__1,axiom,
% 0.27/0.50 ! [A: nat,B: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.27/0.50 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.27/0.50 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 0.27/0.50 => ( ( ( times_times_nat @ A @ B )
% 0.27/0.50 = one_one_nat )
% 0.27/0.50 = ( ( A = one_one_nat )
% 0.27/0.50 & ( B = one_one_nat ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_eq_1
% 0.27/0.50 thf(fact_263_mult__eq__1,axiom,
% 0.27/0.50 ! [A: real,B: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.27/0.50 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.27/0.50 => ( ( ord_less_eq_real @ B @ one_one_real )
% 0.27/0.50 => ( ( ( times_times_real @ A @ B )
% 0.27/0.50 = one_one_real )
% 0.27/0.50 = ( ( A = one_one_real )
% 0.27/0.50 & ( B = one_one_real ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_eq_1
% 0.27/0.50 thf(fact_264_artanh__0,axiom,
% 0.27/0.50 ( ( artanh_real @ zero_zero_real )
% 0.27/0.50 = zero_zero_real ) ).
% 0.27/0.50
% 0.27/0.50 % artanh_0
% 0.27/0.50 thf(fact_265_artanh__0,axiom,
% 0.27/0.50 ( ( artanh_complex @ zero_zero_complex )
% 0.27/0.50 = zero_zero_complex ) ).
% 0.27/0.50
% 0.27/0.50 % artanh_0
% 0.27/0.50 thf(fact_266_arsinh__0,axiom,
% 0.27/0.50 ( ( arsinh_real @ zero_zero_real )
% 0.27/0.50 = zero_zero_real ) ).
% 0.27/0.50
% 0.27/0.50 % arsinh_0
% 0.27/0.50 thf(fact_267_arsinh__0,axiom,
% 0.27/0.50 ( ( arsinh_complex @ zero_zero_complex )
% 0.27/0.50 = zero_zero_complex ) ).
% 0.27/0.50
% 0.27/0.50 % arsinh_0
% 0.27/0.50 thf(fact_268_subset__antisym,axiom,
% 0.27/0.50 ! [A2: set_complex,B3: set_complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ A2 @ B3 )
% 0.27/0.50 => ( ( ord_le701908932omplex @ B3 @ A2 )
% 0.27/0.50 => ( A2 = B3 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % subset_antisym
% 0.27/0.50 thf(fact_269_image__eqI,axiom,
% 0.27/0.50 ! [B: complex,F: complex > complex,X2: complex,A2: set_complex] :
% 0.27/0.50 ( ( B
% 0.27/0.50 = ( F @ X2 ) )
% 0.27/0.50 => ( ( member_complex @ X2 @ A2 )
% 0.27/0.50 => ( member_complex @ B @ ( image_58037603omplex @ F @ A2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % image_eqI
% 0.27/0.50 thf(fact_270_subsetI,axiom,
% 0.27/0.50 ! [A2: set_complex,B3: set_complex] :
% 0.27/0.50 ( ! [X3: complex] :
% 0.27/0.50 ( ( member_complex @ X3 @ A2 )
% 0.27/0.50 => ( member_complex @ X3 @ B3 ) )
% 0.27/0.50 => ( ord_le701908932omplex @ A2 @ B3 ) ) ).
% 0.27/0.50
% 0.27/0.50 % subsetI
% 0.27/0.50 thf(fact_271_le0,axiom,
% 0.27/0.50 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.27/0.50
% 0.27/0.50 % le0
% 0.27/0.50 thf(fact_272_bot__nat__0_Oextremum,axiom,
% 0.27/0.50 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.27/0.50
% 0.27/0.50 % bot_nat_0.extremum
% 0.27/0.50 thf(fact_273_less__eq__nat_Osimps_I1_J,axiom,
% 0.27/0.50 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.27/0.50
% 0.27/0.50 % less_eq_nat.simps(1)
% 0.27/0.50 thf(fact_274_le__0__eq,axiom,
% 0.27/0.50 ! [N: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.27/0.50 = ( N = zero_zero_nat ) ) ).
% 0.27/0.50
% 0.27/0.50 % le_0_eq
% 0.27/0.50 thf(fact_275_bot__nat__0_Oextremum__unique,axiom,
% 0.27/0.50 ! [A: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.27/0.50 = ( A = zero_zero_nat ) ) ).
% 0.27/0.50
% 0.27/0.50 % bot_nat_0.extremum_unique
% 0.27/0.50 thf(fact_276_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.27/0.50 ! [A: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.27/0.50 => ( A = zero_zero_nat ) ) ).
% 0.27/0.50
% 0.27/0.50 % bot_nat_0.extremum_uniqueI
% 0.27/0.50 thf(fact_277_mult__le__mono2,axiom,
% 0.27/0.50 ! [I: nat,J: nat,K: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ I @ J )
% 0.27/0.50 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_le_mono2
% 0.27/0.50 thf(fact_278_mult__le__mono1,axiom,
% 0.27/0.50 ! [I: nat,J: nat,K: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ I @ J )
% 0.27/0.50 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_le_mono1
% 0.27/0.50 thf(fact_279_mult__le__mono,axiom,
% 0.27/0.50 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ I @ J )
% 0.27/0.50 => ( ( ord_less_eq_nat @ K @ L )
% 0.27/0.50 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mult_le_mono
% 0.27/0.50 thf(fact_280_le__square,axiom,
% 0.27/0.50 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 0.27/0.50
% 0.27/0.50 % le_square
% 0.27/0.50 thf(fact_281_le__cube,axiom,
% 0.27/0.50 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % le_cube
% 0.27/0.50 thf(fact_282_imageI,axiom,
% 0.27/0.50 ! [X2: complex,A2: set_complex,F: complex > complex] :
% 0.27/0.50 ( ( member_complex @ X2 @ A2 )
% 0.27/0.50 => ( member_complex @ ( F @ X2 ) @ ( image_58037603omplex @ F @ A2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % imageI
% 0.27/0.50 thf(fact_283_image__iff,axiom,
% 0.27/0.50 ! [Z: complex,F: complex > complex,A2: set_complex] :
% 0.27/0.50 ( ( member_complex @ Z @ ( image_58037603omplex @ F @ A2 ) )
% 0.27/0.50 = ( ? [X: complex] :
% 0.27/0.50 ( ( member_complex @ X @ A2 )
% 0.27/0.50 & ( Z
% 0.27/0.50 = ( F @ X ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % image_iff
% 0.27/0.50 thf(fact_284_bex__imageD,axiom,
% 0.27/0.50 ! [F: complex > complex,A2: set_complex,P: complex > $o] :
% 0.27/0.50 ( ? [X4: complex] :
% 0.27/0.50 ( ( member_complex @ X4 @ ( image_58037603omplex @ F @ A2 ) )
% 0.27/0.50 & ( P @ X4 ) )
% 0.27/0.50 => ? [X3: complex] :
% 0.27/0.50 ( ( member_complex @ X3 @ A2 )
% 0.27/0.50 & ( P @ ( F @ X3 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % bex_imageD
% 0.27/0.50 thf(fact_285_image__cong,axiom,
% 0.27/0.50 ! [M2: set_complex,N2: set_complex,F: complex > complex,G2: complex > complex] :
% 0.27/0.50 ( ( M2 = N2 )
% 0.27/0.50 => ( ! [X3: complex] :
% 0.27/0.50 ( ( member_complex @ X3 @ N2 )
% 0.27/0.50 => ( ( F @ X3 )
% 0.27/0.50 = ( G2 @ X3 ) ) )
% 0.27/0.50 => ( ( image_58037603omplex @ F @ M2 )
% 0.27/0.50 = ( image_58037603omplex @ G2 @ N2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % image_cong
% 0.27/0.50 thf(fact_286_ball__imageD,axiom,
% 0.27/0.50 ! [F: complex > complex,A2: set_complex,P: complex > $o] :
% 0.27/0.50 ( ! [X3: complex] :
% 0.27/0.50 ( ( member_complex @ X3 @ ( image_58037603omplex @ F @ A2 ) )
% 0.27/0.50 => ( P @ X3 ) )
% 0.27/0.50 => ! [X4: complex] :
% 0.27/0.50 ( ( member_complex @ X4 @ A2 )
% 0.27/0.50 => ( P @ ( F @ X4 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ball_imageD
% 0.27/0.50 thf(fact_287_rev__image__eqI,axiom,
% 0.27/0.50 ! [X2: complex,A2: set_complex,B: complex,F: complex > complex] :
% 0.27/0.50 ( ( member_complex @ X2 @ A2 )
% 0.27/0.50 => ( ( B
% 0.27/0.50 = ( F @ X2 ) )
% 0.27/0.50 => ( member_complex @ B @ ( image_58037603omplex @ F @ A2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % rev_image_eqI
% 0.27/0.50 thf(fact_288_in__mono,axiom,
% 0.27/0.50 ! [A2: set_complex,B3: set_complex,X2: complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ A2 @ B3 )
% 0.27/0.50 => ( ( member_complex @ X2 @ A2 )
% 0.27/0.50 => ( member_complex @ X2 @ B3 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % in_mono
% 0.27/0.50 thf(fact_289_subsetD,axiom,
% 0.27/0.50 ! [A2: set_complex,B3: set_complex,C: complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ A2 @ B3 )
% 0.27/0.50 => ( ( member_complex @ C @ A2 )
% 0.27/0.50 => ( member_complex @ C @ B3 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % subsetD
% 0.27/0.50 thf(fact_290_equalityE,axiom,
% 0.27/0.50 ! [A2: set_complex,B3: set_complex] :
% 0.27/0.50 ( ( A2 = B3 )
% 0.27/0.50 => ~ ( ( ord_le701908932omplex @ A2 @ B3 )
% 0.27/0.50 => ~ ( ord_le701908932omplex @ B3 @ A2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % equalityE
% 0.27/0.50 thf(fact_291_subset__eq,axiom,
% 0.27/0.50 ( ord_le701908932omplex
% 0.27/0.50 = ( ^ [A5: set_complex,B5: set_complex] :
% 0.27/0.50 ! [X: complex] :
% 0.27/0.50 ( ( member_complex @ X @ A5 )
% 0.27/0.50 => ( member_complex @ X @ B5 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % subset_eq
% 0.27/0.50 thf(fact_292_equalityD1,axiom,
% 0.27/0.50 ! [A2: set_complex,B3: set_complex] :
% 0.27/0.50 ( ( A2 = B3 )
% 0.27/0.50 => ( ord_le701908932omplex @ A2 @ B3 ) ) ).
% 0.27/0.50
% 0.27/0.50 % equalityD1
% 0.27/0.50 thf(fact_293_equalityD2,axiom,
% 0.27/0.50 ! [A2: set_complex,B3: set_complex] :
% 0.27/0.50 ( ( A2 = B3 )
% 0.27/0.50 => ( ord_le701908932omplex @ B3 @ A2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % equalityD2
% 0.27/0.50 thf(fact_294_subset__iff,axiom,
% 0.27/0.50 ( ord_le701908932omplex
% 0.27/0.50 = ( ^ [A5: set_complex,B5: set_complex] :
% 0.27/0.50 ! [T3: complex] :
% 0.27/0.50 ( ( member_complex @ T3 @ A5 )
% 0.27/0.50 => ( member_complex @ T3 @ B5 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % subset_iff
% 0.27/0.50 thf(fact_295_subset__refl,axiom,
% 0.27/0.50 ! [A2: set_complex] : ( ord_le701908932omplex @ A2 @ A2 ) ).
% 0.27/0.50
% 0.27/0.50 % subset_refl
% 0.27/0.50 thf(fact_296_Collect__mono,axiom,
% 0.27/0.50 ! [P: complex > $o,Q2: complex > $o] :
% 0.27/0.50 ( ! [X3: complex] :
% 0.27/0.50 ( ( P @ X3 )
% 0.27/0.50 => ( Q2 @ X3 ) )
% 0.27/0.50 => ( ord_le701908932omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Collect_mono
% 0.27/0.50 thf(fact_297_subset__trans,axiom,
% 0.27/0.50 ! [A2: set_complex,B3: set_complex,C2: set_complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ A2 @ B3 )
% 0.27/0.50 => ( ( ord_le701908932omplex @ B3 @ C2 )
% 0.27/0.50 => ( ord_le701908932omplex @ A2 @ C2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % subset_trans
% 0.27/0.50 thf(fact_298_set__eq__subset,axiom,
% 0.27/0.50 ( ( ^ [Y2: set_complex,Z4: set_complex] : ( Y2 = Z4 ) )
% 0.27/0.50 = ( ^ [A5: set_complex,B5: set_complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ A5 @ B5 )
% 0.27/0.50 & ( ord_le701908932omplex @ B5 @ A5 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % set_eq_subset
% 0.27/0.50 thf(fact_299_Collect__mono__iff,axiom,
% 0.27/0.50 ! [P: complex > $o,Q2: complex > $o] :
% 0.27/0.50 ( ( ord_le701908932omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q2 ) )
% 0.27/0.50 = ( ! [X: complex] :
% 0.27/0.50 ( ( P @ X )
% 0.27/0.50 => ( Q2 @ X ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Collect_mono_iff
% 0.27/0.50 thf(fact_300_image__mono,axiom,
% 0.27/0.50 ! [A2: set_complex,B3: set_complex,F: complex > complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ A2 @ B3 )
% 0.27/0.50 => ( ord_le701908932omplex @ ( image_58037603omplex @ F @ A2 ) @ ( image_58037603omplex @ F @ B3 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % image_mono
% 0.27/0.50 thf(fact_301_image__subsetI,axiom,
% 0.27/0.50 ! [A2: set_complex,F: complex > complex,B3: set_complex] :
% 0.27/0.50 ( ! [X3: complex] :
% 0.27/0.50 ( ( member_complex @ X3 @ A2 )
% 0.27/0.50 => ( member_complex @ ( F @ X3 ) @ B3 ) )
% 0.27/0.50 => ( ord_le701908932omplex @ ( image_58037603omplex @ F @ A2 ) @ B3 ) ) ).
% 0.27/0.50
% 0.27/0.50 % image_subsetI
% 0.27/0.50 thf(fact_302_subset__imageE,axiom,
% 0.27/0.50 ! [B3: set_complex,F: complex > complex,A2: set_complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ B3 @ ( image_58037603omplex @ F @ A2 ) )
% 0.27/0.50 => ~ ! [C3: set_complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ C3 @ A2 )
% 0.27/0.50 => ( B3
% 0.27/0.50 != ( image_58037603omplex @ F @ C3 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % subset_imageE
% 0.27/0.50 thf(fact_303_image__subset__iff,axiom,
% 0.27/0.50 ! [F: complex > complex,A2: set_complex,B3: set_complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ ( image_58037603omplex @ F @ A2 ) @ B3 )
% 0.27/0.50 = ( ! [X: complex] :
% 0.27/0.50 ( ( member_complex @ X @ A2 )
% 0.27/0.50 => ( member_complex @ ( F @ X ) @ B3 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % image_subset_iff
% 0.27/0.50 thf(fact_304_subset__image__iff,axiom,
% 0.27/0.50 ! [B3: set_complex,F: complex > complex,A2: set_complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ B3 @ ( image_58037603omplex @ F @ A2 ) )
% 0.27/0.50 = ( ? [AA: set_complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ AA @ A2 )
% 0.27/0.50 & ( B3
% 0.27/0.50 = ( image_58037603omplex @ F @ AA ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % subset_image_iff
% 0.27/0.50 thf(fact_305_order__refl,axiom,
% 0.27/0.50 ! [X2: set_complex] : ( ord_le701908932omplex @ X2 @ X2 ) ).
% 0.27/0.50
% 0.27/0.50 % order_refl
% 0.27/0.50 thf(fact_306_order__refl,axiom,
% 0.27/0.50 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 0.27/0.50
% 0.27/0.50 % order_refl
% 0.27/0.50 thf(fact_307_order__refl,axiom,
% 0.27/0.50 ! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% 0.27/0.50
% 0.27/0.50 % order_refl
% 0.27/0.50 thf(fact_308_kuhn__labelling__lemma_H,axiom,
% 0.27/0.50 ! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q2: nat > $o] :
% 0.27/0.50 ( ! [X3: nat > real] :
% 0.27/0.50 ( ( P @ X3 )
% 0.27/0.50 => ( P @ ( F @ X3 ) ) )
% 0.27/0.50 => ( ! [X3: nat > real] :
% 0.27/0.50 ( ( P @ X3 )
% 0.27/0.50 => ! [I2: nat] :
% 0.27/0.50 ( ( Q2 @ I2 )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I2 ) )
% 0.27/0.50 & ( ord_less_eq_real @ ( X3 @ I2 ) @ one_one_real ) ) ) )
% 0.27/0.50 => ? [L3: ( nat > real ) > nat > nat] :
% 0.27/0.50 ( ! [X4: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L3 @ X4 @ I3 ) @ one_one_nat )
% 0.27/0.50 & ! [X4: nat > real,I3: nat] :
% 0.27/0.50 ( ( ( P @ X4 )
% 0.27/0.50 & ( Q2 @ I3 )
% 0.27/0.50 & ( ( X4 @ I3 )
% 0.27/0.50 = zero_zero_real ) )
% 0.27/0.50 => ( ( L3 @ X4 @ I3 )
% 0.27/0.50 = zero_zero_nat ) )
% 0.27/0.50 & ! [X4: nat > real,I3: nat] :
% 0.27/0.50 ( ( ( P @ X4 )
% 0.27/0.50 & ( Q2 @ I3 )
% 0.27/0.50 & ( ( X4 @ I3 )
% 0.27/0.50 = one_one_real ) )
% 0.27/0.50 => ( ( L3 @ X4 @ I3 )
% 0.27/0.50 = one_one_nat ) )
% 0.27/0.50 & ! [X4: nat > real,I3: nat] :
% 0.27/0.50 ( ( ( P @ X4 )
% 0.27/0.50 & ( Q2 @ I3 )
% 0.27/0.50 & ( ( L3 @ X4 @ I3 )
% 0.27/0.50 = zero_zero_nat ) )
% 0.27/0.50 => ( ord_less_eq_real @ ( X4 @ I3 ) @ ( F @ X4 @ I3 ) ) )
% 0.27/0.50 & ! [X4: nat > real,I3: nat] :
% 0.27/0.50 ( ( ( P @ X4 )
% 0.27/0.50 & ( Q2 @ I3 )
% 0.27/0.50 & ( ( L3 @ X4 @ I3 )
% 0.27/0.50 = one_one_nat ) )
% 0.27/0.50 => ( ord_less_eq_real @ ( F @ X4 @ I3 ) @ ( X4 @ I3 ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % kuhn_labelling_lemma'
% 0.27/0.50 thf(fact_309_Sup_OSUP__id__eq,axiom,
% 0.27/0.50 ! [Sup: set_real > real,A2: set_real] :
% 0.27/0.50 ( ( Sup @ ( image_real_real @ id_real @ A2 ) )
% 0.27/0.50 = ( Sup @ A2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % Sup.SUP_id_eq
% 0.27/0.50 thf(fact_310_Sup_OSUP__id__eq,axiom,
% 0.27/0.50 ! [Sup: set_complex_complex > complex > complex,A2: set_complex_complex] :
% 0.27/0.50 ( ( Sup @ ( image_944012797omplex @ id_complex_complex @ A2 ) )
% 0.27/0.50 = ( Sup @ A2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % Sup.SUP_id_eq
% 0.27/0.50 thf(fact_311_Sup_OSUP__id__eq,axiom,
% 0.27/0.50 ! [Sup: set_complex > complex,A2: set_complex] :
% 0.27/0.50 ( ( Sup @ ( image_58037603omplex @ id_complex @ A2 ) )
% 0.27/0.50 = ( Sup @ A2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % Sup.SUP_id_eq
% 0.27/0.50 thf(fact_312_le__refl,axiom,
% 0.27/0.50 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.27/0.50
% 0.27/0.50 % le_refl
% 0.27/0.50 thf(fact_313_le__trans,axiom,
% 0.27/0.50 ! [I: nat,J: nat,K: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ I @ J )
% 0.27/0.50 => ( ( ord_less_eq_nat @ J @ K )
% 0.27/0.50 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % le_trans
% 0.27/0.50 thf(fact_314_eq__imp__le,axiom,
% 0.27/0.50 ! [M: nat,N: nat] :
% 0.27/0.50 ( ( M = N )
% 0.27/0.50 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.27/0.50
% 0.27/0.50 % eq_imp_le
% 0.27/0.50 thf(fact_315_le__antisym,axiom,
% 0.27/0.50 ! [M: nat,N: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ M @ N )
% 0.27/0.50 => ( ( ord_less_eq_nat @ N @ M )
% 0.27/0.50 => ( M = N ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % le_antisym
% 0.27/0.50 thf(fact_316_nat__le__linear,axiom,
% 0.27/0.50 ! [M: nat,N: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ M @ N )
% 0.27/0.50 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.27/0.50
% 0.27/0.50 % nat_le_linear
% 0.27/0.50 thf(fact_317_Nat_Oex__has__greatest__nat,axiom,
% 0.27/0.50 ! [P: nat > $o,K: nat,B: nat] :
% 0.27/0.50 ( ( P @ K )
% 0.27/0.50 => ( ! [Y3: nat] :
% 0.27/0.50 ( ( P @ Y3 )
% 0.27/0.50 => ( ord_less_eq_nat @ Y3 @ B ) )
% 0.27/0.50 => ? [X3: nat] :
% 0.27/0.50 ( ( P @ X3 )
% 0.27/0.50 & ! [Y4: nat] :
% 0.27/0.50 ( ( P @ Y4 )
% 0.27/0.50 => ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Nat.ex_has_greatest_nat
% 0.27/0.50 thf(fact_318_dual__order_Oantisym,axiom,
% 0.27/0.50 ! [B: set_complex,A: set_complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ B @ A )
% 0.27/0.50 => ( ( ord_le701908932omplex @ A @ B )
% 0.27/0.50 => ( A = B ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % dual_order.antisym
% 0.27/0.50 thf(fact_319_dual__order_Oantisym,axiom,
% 0.27/0.50 ! [B: nat,A: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ B @ A )
% 0.27/0.50 => ( ( ord_less_eq_nat @ A @ B )
% 0.27/0.50 => ( A = B ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % dual_order.antisym
% 0.27/0.50 thf(fact_320_dual__order_Oantisym,axiom,
% 0.27/0.50 ! [B: real,A: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ B @ A )
% 0.27/0.50 => ( ( ord_less_eq_real @ A @ B )
% 0.27/0.50 => ( A = B ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % dual_order.antisym
% 0.27/0.50 thf(fact_321_dual__order_Oeq__iff,axiom,
% 0.27/0.50 ( ( ^ [Y2: set_complex,Z4: set_complex] : ( Y2 = Z4 ) )
% 0.27/0.50 = ( ^ [A3: set_complex,B2: set_complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ B2 @ A3 )
% 0.27/0.50 & ( ord_le701908932omplex @ A3 @ B2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % dual_order.eq_iff
% 0.27/0.50 thf(fact_322_dual__order_Oeq__iff,axiom,
% 0.27/0.50 ( ( ^ [Y2: nat,Z4: nat] : ( Y2 = Z4 ) )
% 0.27/0.50 = ( ^ [A3: nat,B2: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ B2 @ A3 )
% 0.27/0.50 & ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % dual_order.eq_iff
% 0.27/0.50 thf(fact_323_dual__order_Oeq__iff,axiom,
% 0.27/0.50 ( ( ^ [Y2: real,Z4: real] : ( Y2 = Z4 ) )
% 0.27/0.50 = ( ^ [A3: real,B2: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ B2 @ A3 )
% 0.27/0.50 & ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % dual_order.eq_iff
% 0.27/0.50 thf(fact_324_dual__order_Otrans,axiom,
% 0.27/0.50 ! [B: set_complex,A: set_complex,C: set_complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ B @ A )
% 0.27/0.50 => ( ( ord_le701908932omplex @ C @ B )
% 0.27/0.50 => ( ord_le701908932omplex @ C @ A ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % dual_order.trans
% 0.27/0.50 thf(fact_325_dual__order_Otrans,axiom,
% 0.27/0.50 ! [B: nat,A: nat,C: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ B @ A )
% 0.27/0.50 => ( ( ord_less_eq_nat @ C @ B )
% 0.27/0.50 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % dual_order.trans
% 0.27/0.50 thf(fact_326_dual__order_Otrans,axiom,
% 0.27/0.50 ! [B: real,A: real,C: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ B @ A )
% 0.27/0.50 => ( ( ord_less_eq_real @ C @ B )
% 0.27/0.50 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % dual_order.trans
% 0.27/0.50 thf(fact_327_linorder__wlog,axiom,
% 0.27/0.50 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.27/0.50 ( ! [A4: nat,B4: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ A4 @ B4 )
% 0.27/0.50 => ( P @ A4 @ B4 ) )
% 0.27/0.50 => ( ! [A4: nat,B4: nat] :
% 0.27/0.50 ( ( P @ B4 @ A4 )
% 0.27/0.50 => ( P @ A4 @ B4 ) )
% 0.27/0.50 => ( P @ A @ B ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % linorder_wlog
% 0.27/0.50 thf(fact_328_linorder__wlog,axiom,
% 0.27/0.50 ! [P: real > real > $o,A: real,B: real] :
% 0.27/0.50 ( ! [A4: real,B4: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ A4 @ B4 )
% 0.27/0.50 => ( P @ A4 @ B4 ) )
% 0.27/0.50 => ( ! [A4: real,B4: real] :
% 0.27/0.50 ( ( P @ B4 @ A4 )
% 0.27/0.50 => ( P @ A4 @ B4 ) )
% 0.27/0.50 => ( P @ A @ B ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % linorder_wlog
% 0.27/0.50 thf(fact_329_dual__order_Orefl,axiom,
% 0.27/0.50 ! [A: set_complex] : ( ord_le701908932omplex @ A @ A ) ).
% 0.27/0.50
% 0.27/0.50 % dual_order.refl
% 0.27/0.50 thf(fact_330_dual__order_Orefl,axiom,
% 0.27/0.50 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.27/0.50
% 0.27/0.50 % dual_order.refl
% 0.27/0.50 thf(fact_331_dual__order_Orefl,axiom,
% 0.27/0.50 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 0.27/0.50
% 0.27/0.50 % dual_order.refl
% 0.27/0.50 thf(fact_332_order__trans,axiom,
% 0.27/0.50 ! [X2: set_complex,Y: set_complex,Z: set_complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ X2 @ Y )
% 0.27/0.50 => ( ( ord_le701908932omplex @ Y @ Z )
% 0.27/0.50 => ( ord_le701908932omplex @ X2 @ Z ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % order_trans
% 0.27/0.50 thf(fact_333_order__trans,axiom,
% 0.27/0.50 ! [X2: nat,Y: nat,Z: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.27/0.50 => ( ( ord_less_eq_nat @ Y @ Z )
% 0.27/0.50 => ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % order_trans
% 0.27/0.50 thf(fact_334_order__trans,axiom,
% 0.27/0.50 ! [X2: real,Y: real,Z: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ X2 @ Y )
% 0.27/0.50 => ( ( ord_less_eq_real @ Y @ Z )
% 0.27/0.50 => ( ord_less_eq_real @ X2 @ Z ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % order_trans
% 0.27/0.50 thf(fact_335_order__class_Oorder_Oantisym,axiom,
% 0.27/0.50 ! [A: set_complex,B: set_complex] :
% 0.27/0.50 ( ( ord_le701908932omplex @ A @ B )
% 0.27/0.50 => ( ( ord_le701908932omplex @ B @ A )
% 0.27/0.50 => ( A = B ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % order_class.order.antisym
% 0.27/0.50 thf(fact_336_order__class_Oorder_Oantisym,axiom,
% 0.27/0.50 ! [A: nat,B: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.27/0.50 => ( ( ord_less_eq_nat @ B @ A )
% 0.27/0.50 => ( A = B ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % order_class.order.antisym
% 0.27/0.50 thf(fact_337_order__class_Oorder_Oantisym,axiom,
% 0.27/0.50 ! [A: real,B: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.27/0.50 => ( ( ord_less_eq_real @ B @ A )
% 0.27/0.50 => ( A = B ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % order_class.order.antisym
% 0.27/0.50 thf(fact_338_ord__le__eq__trans,axiom,
% 0.27/0.50 ! [A: nat,B: nat,C: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.27/0.50 => ( ( B = C )
% 0.27/0.50 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ord_le_eq_trans
% 0.27/0.50 thf(fact_339_ord__le__eq__trans,axiom,
% 0.27/0.50 ! [A: real,B: real,C: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.27/0.50 => ( ( B = C )
% 0.27/0.50 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ord_le_eq_trans
% 0.27/0.50 thf(fact_340_landau__omega_OR__mult__right__mono,axiom,
% 0.27/0.50 ! [B: real,A: real,C: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ B @ A )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ B @ C ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % landau_omega.R_mult_right_mono
% 0.27/0.50 thf(fact_341_landau__omega_OR__mult__left__mono,axiom,
% 0.27/0.50 ! [B: real,A: real,C: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ B @ A )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ C @ B ) @ ( times_times_real @ C @ A ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % landau_omega.R_mult_left_mono
% 0.27/0.50 thf(fact_342_landau__omega_OR__linear,axiom,
% 0.27/0.50 ! [Y: real,X2: real] :
% 0.27/0.50 ( ~ ( ord_less_eq_real @ Y @ X2 )
% 0.27/0.50 => ( ord_less_eq_real @ X2 @ Y ) ) ).
% 0.27/0.50
% 0.27/0.50 % landau_omega.R_linear
% 0.27/0.50 thf(fact_343_landau__omega_OR__trans,axiom,
% 0.27/0.50 ! [B: real,A: real,C: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ B @ A )
% 0.27/0.50 => ( ( ord_less_eq_real @ C @ B )
% 0.27/0.50 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % landau_omega.R_trans
% 0.27/0.50 thf(fact_344_landau__omega_OR__refl,axiom,
% 0.27/0.50 ! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% 0.27/0.50
% 0.27/0.50 % landau_omega.R_refl
% 0.27/0.50 thf(fact_345_landau__o_OR__linear,axiom,
% 0.27/0.50 ! [X2: real,Y: real] :
% 0.27/0.50 ( ~ ( ord_less_eq_real @ X2 @ Y )
% 0.27/0.50 => ( ord_less_eq_real @ Y @ X2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % landau_o.R_linear
% 0.27/0.50 thf(fact_346_landau__o_OR__trans,axiom,
% 0.27/0.50 ! [A: real,B: real,C: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.27/0.50 => ( ( ord_less_eq_real @ B @ C )
% 0.27/0.50 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % landau_o.R_trans
% 0.27/0.50 thf(fact_347_landau__o_OR__mult__left__mono,axiom,
% 0.27/0.50 ! [A: real,B: real,C: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % landau_o.R_mult_left_mono
% 0.27/0.50 thf(fact_348_landau__o_OR__mult__right__mono,axiom,
% 0.27/0.50 ! [A: real,B: real,C: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.27/0.50 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.27/0.50 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % landau_o.R_mult_right_mono
% 0.27/0.50 thf(fact_349_ln__ge__zero,axiom,
% 0.27/0.50 ! [X2: real] :
% 0.27/0.50 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.27/0.50 => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ln_ge_zero
% 0.27/0.50 thf(fact_350_Ln__1,axiom,
% 0.27/0.50 ( ( ln_ln_complex @ one_one_complex )
% 0.27/0.50 = zero_zero_complex ) ).
% 0.27/0.50
% 0.27/0.50 % Ln_1
% 0.27/0.50 thf(fact_351_holomorphic__fun__eq__0__on__connected,axiom,
% 0.27/0.50 ! [F: complex > complex,S: set_complex,Z: complex,W2: complex] :
% 0.27/0.50 ( ( comple372758642hic_on @ F @ S )
% 0.27/0.50 => ( ( topolo935673511omplex @ S )
% 0.27/0.50 => ( ( topolo2127351575omplex @ S )
% 0.27/0.50 => ( ! [N3: nat] :
% 0.27/0.50 ( ( compow1098280738omplex @ N3 @ deriv_complex @ F @ Z )
% 0.27/0.50 = zero_zero_complex )
% 0.27/0.50 => ( ( member_complex @ Z @ S )
% 0.27/0.50 => ( ( member_complex @ W2 @ S )
% 0.27/0.50 => ( ( F @ W2 )
% 0.27/0.50 = zero_zero_complex ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % holomorphic_fun_eq_0_on_connected
% 0.27/0.50
% 0.27/0.50 % Helper facts (7)
% 0.27/0.50 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.27/0.50 ! [X2: nat,Y: nat] :
% 0.27/0.50 ( ( if_nat @ $false @ X2 @ Y )
% 0.27/0.50 = Y ) ).
% 0.27/0.50
% 0.27/0.50 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.27/0.50 ! [X2: nat,Y: nat] :
% 0.27/0.50 ( ( if_nat @ $true @ X2 @ Y )
% 0.27/0.50 = X2 ) ).
% 0.27/0.50
% 0.27/0.50 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.27/0.50 ! [X2: real,Y: real] :
% 0.27/0.50 ( ( if_real @ $false @ X2 @ Y )
% 0.27/0.50 = Y ) ).
% 0.27/0.50
% 0.27/0.50 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.27/0.50 ! [X2: real,Y: real] :
% 0.27/0.50 ( ( if_real @ $true @ X2 @ Y )
% 0.27/0.50 = X2 ) ).
% 0.27/0.50
% 0.27/0.50 thf(help_If_3_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.27/0.50 ! [P: $o] :
% 0.27/0.50 ( ( P = $true )
% 0.27/0.50 | ( P = $false ) ) ).
% 0.27/0.50
% 0.27/0.50 thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.27/0.50 ! [X2: complex,Y: complex] :
% 0.27/0.50 ( ( if_complex @ $false @ X2 @ Y )
% 0.27/0.50 = Y ) ).
% 0.27/0.50
% 0.27/0.50 thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.27/0.50 ! [X2: complex,Y: complex] :
% 0.27/0.50 ( ( if_complex @ $true @ X2 @ Y )
% 0.27/0.50 = X2 ) ).
% 0.27/0.50
% 0.27/0.50 % Conjectures (1)
% 0.27/0.50 thf(conj_0,conjecture,
% 0.27/0.50 ( ( deriv_complex @ id_complex @ w )
% 0.27/0.50 = one_one_complex ) ).
% 0.27/0.50
% 0.27/0.50 %------------------------------------------------------------------------------
% 0.27/0.50 ------------------------
% 2.62/1.51 ---- Embedded in THF ---
% 2.62/1.51 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.62/1.51 %%% Generated on Tue Feb 24 08:24:22 EST 2026
% 2.62/1.51 %%% using '$$dhol' embedding, version 1.3.0.
% 2.62/1.51 %%% Logic specification used:
% 2.62/1.51 %%% thf(spec, logic, $$dhol).
% 2.62/1.51
% 2.62/1.51 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.gvY0VWrXGK/DTF2DTF15200.p
% See solution above
% 2.62/1.51 ------------------------
% 2.62/1.54 ---- Cleaned THF ---
% 2.62/1.54 thf(ty_n_t__Set__Oset_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 2.62/1.54 set_complex_complex: $tType ).
% 2.62/1.54
% 2.62/1.54 thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
% 2.62/1.54 set_complex: $tType ).
% 2.62/1.54
% 2.62/1.54 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 2.62/1.54 set_real: $tType ).
% 2.62/1.54
% 2.62/1.54 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 2.62/1.54 set_nat: $tType ).
% 2.62/1.54
% 2.62/1.54 thf(ty_n_t__Complex__Ocomplex,type,
% 2.62/1.54 complex: $tType ).
% 2.62/1.54
% 2.62/1.54 thf(ty_n_t__Real__Oreal,type,
% 2.62/1.54 real: $tType ).
% 2.62/1.54
% 2.62/1.54 thf(ty_n_t__Nat__Onat,type,
% 2.62/1.54 nat: $tType ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Complex__Analysis__Basics_Oholomorphic__on,type,
% 2.62/1.54 comple372758642hic_on: ( complex > complex ) > set_complex > $o ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Derivative_Oderiv_001t__Complex__Ocomplex,type,
% 2.62/1.54 deriv_complex: ( complex > complex ) > complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Derivative_Oderiv_001t__Real__Oreal,type,
% 2.62/1.54 deriv_real: ( real > real ) > real > real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Fun_Ocomp_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 2.62/1.54 comp_c1610621014omplex: ( ( complex > complex ) > complex > complex ) > ( ( complex > complex ) > complex > complex ) > ( complex > complex ) > complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Fun_Ocomp_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 2.62/1.54 comp_c881053372omplex: ( ( complex > complex ) > complex ) > ( complex > complex > complex ) > complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Fun_Ocomp_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001t__Real__Oreal_001t__Real__Oreal,type,
% 2.62/1.54 comp_c1884694328l_real: ( ( complex > complex ) > real ) > ( real > complex > complex ) > real > real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 2.62/1.54 comp_c606622857omplex: ( complex > complex > complex ) > ( ( complex > complex ) > complex ) > ( complex > complex ) > complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 2.62/1.54 comp_c130555887omplex: ( complex > complex ) > ( complex > complex ) > complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Real__Oreal,type,
% 2.62/1.54 comp_c819638635l_real: ( complex > real ) > ( real > complex ) > real > real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 2.62/1.54 comp_r2009261319omplex: ( real > complex > complex ) > ( ( complex > complex ) > real ) > ( complex > complex ) > complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 2.62/1.54 comp_r667767405omplex: ( real > complex ) > ( complex > real ) > complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
% 2.62/1.54 comp_real_real_real: ( real > real ) > ( real > real ) > real > real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Fun_Oid_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 2.62/1.54 id_complex_complex: ( complex > complex ) > complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Fun_Oid_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 2.62/1.54 id_real_real: ( real > real ) > real > real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Fun_Oid_001t__Complex__Ocomplex,type,
% 2.62/1.54 id_complex: complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Fun_Oid_001t__Real__Oreal,type,
% 2.62/1.54 id_real: real > real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Fun_Oid_001t__Set__Oset_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 2.62/1.54 id_set1618538368omplex: set_complex_complex > set_complex_complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Fun_Oid_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 2.62/1.54 id_set_complex: set_complex > set_complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Fun_Oid_001t__Set__Oset_It__Real__Oreal_J,type,
% 2.62/1.54 id_set_real: set_real > set_real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
% 2.62/1.54 one_one_complex: complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 2.62/1.54 one_one_nat: nat ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 2.62/1.54 one_one_real: real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
% 2.62/1.54 times_times_complex: complex > complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 2.62/1.54 times_times_nat: nat > nat > nat ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
% 2.62/1.54 times_times_real: real > real > real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 2.62/1.54 times_1316095593omplex: set_complex > set_complex > set_complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.62/1.54 times_times_set_nat: set_nat > set_nat > set_nat ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Real__Oreal_J,type,
% 2.62/1.54 times_times_set_real: set_real > set_real > set_real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
% 2.62/1.54 zero_zero_complex: complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 2.62/1.54 zero_zero_nat: nat ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 2.62/1.54 zero_zero_real: real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_If_001t__Complex__Ocomplex,type,
% 2.62/1.54 if_complex: $o > complex > complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_If_001t__Nat__Onat,type,
% 2.62/1.54 if_nat: $o > nat > nat > nat ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_If_001t__Real__Oreal,type,
% 2.62/1.54 if_real: $o > real > real > real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Nat_Ocompow_001_062_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_M_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 2.62/1.54 compow1098280738omplex: nat > ( ( complex > complex ) > complex > complex ) > ( complex > complex ) > complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Nat_Ocompow_001_062_I_062_It__Real__Oreal_Mt__Real__Oreal_J_M_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 2.62/1.54 compow1723822618l_real: nat > ( ( real > real ) > real > real ) > ( real > real ) > real > real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Nat_Ocompow_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 2.62/1.54 compow1667379464omplex: nat > ( complex > complex ) > complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Nat_Ocompow_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 2.62/1.54 compow_real_real: nat > ( real > real ) > real > real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Nat_Ofunpow_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 2.62/1.54 funpow1854104714omplex: nat > ( ( complex > complex ) > complex > complex ) > ( complex > complex ) > complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Nat_Ofunpow_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 2.62/1.54 funpow_real_real: nat > ( ( real > real ) > real > real ) > ( real > real ) > real > real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Nat_Ofunpow_001t__Complex__Ocomplex,type,
% 2.62/1.54 funpow_complex: nat > ( complex > complex ) > complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.62/1.54 ord_less_eq_nat: nat > nat > $o ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 2.62/1.54 ord_less_eq_real: real > real > $o ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 2.62/1.54 ord_le701908932omplex: set_complex > set_complex > $o ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
% 2.62/1.54 collect_complex: ( complex > $o ) > set_complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Set_Oimage_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 2.62/1.54 image_944012797omplex: ( ( complex > complex ) > complex > complex ) > set_complex_complex > set_complex_complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 2.62/1.54 image_58037603omplex: ( complex > complex ) > set_complex > set_complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
% 2.62/1.54 image_real_real: ( real > real ) > set_real > set_real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Complex__Ocomplex,type,
% 2.62/1.54 topolo935673511omplex: set_complex > $o ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Topological__Spaces_Otopological__space__class_Oconnected_001t__Complex__Ocomplex,type,
% 2.62/1.54 topolo2127351575omplex: set_complex > $o ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Transcendental_Oarcosh_001t__Complex__Ocomplex,type,
% 2.62/1.54 arcosh_complex: complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 2.62/1.54 arcosh_real: real > real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Transcendental_Oarsinh_001t__Complex__Ocomplex,type,
% 2.62/1.54 arsinh_complex: complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 2.62/1.54 arsinh_real: real > real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Transcendental_Oartanh_001t__Complex__Ocomplex,type,
% 2.62/1.54 artanh_complex: complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 2.62/1.54 artanh_real: real > real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Transcendental_Oln__class_Oln_001t__Complex__Ocomplex,type,
% 2.62/1.54 ln_ln_complex: complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 2.62/1.54 ln_ln_real: real > real ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_member_001t__Complex__Ocomplex,type,
% 2.62/1.54 member_complex: complex > set_complex > $o ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_member_001t__Nat__Onat,type,
% 2.62/1.54 member_nat: nat > set_nat > $o ).
% 2.62/1.54
% 2.62/1.54 thf(sy_c_member_001t__Real__Oreal,type,
% 2.62/1.54 member_real: real > set_real > $o ).
% 2.62/1.54
% 2.62/1.54 thf(sy_v_S,type,
% 2.62/1.54 s: set_complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_v_T,type,
% 2.62/1.54 t: set_complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_v_f,type,
% 2.62/1.54 f: complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_v_g,type,
% 2.62/1.54 g: complex > complex ).
% 2.62/1.54
% 2.62/1.54 thf(sy_v_w,type,
% 2.62/1.54 w: complex ).
% 2.62/1.54
% 2.62/1.54 thf(fact_0__092_060open_062_092_060And_062z_O_A_Ideriv_A_094_094_A1_J_Aid_Az_A_061_A_Iif_A1_A_061_A0_Athen_Az_Aelse_Aif_A1_A_061_A1_Athen_A1_058_058_063_Ha_Aelse_A_I0_058_058_063_Ha_J_J_092_060close_062,axiom,
% 2.62/1.54 ! [Z: real] :
% 2.62/1.54 ( ( ( one_one_nat = zero_zero_nat )
% 2.62/1.54 => ( ( compow1723822618l_real @ one_one_nat @ deriv_real @ id_real @ Z )
% 2.62/1.54 = Z ) )
% 2.62/1.54 & ( ( one_one_nat != zero_zero_nat )
% 2.62/1.54 => ( ( compow1723822618l_real @ one_one_nat @ deriv_real @ id_real @ Z )
% 2.62/1.54 = one_one_real ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_1__092_060open_062_092_060And_062z_O_A_Ideriv_A_094_094_A1_J_Aid_Az_A_061_A_Iif_A1_A_061_A0_Athen_Az_Aelse_Aif_A1_A_061_A1_Athen_A1_058_058_063_Ha_Aelse_A_I0_058_058_063_Ha_J_J_092_060close_062,axiom,
% 2.62/1.54 ! [Z: complex] :
% 2.62/1.54 ( ( ( one_one_nat = zero_zero_nat )
% 2.62/1.54 => ( ( compow1098280738omplex @ one_one_nat @ deriv_complex @ id_complex @ Z )
% 2.62/1.54 = Z ) )
% 2.62/1.54 & ( ( one_one_nat != zero_zero_nat )
% 2.62/1.54 => ( ( compow1098280738omplex @ one_one_nat @ deriv_complex @ id_complex @ Z )
% 2.62/1.54 = one_one_complex ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_2_assms_I7_J,axiom,
% 2.62/1.54 member_complex @ w @ s ).
% 2.62/1.54
% 2.62/1.54 thf(fact_3_deriv__id,axiom,
% 2.62/1.54 ( ( deriv_complex @ id_complex )
% 2.62/1.54 = ( ^ [Z2: complex] : one_one_complex ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_4_deriv__id,axiom,
% 2.62/1.54 ( ( deriv_real @ id_real )
% 2.62/1.54 = ( ^ [Z2: real] : one_one_real ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_5_id__apply,axiom,
% 2.62/1.54 ( id_real
% 2.62/1.54 = ( ^ [X: real] : X ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_6_id__apply,axiom,
% 2.62/1.54 ( id_complex_complex
% 2.62/1.54 = ( ^ [X: complex > complex] : X ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_7_id__apply,axiom,
% 2.62/1.54 ( id_complex
% 2.62/1.54 = ( ^ [X: complex] : X ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_8_id__def,axiom,
% 2.62/1.54 ( id_real
% 2.62/1.54 = ( ^ [X: real] : X ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_9_id__def,axiom,
% 2.62/1.54 ( id_complex_complex
% 2.62/1.54 = ( ^ [X: complex > complex] : X ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_10_id__def,axiom,
% 2.62/1.54 ( id_complex
% 2.62/1.54 = ( ^ [X: complex] : X ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_11_eq__id__iff,axiom,
% 2.62/1.54 ! [F: real > real] :
% 2.62/1.54 ( ( ! [X: real] :
% 2.62/1.54 ( ( F @ X )
% 2.62/1.54 = X ) )
% 2.62/1.54 = ( F = id_real ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_12_eq__id__iff,axiom,
% 2.62/1.54 ! [F: ( complex > complex ) > complex > complex] :
% 2.62/1.54 ( ( ! [X: complex > complex] :
% 2.62/1.54 ( ( F @ X )
% 2.62/1.54 = X ) )
% 2.62/1.54 = ( F = id_complex_complex ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_13_eq__id__iff,axiom,
% 2.62/1.54 ! [F: complex > complex] :
% 2.62/1.54 ( ( ! [X: complex] :
% 2.62/1.54 ( ( F @ X )
% 2.62/1.54 = X ) )
% 2.62/1.54 = ( F = id_complex ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_14_higher__deriv__id,axiom,
% 2.62/1.54 ! [N: nat,Z: real] :
% 2.62/1.54 ( ( ( N = zero_zero_nat )
% 2.62/1.54 => ( ( compow1723822618l_real @ N @ deriv_real @ id_real @ Z )
% 2.62/1.54 = Z ) )
% 2.62/1.54 & ( ( N != zero_zero_nat )
% 2.62/1.54 => ( ( ( N = one_one_nat )
% 2.62/1.54 => ( ( compow1723822618l_real @ N @ deriv_real @ id_real @ Z )
% 2.62/1.54 = one_one_real ) )
% 2.62/1.54 & ( ( N != one_one_nat )
% 2.62/1.54 => ( ( compow1723822618l_real @ N @ deriv_real @ id_real @ Z )
% 2.62/1.54 = zero_zero_real ) ) ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_15_higher__deriv__id,axiom,
% 2.62/1.54 ! [N: nat,Z: complex] :
% 2.62/1.54 ( ( ( N = zero_zero_nat )
% 2.62/1.54 => ( ( compow1098280738omplex @ N @ deriv_complex @ id_complex @ Z )
% 2.62/1.54 = Z ) )
% 2.62/1.54 & ( ( N != zero_zero_nat )
% 2.62/1.54 => ( ( ( N = one_one_nat )
% 2.62/1.54 => ( ( compow1098280738omplex @ N @ deriv_complex @ id_complex @ Z )
% 2.62/1.54 = one_one_complex ) )
% 2.62/1.54 & ( ( N != one_one_nat )
% 2.62/1.54 => ( ( compow1098280738omplex @ N @ deriv_complex @ id_complex @ Z )
% 2.62/1.54 = zero_zero_complex ) ) ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_16_one__reorient,axiom,
% 2.62/1.54 ! [X2: complex] :
% 2.62/1.54 ( ( one_one_complex = X2 )
% 2.62/1.54 = ( X2 = one_one_complex ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_17_one__reorient,axiom,
% 2.62/1.54 ! [X2: nat] :
% 2.62/1.54 ( ( one_one_nat = X2 )
% 2.62/1.54 = ( X2 = one_one_nat ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_18_one__reorient,axiom,
% 2.62/1.54 ! [X2: real] :
% 2.62/1.54 ( ( one_one_real = X2 )
% 2.62/1.54 = ( X2 = one_one_real ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_19_id__funpow,axiom,
% 2.62/1.54 ! [N: nat] :
% 2.62/1.54 ( ( compow_real_real @ N @ id_real )
% 2.62/1.54 = id_real ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_20_id__funpow,axiom,
% 2.62/1.54 ! [N: nat] :
% 2.62/1.54 ( ( compow1723822618l_real @ N @ id_real_real )
% 2.62/1.54 = id_real_real ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_21_id__funpow,axiom,
% 2.62/1.54 ! [N: nat] :
% 2.62/1.54 ( ( compow1667379464omplex @ N @ id_complex )
% 2.62/1.54 = id_complex ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_22_id__funpow,axiom,
% 2.62/1.54 ! [N: nat] :
% 2.62/1.54 ( ( compow1098280738omplex @ N @ id_complex_complex )
% 2.62/1.54 = id_complex_complex ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_23_calculation,axiom,
% 2.62/1.54 ( ( times_times_complex @ ( deriv_complex @ f @ w ) @ ( deriv_complex @ g @ ( f @ w ) ) )
% 2.62/1.54 = ( deriv_complex @ id_complex @ w ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_24__092_060open_062deriv_A_Ig_A_092_060circ_062_Af_J_Aw_A_061_Aderiv_Aid_Aw_092_060close_062,axiom,
% 2.62/1.54 ( ( deriv_complex @ ( comp_c130555887omplex @ g @ f ) @ w )
% 2.62/1.54 = ( deriv_complex @ id_complex @ w ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_25_zero__neq__one,axiom,
% 2.62/1.54 zero_zero_nat != one_one_nat ).
% 2.62/1.54
% 2.62/1.54 thf(fact_26_zero__neq__one,axiom,
% 2.62/1.54 zero_zero_real != one_one_real ).
% 2.62/1.54
% 2.62/1.54 thf(fact_27_zero__neq__one,axiom,
% 2.62/1.54 zero_zero_complex != one_one_complex ).
% 2.62/1.54
% 2.62/1.54 thf(fact_28__092_060open_062deriv_Af_Aw_A_K_Aderiv_Ag_A_If_Aw_J_A_061_Aderiv_Ag_A_If_Aw_J_A_K_Aderiv_Af_Aw_092_060close_062,axiom,
% 2.62/1.54 ( ( times_times_complex @ ( deriv_complex @ f @ w ) @ ( deriv_complex @ g @ ( f @ w ) ) )
% 2.62/1.54 = ( times_times_complex @ ( deriv_complex @ g @ ( f @ w ) ) @ ( deriv_complex @ f @ w ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_29_comp__apply,axiom,
% 2.62/1.54 ( comp_c130555887omplex
% 2.62/1.54 = ( ^ [F2: complex > complex,G: complex > complex,X: complex] : ( F2 @ ( G @ X ) ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_30_mult__zero__left,axiom,
% 2.62/1.54 ! [A: complex] :
% 2.62/1.54 ( ( times_times_complex @ zero_zero_complex @ A )
% 2.62/1.54 = zero_zero_complex ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_31_mult__zero__left,axiom,
% 2.62/1.54 ! [A: nat] :
% 2.62/1.54 ( ( times_times_nat @ zero_zero_nat @ A )
% 2.62/1.54 = zero_zero_nat ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_32_mult__zero__left,axiom,
% 2.62/1.54 ! [A: real] :
% 2.62/1.54 ( ( times_times_real @ zero_zero_real @ A )
% 2.62/1.54 = zero_zero_real ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_33_mult__zero__right,axiom,
% 2.62/1.54 ! [A: complex] :
% 2.62/1.54 ( ( times_times_complex @ A @ zero_zero_complex )
% 2.62/1.54 = zero_zero_complex ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_34_mult__zero__right,axiom,
% 2.62/1.54 ! [A: nat] :
% 2.62/1.54 ( ( times_times_nat @ A @ zero_zero_nat )
% 2.62/1.54 = zero_zero_nat ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_35_mult__zero__right,axiom,
% 2.62/1.54 ! [A: real] :
% 2.62/1.54 ( ( times_times_real @ A @ zero_zero_real )
% 2.62/1.54 = zero_zero_real ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_36_mult__eq__0__iff,axiom,
% 2.62/1.54 ! [A: complex,B: complex] :
% 2.62/1.54 ( ( ( times_times_complex @ A @ B )
% 2.62/1.54 = zero_zero_complex )
% 2.62/1.54 = ( ( A = zero_zero_complex )
% 2.62/1.54 | ( B = zero_zero_complex ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_37_mult__eq__0__iff,axiom,
% 2.62/1.54 ! [A: nat,B: nat] :
% 2.62/1.54 ( ( ( times_times_nat @ A @ B )
% 2.62/1.54 = zero_zero_nat )
% 2.62/1.54 = ( ( A = zero_zero_nat )
% 2.62/1.54 | ( B = zero_zero_nat ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_38_mult__eq__0__iff,axiom,
% 2.62/1.54 ! [A: real,B: real] :
% 2.62/1.54 ( ( ( times_times_real @ A @ B )
% 2.62/1.54 = zero_zero_real )
% 2.62/1.54 = ( ( A = zero_zero_real )
% 2.62/1.54 | ( B = zero_zero_real ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_39_mult__cancel__left,axiom,
% 2.62/1.54 ! [C: complex,A: complex,B: complex] :
% 2.62/1.54 ( ( ( times_times_complex @ C @ A )
% 2.62/1.54 = ( times_times_complex @ C @ B ) )
% 2.62/1.54 = ( ( C = zero_zero_complex )
% 2.62/1.54 | ( A = B ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_40_mult__cancel__left,axiom,
% 2.62/1.54 ! [C: nat,A: nat,B: nat] :
% 2.62/1.54 ( ( ( times_times_nat @ C @ A )
% 2.62/1.54 = ( times_times_nat @ C @ B ) )
% 2.62/1.54 = ( ( C = zero_zero_nat )
% 2.62/1.54 | ( A = B ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_41_mult__cancel__left,axiom,
% 2.62/1.54 ! [C: real,A: real,B: real] :
% 2.62/1.54 ( ( ( times_times_real @ C @ A )
% 2.62/1.54 = ( times_times_real @ C @ B ) )
% 2.62/1.54 = ( ( C = zero_zero_real )
% 2.62/1.54 | ( A = B ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_42_mult__cancel__right,axiom,
% 2.62/1.54 ! [A: complex,C: complex,B: complex] :
% 2.62/1.54 ( ( ( times_times_complex @ A @ C )
% 2.62/1.54 = ( times_times_complex @ B @ C ) )
% 2.62/1.54 = ( ( C = zero_zero_complex )
% 2.62/1.54 | ( A = B ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_43_mult__cancel__right,axiom,
% 2.62/1.54 ! [A: nat,C: nat,B: nat] :
% 2.62/1.54 ( ( ( times_times_nat @ A @ C )
% 2.62/1.54 = ( times_times_nat @ B @ C ) )
% 2.62/1.54 = ( ( C = zero_zero_nat )
% 2.62/1.54 | ( A = B ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_44_mult__cancel__right,axiom,
% 2.62/1.54 ! [A: real,C: real,B: real] :
% 2.62/1.54 ( ( ( times_times_real @ A @ C )
% 2.62/1.54 = ( times_times_real @ B @ C ) )
% 2.62/1.54 = ( ( C = zero_zero_real )
% 2.62/1.54 | ( A = B ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_45_mult_Oright__neutral,axiom,
% 2.62/1.54 ! [A: complex] :
% 2.62/1.54 ( ( times_times_complex @ A @ one_one_complex )
% 2.62/1.54 = A ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_46_mult_Oright__neutral,axiom,
% 2.62/1.54 ! [A: nat] :
% 2.62/1.54 ( ( times_times_nat @ A @ one_one_nat )
% 2.62/1.54 = A ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_47_mult_Oright__neutral,axiom,
% 2.62/1.54 ! [A: real] :
% 2.62/1.54 ( ( times_times_real @ A @ one_one_real )
% 2.62/1.54 = A ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_48_mult_Oleft__neutral,axiom,
% 2.62/1.54 ! [A: complex] :
% 2.62/1.54 ( ( times_times_complex @ one_one_complex @ A )
% 2.62/1.54 = A ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_49_mult_Oleft__neutral,axiom,
% 2.62/1.54 ! [A: nat] :
% 2.62/1.54 ( ( times_times_nat @ one_one_nat @ A )
% 2.62/1.54 = A ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_50_mult_Oleft__neutral,axiom,
% 2.62/1.54 ! [A: real] :
% 2.62/1.54 ( ( times_times_real @ one_one_real @ A )
% 2.62/1.54 = A ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_51_id__comp,axiom,
% 2.62/1.54 ! [G2: complex > complex] :
% 2.62/1.54 ( ( comp_c130555887omplex @ id_complex @ G2 )
% 2.62/1.54 = G2 ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_52_comp__id,axiom,
% 2.62/1.54 ! [F: complex > complex] :
% 2.62/1.54 ( ( comp_c130555887omplex @ F @ id_complex )
% 2.62/1.54 = F ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_53_funpow__0,axiom,
% 2.62/1.54 ! [F: ( complex > complex ) > complex > complex,X2: complex > complex] :
% 2.62/1.54 ( ( compow1098280738omplex @ zero_zero_nat @ F @ X2 )
% 2.62/1.54 = X2 ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_54_funpow__0,axiom,
% 2.62/1.54 ! [F: ( real > real ) > real > real,X2: real > real] :
% 2.62/1.54 ( ( compow1723822618l_real @ zero_zero_nat @ F @ X2 )
% 2.62/1.54 = X2 ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_55_funpow__0,axiom,
% 2.62/1.54 ! [F: complex > complex,X2: complex] :
% 2.62/1.54 ( ( compow1667379464omplex @ zero_zero_nat @ F @ X2 )
% 2.62/1.54 = X2 ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_56_assms_I3_J,axiom,
% 2.62/1.54 topolo935673511omplex @ s ).
% 2.62/1.54
% 2.62/1.54 thf(fact_57__092_060open_062deriv_Ag_A_If_Aw_J_A_K_Aderiv_Af_Aw_A_061_Aderiv_A_Ig_A_092_060circ_062_Af_J_Aw_092_060close_062,axiom,
% 2.62/1.54 ( ( times_times_complex @ ( deriv_complex @ g @ ( f @ w ) ) @ ( deriv_complex @ f @ w ) )
% 2.62/1.54 = ( deriv_complex @ ( comp_c130555887omplex @ g @ f ) @ w ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_58_mult__cancel__right2,axiom,
% 2.62/1.54 ! [A: complex,C: complex] :
% 2.62/1.54 ( ( ( times_times_complex @ A @ C )
% 2.62/1.54 = C )
% 2.62/1.54 = ( ( C = zero_zero_complex )
% 2.62/1.54 | ( A = one_one_complex ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_59_mult__cancel__right2,axiom,
% 2.62/1.54 ! [A: real,C: real] :
% 2.62/1.54 ( ( ( times_times_real @ A @ C )
% 2.62/1.54 = C )
% 2.62/1.54 = ( ( C = zero_zero_real )
% 2.62/1.54 | ( A = one_one_real ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_60_mult__cancel__right1,axiom,
% 2.62/1.54 ! [C: complex,B: complex] :
% 2.62/1.54 ( ( C
% 2.62/1.54 = ( times_times_complex @ B @ C ) )
% 2.62/1.54 = ( ( C = zero_zero_complex )
% 2.62/1.54 | ( B = one_one_complex ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_61_mult__cancel__right1,axiom,
% 2.62/1.54 ! [C: real,B: real] :
% 2.62/1.54 ( ( C
% 2.62/1.54 = ( times_times_real @ B @ C ) )
% 2.62/1.54 = ( ( C = zero_zero_real )
% 2.62/1.54 | ( B = one_one_real ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_62_mult__cancel__left2,axiom,
% 2.62/1.54 ! [C: complex,A: complex] :
% 2.62/1.54 ( ( ( times_times_complex @ C @ A )
% 2.62/1.54 = C )
% 2.62/1.54 = ( ( C = zero_zero_complex )
% 2.62/1.54 | ( A = one_one_complex ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_63_mult__cancel__left2,axiom,
% 2.62/1.54 ! [C: real,A: real] :
% 2.62/1.54 ( ( ( times_times_real @ C @ A )
% 2.62/1.54 = C )
% 2.62/1.54 = ( ( C = zero_zero_real )
% 2.62/1.54 | ( A = one_one_real ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_64_mult__cancel__left1,axiom,
% 2.62/1.54 ! [C: complex,B: complex] :
% 2.62/1.54 ( ( C
% 2.62/1.54 = ( times_times_complex @ C @ B ) )
% 2.62/1.54 = ( ( C = zero_zero_complex )
% 2.62/1.54 | ( B = one_one_complex ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_65_mult__cancel__left1,axiom,
% 2.62/1.54 ! [C: real,B: real] :
% 2.62/1.54 ( ( C
% 2.62/1.54 = ( times_times_real @ C @ B ) )
% 2.62/1.54 = ( ( C = zero_zero_real )
% 2.62/1.54 | ( B = one_one_real ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_66_assms_I6_J,axiom,
% 2.62/1.54 ! [Z: complex] :
% 2.62/1.54 ( ( member_complex @ Z @ s )
% 2.62/1.54 => ( ( g @ ( f @ Z ) )
% 2.62/1.54 = Z ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_67_assms_I1_J,axiom,
% 2.62/1.54 comple372758642hic_on @ f @ s ).
% 2.62/1.54
% 2.62/1.54 thf(fact_68_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 2.62/1.54 ! [A: complex,B: complex,C: complex] :
% 2.62/1.54 ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
% 2.62/1.54 = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_69_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 2.62/1.54 ! [A: nat,B: nat,C: nat] :
% 2.62/1.54 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 2.62/1.54 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_70_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 2.62/1.54 ! [A: real,B: real,C: real] :
% 2.62/1.54 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 2.62/1.54 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_71_comp__def,axiom,
% 2.62/1.54 ( comp_c130555887omplex
% 2.62/1.54 = ( ^ [F2: complex > complex,G: complex > complex,X: complex] : ( F2 @ ( G @ X ) ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_72_comp__assoc,axiom,
% 2.62/1.54 ! [F: complex > complex,G2: complex > complex,H: complex > complex] :
% 2.62/1.54 ( ( comp_c130555887omplex @ ( comp_c130555887omplex @ F @ G2 ) @ H )
% 2.62/1.54 = ( comp_c130555887omplex @ F @ ( comp_c130555887omplex @ G2 @ H ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_73_comp__funpow,axiom,
% 2.62/1.54 ! [N: nat,F: complex > complex] :
% 2.62/1.54 ( ( compow1098280738omplex @ N @ ( comp_c130555887omplex @ F ) )
% 2.62/1.54 = ( comp_c130555887omplex @ ( compow1667379464omplex @ N @ F ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_74_comp__funpow,axiom,
% 2.62/1.54 ! [N: nat,F: real > real] :
% 2.62/1.54 ( ( compow1723822618l_real @ N @ ( comp_real_real_real @ F ) )
% 2.62/1.54 = ( comp_real_real_real @ ( compow_real_real @ N @ F ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_75_comp__eq__dest,axiom,
% 2.62/1.54 ! [A: complex > complex,B: complex > complex,C: complex > complex,D: complex > complex,V: complex] :
% 2.62/1.54 ( ( ( comp_c130555887omplex @ A @ B )
% 2.62/1.54 = ( comp_c130555887omplex @ C @ D ) )
% 2.62/1.54 => ( ( A @ ( B @ V ) )
% 2.62/1.54 = ( C @ ( D @ V ) ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_76_comp__eq__elim,axiom,
% 2.62/1.54 ! [A: complex > complex,B: complex > complex,C: complex > complex,D: complex > complex] :
% 2.62/1.54 ( ( ( comp_c130555887omplex @ A @ B )
% 2.62/1.54 = ( comp_c130555887omplex @ C @ D ) )
% 2.62/1.54 => ! [V2: complex] :
% 2.62/1.54 ( ( A @ ( B @ V2 ) )
% 2.62/1.54 = ( C @ ( D @ V2 ) ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_77_funpow__swap1,axiom,
% 2.62/1.54 ! [F: ( complex > complex ) > complex > complex,N: nat,X2: complex > complex] :
% 2.62/1.54 ( ( F @ ( compow1098280738omplex @ N @ F @ X2 ) )
% 2.62/1.54 = ( compow1098280738omplex @ N @ F @ ( F @ X2 ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_78_funpow__swap1,axiom,
% 2.62/1.54 ! [F: ( real > real ) > real > real,N: nat,X2: real > real] :
% 2.62/1.54 ( ( F @ ( compow1723822618l_real @ N @ F @ X2 ) )
% 2.62/1.54 = ( compow1723822618l_real @ N @ F @ ( F @ X2 ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_79_funpow__swap1,axiom,
% 2.62/1.54 ! [F: complex > complex,N: nat,X2: complex] :
% 2.62/1.54 ( ( F @ ( compow1667379464omplex @ N @ F @ X2 ) )
% 2.62/1.54 = ( compow1667379464omplex @ N @ F @ ( F @ X2 ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_80_comp__eq__dest__lhs,axiom,
% 2.62/1.54 ! [A: complex > complex,B: complex > complex,C: complex > complex,V: complex] :
% 2.62/1.54 ( ( ( comp_c130555887omplex @ A @ B )
% 2.62/1.54 = C )
% 2.62/1.54 => ( ( A @ ( B @ V ) )
% 2.62/1.54 = ( C @ V ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_81_zero__reorient,axiom,
% 2.62/1.54 ! [X2: nat] :
% 2.62/1.54 ( ( zero_zero_nat = X2 )
% 2.62/1.54 = ( X2 = zero_zero_nat ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_82_zero__reorient,axiom,
% 2.62/1.54 ! [X2: real] :
% 2.62/1.54 ( ( zero_zero_real = X2 )
% 2.62/1.54 = ( X2 = zero_zero_real ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_83_zero__reorient,axiom,
% 2.62/1.54 ! [X2: complex] :
% 2.62/1.54 ( ( zero_zero_complex = X2 )
% 2.62/1.54 = ( X2 = zero_zero_complex ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_84_mult__not__zero,axiom,
% 2.62/1.54 ! [A: complex,B: complex] :
% 2.62/1.54 ( ( ( times_times_complex @ A @ B )
% 2.62/1.54 != zero_zero_complex )
% 2.62/1.54 => ( ( A != zero_zero_complex )
% 2.62/1.54 & ( B != zero_zero_complex ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_85_mult__not__zero,axiom,
% 2.62/1.54 ! [A: nat,B: nat] :
% 2.62/1.54 ( ( ( times_times_nat @ A @ B )
% 2.62/1.54 != zero_zero_nat )
% 2.62/1.54 => ( ( A != zero_zero_nat )
% 2.62/1.54 & ( B != zero_zero_nat ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_86_mult__not__zero,axiom,
% 2.62/1.54 ! [A: real,B: real] :
% 2.62/1.54 ( ( ( times_times_real @ A @ B )
% 2.62/1.54 != zero_zero_real )
% 2.62/1.54 => ( ( A != zero_zero_real )
% 2.62/1.54 & ( B != zero_zero_real ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_87_mem__Collect__eq,axiom,
% 2.62/1.54 ! [A: complex,P: complex > $o] :
% 2.62/1.54 ( ( member_complex @ A @ ( collect_complex @ P ) )
% 2.62/1.54 = ( P @ A ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_88_Collect__mem__eq,axiom,
% 2.62/1.54 ! [A2: set_complex] :
% 2.62/1.54 ( ( collect_complex
% 2.62/1.54 @ ^ [X: complex] : ( member_complex @ X @ A2 ) )
% 2.62/1.54 = A2 ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_89_mult_Oassoc,axiom,
% 2.62/1.54 ! [A: complex,B: complex,C: complex] :
% 2.62/1.54 ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
% 2.62/1.54 = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_90_mult_Oassoc,axiom,
% 2.62/1.54 ! [A: nat,B: nat,C: nat] :
% 2.62/1.54 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 2.62/1.54 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_91_mult_Oassoc,axiom,
% 2.62/1.54 ! [A: real,B: real,C: real] :
% 2.62/1.54 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 2.62/1.54 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_92_mult_Ocommute,axiom,
% 2.62/1.54 ( times_times_complex
% 2.62/1.54 = ( ^ [A3: complex,B2: complex] : ( times_times_complex @ B2 @ A3 ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_93_mult_Ocommute,axiom,
% 2.62/1.54 ( times_times_nat
% 2.62/1.54 = ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_94_mult_Ocommute,axiom,
% 2.62/1.54 ( times_times_real
% 2.62/1.54 = ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_95_mult_Oleft__commute,axiom,
% 2.62/1.54 ! [B: complex,A: complex,C: complex] :
% 2.62/1.54 ( ( times_times_complex @ B @ ( times_times_complex @ A @ C ) )
% 2.62/1.54 = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_96_mult_Oleft__commute,axiom,
% 2.62/1.54 ! [B: nat,A: nat,C: nat] :
% 2.62/1.54 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 2.62/1.54 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_97_mult_Oleft__commute,axiom,
% 2.62/1.54 ! [B: real,A: real,C: real] :
% 2.62/1.54 ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 2.62/1.54 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_98_divisors__zero,axiom,
% 2.62/1.54 ! [A: complex,B: complex] :
% 2.62/1.54 ( ( ( times_times_complex @ A @ B )
% 2.62/1.54 = zero_zero_complex )
% 2.62/1.54 => ( ( A = zero_zero_complex )
% 2.62/1.54 | ( B = zero_zero_complex ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_99_divisors__zero,axiom,
% 2.62/1.54 ! [A: nat,B: nat] :
% 2.62/1.54 ( ( ( times_times_nat @ A @ B )
% 2.62/1.54 = zero_zero_nat )
% 2.62/1.54 => ( ( A = zero_zero_nat )
% 2.62/1.54 | ( B = zero_zero_nat ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_100_divisors__zero,axiom,
% 2.62/1.54 ! [A: real,B: real] :
% 2.62/1.54 ( ( ( times_times_real @ A @ B )
% 2.62/1.54 = zero_zero_real )
% 2.62/1.54 => ( ( A = zero_zero_real )
% 2.62/1.54 | ( B = zero_zero_real ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_101_no__zero__divisors,axiom,
% 2.62/1.54 ! [A: complex,B: complex] :
% 2.62/1.54 ( ( A != zero_zero_complex )
% 2.62/1.54 => ( ( B != zero_zero_complex )
% 2.62/1.54 => ( ( times_times_complex @ A @ B )
% 2.62/1.54 != zero_zero_complex ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_102_no__zero__divisors,axiom,
% 2.62/1.54 ! [A: nat,B: nat] :
% 2.62/1.54 ( ( A != zero_zero_nat )
% 2.62/1.54 => ( ( B != zero_zero_nat )
% 2.62/1.54 => ( ( times_times_nat @ A @ B )
% 2.62/1.54 != zero_zero_nat ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_103_no__zero__divisors,axiom,
% 2.62/1.54 ! [A: real,B: real] :
% 2.62/1.54 ( ( A != zero_zero_real )
% 2.62/1.54 => ( ( B != zero_zero_real )
% 2.62/1.54 => ( ( times_times_real @ A @ B )
% 2.62/1.54 != zero_zero_real ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_104_mult__left__cancel,axiom,
% 2.62/1.54 ! [C: complex,A: complex,B: complex] :
% 2.62/1.54 ( ( C != zero_zero_complex )
% 2.62/1.54 => ( ( ( times_times_complex @ C @ A )
% 2.62/1.54 = ( times_times_complex @ C @ B ) )
% 2.62/1.54 = ( A = B ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_105_mult__left__cancel,axiom,
% 2.62/1.54 ! [C: nat,A: nat,B: nat] :
% 2.62/1.54 ( ( C != zero_zero_nat )
% 2.62/1.54 => ( ( ( times_times_nat @ C @ A )
% 2.62/1.54 = ( times_times_nat @ C @ B ) )
% 2.62/1.54 = ( A = B ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_106_mult__left__cancel,axiom,
% 2.62/1.54 ! [C: real,A: real,B: real] :
% 2.62/1.54 ( ( C != zero_zero_real )
% 2.62/1.54 => ( ( ( times_times_real @ C @ A )
% 2.62/1.54 = ( times_times_real @ C @ B ) )
% 2.62/1.54 = ( A = B ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_107_mult__right__cancel,axiom,
% 2.62/1.54 ! [C: complex,A: complex,B: complex] :
% 2.62/1.54 ( ( C != zero_zero_complex )
% 2.62/1.54 => ( ( ( times_times_complex @ A @ C )
% 2.62/1.54 = ( times_times_complex @ B @ C ) )
% 2.62/1.54 = ( A = B ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_108_mult__right__cancel,axiom,
% 2.62/1.54 ! [C: nat,A: nat,B: nat] :
% 2.62/1.54 ( ( C != zero_zero_nat )
% 2.62/1.54 => ( ( ( times_times_nat @ A @ C )
% 2.62/1.54 = ( times_times_nat @ B @ C ) )
% 2.62/1.54 = ( A = B ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_109_mult__right__cancel,axiom,
% 2.62/1.54 ! [C: real,A: real,B: real] :
% 2.62/1.54 ( ( C != zero_zero_real )
% 2.62/1.54 => ( ( ( times_times_real @ A @ C )
% 2.62/1.54 = ( times_times_real @ B @ C ) )
% 2.62/1.54 = ( A = B ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_110_comp__eq__id__dest,axiom,
% 2.62/1.54 ! [A: complex > complex,B: complex > complex,C: complex > complex,V: complex] :
% 2.62/1.54 ( ( ( comp_c130555887omplex @ A @ B )
% 2.62/1.54 = ( comp_c130555887omplex @ id_complex @ C ) )
% 2.62/1.54 => ( ( A @ ( B @ V ) )
% 2.62/1.54 = ( C @ V ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_111_mult_Ocomm__neutral,axiom,
% 2.62/1.54 ! [A: complex] :
% 2.62/1.54 ( ( times_times_complex @ A @ one_one_complex )
% 2.62/1.54 = A ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_112_mult_Ocomm__neutral,axiom,
% 2.62/1.54 ! [A: nat] :
% 2.62/1.54 ( ( times_times_nat @ A @ one_one_nat )
% 2.62/1.54 = A ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_113_mult_Ocomm__neutral,axiom,
% 2.62/1.54 ! [A: real] :
% 2.62/1.54 ( ( times_times_real @ A @ one_one_real )
% 2.62/1.54 = A ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_114_comm__monoid__mult__class_Omult__1,axiom,
% 2.62/1.54 ! [A: complex] :
% 2.62/1.54 ( ( times_times_complex @ one_one_complex @ A )
% 2.62/1.54 = A ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_115_comm__monoid__mult__class_Omult__1,axiom,
% 2.62/1.54 ! [A: nat] :
% 2.62/1.54 ( ( times_times_nat @ one_one_nat @ A )
% 2.62/1.54 = A ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_116_comm__monoid__mult__class_Omult__1,axiom,
% 2.62/1.54 ! [A: real] :
% 2.62/1.54 ( ( times_times_real @ one_one_real @ A )
% 2.62/1.54 = A ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_117_funpow__simps__right_I1_J,axiom,
% 2.62/1.54 ! [F: real > real] :
% 2.62/1.54 ( ( compow_real_real @ zero_zero_nat @ F )
% 2.62/1.54 = id_real ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_118_funpow__simps__right_I1_J,axiom,
% 2.62/1.54 ! [F: ( complex > complex ) > complex > complex] :
% 2.62/1.54 ( ( compow1098280738omplex @ zero_zero_nat @ F )
% 2.62/1.54 = id_complex_complex ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_119_funpow__simps__right_I1_J,axiom,
% 2.62/1.54 ! [F: ( real > real ) > real > real] :
% 2.62/1.54 ( ( compow1723822618l_real @ zero_zero_nat @ F )
% 2.62/1.54 = id_real_real ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_120_funpow__simps__right_I1_J,axiom,
% 2.62/1.54 ! [F: complex > complex] :
% 2.62/1.54 ( ( compow1667379464omplex @ zero_zero_nat @ F )
% 2.62/1.54 = id_complex ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_121_fun_Omap__id,axiom,
% 2.62/1.54 ! [T: complex > complex] :
% 2.62/1.54 ( ( comp_c130555887omplex @ id_complex @ T )
% 2.62/1.54 = T ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_122_vector__space__over__itself_Oscale__one,axiom,
% 2.62/1.54 ! [X2: complex] :
% 2.62/1.54 ( ( times_times_complex @ one_one_complex @ X2 )
% 2.62/1.54 = X2 ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_123_vector__space__over__itself_Oscale__one,axiom,
% 2.62/1.54 ! [X2: real] :
% 2.62/1.54 ( ( times_times_real @ one_one_real @ X2 )
% 2.62/1.54 = X2 ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_124_vector__space__over__itself_Oscale__eq__0__iff,axiom,
% 2.62/1.54 ! [A: complex,X2: complex] :
% 2.62/1.54 ( ( ( times_times_complex @ A @ X2 )
% 2.62/1.54 = zero_zero_complex )
% 2.62/1.54 = ( ( A = zero_zero_complex )
% 2.62/1.54 | ( X2 = zero_zero_complex ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_125_vector__space__over__itself_Oscale__eq__0__iff,axiom,
% 2.62/1.54 ! [A: real,X2: real] :
% 2.62/1.54 ( ( ( times_times_real @ A @ X2 )
% 2.62/1.54 = zero_zero_real )
% 2.62/1.54 = ( ( A = zero_zero_real )
% 2.62/1.54 | ( X2 = zero_zero_real ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_126_vector__space__over__itself_Oscale__zero__left,axiom,
% 2.62/1.54 ! [X2: complex] :
% 2.62/1.54 ( ( times_times_complex @ zero_zero_complex @ X2 )
% 2.62/1.54 = zero_zero_complex ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_127_vector__space__over__itself_Oscale__zero__left,axiom,
% 2.62/1.54 ! [X2: real] :
% 2.62/1.54 ( ( times_times_real @ zero_zero_real @ X2 )
% 2.62/1.54 = zero_zero_real ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_128_vector__space__over__itself_Oscale__zero__right,axiom,
% 2.62/1.54 ! [A: complex] :
% 2.62/1.54 ( ( times_times_complex @ A @ zero_zero_complex )
% 2.62/1.54 = zero_zero_complex ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_129_vector__space__over__itself_Oscale__zero__right,axiom,
% 2.62/1.54 ! [A: real] :
% 2.62/1.54 ( ( times_times_real @ A @ zero_zero_real )
% 2.62/1.54 = zero_zero_real ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_130_vector__space__over__itself_Oscale__cancel__left,axiom,
% 2.62/1.54 ! [A: complex,X2: complex,Y: complex] :
% 2.62/1.54 ( ( ( times_times_complex @ A @ X2 )
% 2.62/1.54 = ( times_times_complex @ A @ Y ) )
% 2.62/1.54 = ( ( X2 = Y )
% 2.62/1.54 | ( A = zero_zero_complex ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_131_vector__space__over__itself_Oscale__cancel__left,axiom,
% 2.62/1.54 ! [A: real,X2: real,Y: real] :
% 2.62/1.54 ( ( ( times_times_real @ A @ X2 )
% 2.62/1.54 = ( times_times_real @ A @ Y ) )
% 2.62/1.54 = ( ( X2 = Y )
% 2.62/1.54 | ( A = zero_zero_real ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_132_vector__space__over__itself_Oscale__cancel__right,axiom,
% 2.62/1.54 ! [A: complex,X2: complex,B: complex] :
% 2.62/1.54 ( ( ( times_times_complex @ A @ X2 )
% 2.62/1.54 = ( times_times_complex @ B @ X2 ) )
% 2.62/1.54 = ( ( A = B )
% 2.62/1.54 | ( X2 = zero_zero_complex ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_133_vector__space__over__itself_Oscale__cancel__right,axiom,
% 2.62/1.54 ! [A: real,X2: real,B: real] :
% 2.62/1.54 ( ( ( times_times_real @ A @ X2 )
% 2.62/1.54 = ( times_times_real @ B @ X2 ) )
% 2.62/1.54 = ( ( A = B )
% 2.62/1.54 | ( X2 = zero_zero_real ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_134_mult__if__delta,axiom,
% 2.62/1.54 ! [P: $o,Q: complex] :
% 2.62/1.54 ( ( P
% 2.62/1.54 => ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q )
% 2.62/1.54 = Q ) )
% 2.62/1.54 & ( ~ P
% 2.62/1.54 => ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q )
% 2.62/1.54 = zero_zero_complex ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_135_mult__if__delta,axiom,
% 2.62/1.54 ! [P: $o,Q: nat] :
% 2.62/1.54 ( ( P
% 2.62/1.54 => ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
% 2.62/1.54 = Q ) )
% 2.62/1.54 & ( ~ P
% 2.62/1.54 => ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
% 2.62/1.54 = zero_zero_nat ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_136_mult__if__delta,axiom,
% 2.62/1.54 ! [P: $o,Q: real] :
% 2.62/1.54 ( ( P
% 2.62/1.54 => ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
% 2.62/1.54 = Q ) )
% 2.62/1.54 & ( ~ P
% 2.62/1.54 => ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
% 2.62/1.54 = zero_zero_real ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_137_set__times__intro,axiom,
% 2.62/1.54 ! [A: complex,C2: set_complex,B: complex,D2: set_complex] :
% 2.62/1.54 ( ( member_complex @ A @ C2 )
% 2.62/1.54 => ( ( member_complex @ B @ D2 )
% 2.62/1.54 => ( member_complex @ ( times_times_complex @ A @ B ) @ ( times_1316095593omplex @ C2 @ D2 ) ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_138_set__times__intro,axiom,
% 2.62/1.54 ! [A: nat,C2: set_nat,B: nat,D2: set_nat] :
% 2.62/1.54 ( ( member_nat @ A @ C2 )
% 2.62/1.54 => ( ( member_nat @ B @ D2 )
% 2.62/1.54 => ( member_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C2 @ D2 ) ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_139_set__times__intro,axiom,
% 2.62/1.54 ! [A: real,C2: set_real,B: real,D2: set_real] :
% 2.62/1.54 ( ( member_real @ A @ C2 )
% 2.62/1.54 => ( ( member_real @ B @ D2 )
% 2.62/1.54 => ( member_real @ ( times_times_real @ A @ B ) @ ( times_times_set_real @ C2 @ D2 ) ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_140_funpow__code__def,axiom,
% 2.62/1.54 funpow1854104714omplex = compow1098280738omplex ).
% 2.62/1.54
% 2.62/1.54 thf(fact_141_funpow__code__def,axiom,
% 2.62/1.54 funpow_real_real = compow1723822618l_real ).
% 2.62/1.54
% 2.62/1.54 thf(fact_142_funpow__code__def,axiom,
% 2.62/1.54 funpow_complex = compow1667379464omplex ).
% 2.62/1.54
% 2.62/1.54 thf(fact_143_assms_I4_J,axiom,
% 2.62/1.54 topolo935673511omplex @ t ).
% 2.62/1.54
% 2.62/1.54 thf(fact_144_assms_I2_J,axiom,
% 2.62/1.54 comple372758642hic_on @ g @ t ).
% 2.62/1.54
% 2.62/1.54 thf(fact_145_mult__cancel2,axiom,
% 2.62/1.54 ! [M: nat,K: nat,N: nat] :
% 2.62/1.54 ( ( ( times_times_nat @ M @ K )
% 2.62/1.54 = ( times_times_nat @ N @ K ) )
% 2.62/1.54 = ( ( M = N )
% 2.62/1.54 | ( K = zero_zero_nat ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_146_mult__cancel1,axiom,
% 2.62/1.54 ! [K: nat,M: nat,N: nat] :
% 2.62/1.54 ( ( ( times_times_nat @ K @ M )
% 2.62/1.54 = ( times_times_nat @ K @ N ) )
% 2.62/1.54 = ( ( M = N )
% 2.62/1.54 | ( K = zero_zero_nat ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_147_mult__0__right,axiom,
% 2.62/1.54 ! [M: nat] :
% 2.62/1.54 ( ( times_times_nat @ M @ zero_zero_nat )
% 2.62/1.54 = zero_zero_nat ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_148_mult__is__0,axiom,
% 2.62/1.54 ! [M: nat,N: nat] :
% 2.62/1.54 ( ( ( times_times_nat @ M @ N )
% 2.62/1.54 = zero_zero_nat )
% 2.62/1.54 = ( ( M = zero_zero_nat )
% 2.62/1.54 | ( N = zero_zero_nat ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_149_nat__mult__eq__1__iff,axiom,
% 2.62/1.54 ! [M: nat,N: nat] :
% 2.62/1.54 ( ( ( times_times_nat @ M @ N )
% 2.62/1.54 = one_one_nat )
% 2.62/1.54 = ( ( M = one_one_nat )
% 2.62/1.54 & ( N = one_one_nat ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_150_nat__1__eq__mult__iff,axiom,
% 2.62/1.54 ! [M: nat,N: nat] :
% 2.62/1.54 ( ( one_one_nat
% 2.62/1.54 = ( times_times_nat @ M @ N ) )
% 2.62/1.54 = ( ( M = one_one_nat )
% 2.62/1.54 & ( N = one_one_nat ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_151_higher__deriv__transform__within__open,axiom,
% 2.62/1.54 ! [F: complex > complex,S: set_complex,G2: complex > complex,Z: complex,I: nat] :
% 2.62/1.54 ( ( comple372758642hic_on @ F @ S )
% 2.62/1.54 => ( ( comple372758642hic_on @ G2 @ S )
% 2.62/1.54 => ( ( topolo935673511omplex @ S )
% 2.62/1.54 => ( ( member_complex @ Z @ S )
% 2.62/1.54 => ( ! [W: complex] :
% 2.62/1.54 ( ( member_complex @ W @ S )
% 2.62/1.54 => ( ( F @ W )
% 2.62/1.54 = ( G2 @ W ) ) )
% 2.62/1.54 => ( ( compow1098280738omplex @ I @ deriv_complex @ F @ Z )
% 2.62/1.54 = ( compow1098280738omplex @ I @ deriv_complex @ G2 @ Z ) ) ) ) ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_152_holomorphic__higher__deriv,axiom,
% 2.62/1.54 ! [F: complex > complex,S: set_complex,N: nat] :
% 2.62/1.54 ( ( comple372758642hic_on @ F @ S )
% 2.62/1.54 => ( ( topolo935673511omplex @ S )
% 2.62/1.54 => ( comple372758642hic_on @ ( compow1098280738omplex @ N @ deriv_complex @ F ) @ S ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_153_holomorphic__deriv,axiom,
% 2.62/1.54 ! [F: complex > complex,S: set_complex] :
% 2.62/1.54 ( ( comple372758642hic_on @ F @ S )
% 2.62/1.54 => ( ( topolo935673511omplex @ S )
% 2.62/1.54 => ( comple372758642hic_on @ ( deriv_complex @ F ) @ S ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_154_mult__0,axiom,
% 2.62/1.54 ! [N: nat] :
% 2.62/1.54 ( ( times_times_nat @ zero_zero_nat @ N )
% 2.62/1.54 = zero_zero_nat ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_155_nat__mult__1__right,axiom,
% 2.62/1.54 ! [N: nat] :
% 2.62/1.54 ( ( times_times_nat @ N @ one_one_nat )
% 2.62/1.54 = N ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_156_nat__mult__1,axiom,
% 2.62/1.54 ! [N: nat] :
% 2.62/1.54 ( ( times_times_nat @ one_one_nat @ N )
% 2.62/1.54 = N ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_157_funpow__mult,axiom,
% 2.62/1.54 ! [N: nat,M: nat,F: ( complex > complex ) > complex > complex] :
% 2.62/1.54 ( ( compow1098280738omplex @ N @ ( compow1098280738omplex @ M @ F ) )
% 2.62/1.54 = ( compow1098280738omplex @ ( times_times_nat @ M @ N ) @ F ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_158_funpow__mult,axiom,
% 2.62/1.54 ! [N: nat,M: nat,F: ( real > real ) > real > real] :
% 2.62/1.54 ( ( compow1723822618l_real @ N @ ( compow1723822618l_real @ M @ F ) )
% 2.62/1.54 = ( compow1723822618l_real @ ( times_times_nat @ M @ N ) @ F ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_159_funpow__mult,axiom,
% 2.62/1.54 ! [N: nat,M: nat,F: complex > complex] :
% 2.62/1.54 ( ( compow1667379464omplex @ N @ ( compow1667379464omplex @ M @ F ) )
% 2.62/1.54 = ( compow1667379464omplex @ ( times_times_nat @ M @ N ) @ F ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_160_mult__eq__self__implies__10,axiom,
% 2.62/1.54 ! [M: nat,N: nat] :
% 2.62/1.54 ( ( M
% 2.62/1.54 = ( times_times_nat @ M @ N ) )
% 2.62/1.54 => ( ( N = one_one_nat )
% 2.62/1.54 | ( M = zero_zero_nat ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_161_vector__space__over__itself_Oscale__left__commute,axiom,
% 2.62/1.54 ! [A: complex,B: complex,X2: complex] :
% 2.62/1.54 ( ( times_times_complex @ A @ ( times_times_complex @ B @ X2 ) )
% 2.62/1.54 = ( times_times_complex @ B @ ( times_times_complex @ A @ X2 ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_162_vector__space__over__itself_Oscale__left__commute,axiom,
% 2.62/1.54 ! [A: real,B: real,X2: real] :
% 2.62/1.54 ( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
% 2.62/1.54 = ( times_times_real @ B @ ( times_times_real @ A @ X2 ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_163_vector__space__over__itself_Oscale__scale,axiom,
% 2.62/1.54 ! [A: complex,B: complex,X2: complex] :
% 2.62/1.54 ( ( times_times_complex @ A @ ( times_times_complex @ B @ X2 ) )
% 2.62/1.54 = ( times_times_complex @ ( times_times_complex @ A @ B ) @ X2 ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_164_vector__space__over__itself_Oscale__scale,axiom,
% 2.62/1.54 ! [A: real,B: real,X2: real] :
% 2.62/1.54 ( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
% 2.62/1.54 = ( times_times_real @ ( times_times_real @ A @ B ) @ X2 ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_165_set__times__elim,axiom,
% 2.62/1.54 ! [X2: complex,A2: set_complex,B3: set_complex] :
% 2.62/1.54 ( ( member_complex @ X2 @ ( times_1316095593omplex @ A2 @ B3 ) )
% 2.62/1.54 => ~ ! [A4: complex,B4: complex] :
% 2.62/1.54 ( ( X2
% 2.62/1.54 = ( times_times_complex @ A4 @ B4 ) )
% 2.62/1.54 => ( ( member_complex @ A4 @ A2 )
% 2.62/1.54 => ~ ( member_complex @ B4 @ B3 ) ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_166_set__times__elim,axiom,
% 2.62/1.54 ! [X2: nat,A2: set_nat,B3: set_nat] :
% 2.62/1.54 ( ( member_nat @ X2 @ ( times_times_set_nat @ A2 @ B3 ) )
% 2.62/1.54 => ~ ! [A4: nat,B4: nat] :
% 2.62/1.54 ( ( X2
% 2.62/1.54 = ( times_times_nat @ A4 @ B4 ) )
% 2.62/1.54 => ( ( member_nat @ A4 @ A2 )
% 2.62/1.54 => ~ ( member_nat @ B4 @ B3 ) ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_167_set__times__elim,axiom,
% 2.62/1.54 ! [X2: real,A2: set_real,B3: set_real] :
% 2.62/1.54 ( ( member_real @ X2 @ ( times_times_set_real @ A2 @ B3 ) )
% 2.62/1.54 => ~ ! [A4: real,B4: real] :
% 2.62/1.54 ( ( X2
% 2.62/1.54 = ( times_times_real @ A4 @ B4 ) )
% 2.62/1.54 => ( ( member_real @ A4 @ A2 )
% 2.62/1.54 => ~ ( member_real @ B4 @ B3 ) ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_168_fun_Omap__comp,axiom,
% 2.62/1.54 ! [G2: complex > complex,F: complex > complex,V: complex > complex] :
% 2.62/1.54 ( ( comp_c130555887omplex @ G2 @ ( comp_c130555887omplex @ F @ V ) )
% 2.62/1.54 = ( comp_c130555887omplex @ ( comp_c130555887omplex @ G2 @ F ) @ V ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_169_vector__space__over__itself_Oscale__right__imp__eq,axiom,
% 2.62/1.54 ! [X2: complex,A: complex,B: complex] :
% 2.62/1.54 ( ( X2 != zero_zero_complex )
% 2.62/1.54 => ( ( ( times_times_complex @ A @ X2 )
% 2.62/1.54 = ( times_times_complex @ B @ X2 ) )
% 2.62/1.54 => ( A = B ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_170_vector__space__over__itself_Oscale__right__imp__eq,axiom,
% 2.62/1.54 ! [X2: real,A: real,B: real] :
% 2.62/1.54 ( ( X2 != zero_zero_real )
% 2.62/1.54 => ( ( ( times_times_real @ A @ X2 )
% 2.62/1.54 = ( times_times_real @ B @ X2 ) )
% 2.62/1.54 => ( A = B ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_171_vector__space__over__itself_Oscale__left__imp__eq,axiom,
% 2.62/1.54 ! [A: complex,X2: complex,Y: complex] :
% 2.62/1.54 ( ( A != zero_zero_complex )
% 2.62/1.54 => ( ( ( times_times_complex @ A @ X2 )
% 2.62/1.54 = ( times_times_complex @ A @ Y ) )
% 2.62/1.54 => ( X2 = Y ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_172_vector__space__over__itself_Oscale__left__imp__eq,axiom,
% 2.62/1.54 ! [A: real,X2: real,Y: real] :
% 2.62/1.54 ( ( A != zero_zero_real )
% 2.62/1.54 => ( ( ( times_times_real @ A @ X2 )
% 2.62/1.54 = ( times_times_real @ A @ Y ) )
% 2.62/1.54 => ( X2 = Y ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_173_fun_Omap__id0,axiom,
% 2.62/1.54 ( ( comp_c130555887omplex @ id_complex )
% 2.62/1.54 = id_complex_complex ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_174_complex__derivative__transform__within__open,axiom,
% 2.62/1.54 ! [F: complex > complex,S2: set_complex,G2: complex > complex,Z: complex] :
% 2.62/1.54 ( ( comple372758642hic_on @ F @ S2 )
% 2.62/1.54 => ( ( comple372758642hic_on @ G2 @ S2 )
% 2.62/1.54 => ( ( topolo935673511omplex @ S2 )
% 2.62/1.54 => ( ( member_complex @ Z @ S2 )
% 2.62/1.54 => ( ! [W: complex] :
% 2.62/1.54 ( ( member_complex @ W @ S2 )
% 2.62/1.54 => ( ( F @ W )
% 2.62/1.54 = ( G2 @ W ) ) )
% 2.62/1.54 => ( ( deriv_complex @ F @ Z )
% 2.62/1.54 = ( deriv_complex @ G2 @ Z ) ) ) ) ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_175_holomorphic__on__id,axiom,
% 2.62/1.54 ! [S2: set_complex] : ( comple372758642hic_on @ id_complex @ S2 ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_176_holomorphic__on__linear,axiom,
% 2.62/1.54 ! [C: complex,S2: set_complex] : ( comple372758642hic_on @ ( times_times_complex @ C ) @ S2 ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_177_pointfree__idE,axiom,
% 2.62/1.54 ! [F: complex > complex,G2: complex > complex,X2: complex] :
% 2.62/1.54 ( ( ( comp_c130555887omplex @ F @ G2 )
% 2.62/1.54 = id_complex )
% 2.62/1.54 => ( ( F @ ( G2 @ X2 ) )
% 2.62/1.54 = X2 ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_178_isomorphism__expand,axiom,
% 2.62/1.54 ! [F: real > complex,G2: complex > real] :
% 2.62/1.54 ( ( ( ( comp_r667767405omplex @ F @ G2 )
% 2.62/1.54 = id_complex )
% 2.62/1.54 & ( ( comp_c819638635l_real @ G2 @ F )
% 2.62/1.54 = id_real ) )
% 2.62/1.54 = ( ! [X: complex] :
% 2.62/1.54 ( ( F @ ( G2 @ X ) )
% 2.62/1.54 = X )
% 2.62/1.54 & ! [X: real] :
% 2.62/1.54 ( ( G2 @ ( F @ X ) )
% 2.62/1.54 = X ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_179_isomorphism__expand,axiom,
% 2.62/1.54 ! [F: ( complex > complex ) > complex,G2: complex > complex > complex] :
% 2.62/1.54 ( ( ( ( comp_c881053372omplex @ F @ G2 )
% 2.62/1.54 = id_complex )
% 2.62/1.54 & ( ( comp_c606622857omplex @ G2 @ F )
% 2.62/1.54 = id_complex_complex ) )
% 2.62/1.54 = ( ! [X: complex] :
% 2.62/1.54 ( ( F @ ( G2 @ X ) )
% 2.62/1.54 = X )
% 2.62/1.54 & ! [X: complex > complex] :
% 2.62/1.54 ( ( G2 @ ( F @ X ) )
% 2.62/1.54 = X ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_180_isomorphism__expand,axiom,
% 2.62/1.54 ! [F: complex > real,G2: real > complex] :
% 2.62/1.54 ( ( ( ( comp_c819638635l_real @ F @ G2 )
% 2.62/1.54 = id_real )
% 2.62/1.54 & ( ( comp_r667767405omplex @ G2 @ F )
% 2.62/1.54 = id_complex ) )
% 2.62/1.54 = ( ! [X: real] :
% 2.62/1.54 ( ( F @ ( G2 @ X ) )
% 2.62/1.54 = X )
% 2.62/1.54 & ! [X: complex] :
% 2.62/1.54 ( ( G2 @ ( F @ X ) )
% 2.62/1.54 = X ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_181_isomorphism__expand,axiom,
% 2.62/1.54 ! [F: real > real,G2: real > real] :
% 2.62/1.54 ( ( ( ( comp_real_real_real @ F @ G2 )
% 2.62/1.54 = id_real )
% 2.62/1.54 & ( ( comp_real_real_real @ G2 @ F )
% 2.62/1.54 = id_real ) )
% 2.62/1.54 = ( ! [X: real] :
% 2.62/1.54 ( ( F @ ( G2 @ X ) )
% 2.62/1.54 = X )
% 2.62/1.54 & ! [X: real] :
% 2.62/1.54 ( ( G2 @ ( F @ X ) )
% 2.62/1.54 = X ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_182_isomorphism__expand,axiom,
% 2.62/1.54 ! [F: ( complex > complex ) > real,G2: real > complex > complex] :
% 2.62/1.54 ( ( ( ( comp_c1884694328l_real @ F @ G2 )
% 2.62/1.54 = id_real )
% 2.62/1.54 & ( ( comp_r2009261319omplex @ G2 @ F )
% 2.62/1.54 = id_complex_complex ) )
% 2.62/1.54 = ( ! [X: real] :
% 2.62/1.54 ( ( F @ ( G2 @ X ) )
% 2.62/1.54 = X )
% 2.62/1.54 & ! [X: complex > complex] :
% 2.62/1.54 ( ( G2 @ ( F @ X ) )
% 2.62/1.54 = X ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_183_isomorphism__expand,axiom,
% 2.62/1.54 ! [F: complex > complex > complex,G2: ( complex > complex ) > complex] :
% 2.62/1.54 ( ( ( ( comp_c606622857omplex @ F @ G2 )
% 2.62/1.54 = id_complex_complex )
% 2.62/1.54 & ( ( comp_c881053372omplex @ G2 @ F )
% 2.62/1.54 = id_complex ) )
% 2.62/1.54 = ( ! [X: complex > complex] :
% 2.62/1.54 ( ( F @ ( G2 @ X ) )
% 2.62/1.54 = X )
% 2.62/1.54 & ! [X: complex] :
% 2.62/1.54 ( ( G2 @ ( F @ X ) )
% 2.62/1.54 = X ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_184_isomorphism__expand,axiom,
% 2.62/1.54 ! [F: real > complex > complex,G2: ( complex > complex ) > real] :
% 2.62/1.54 ( ( ( ( comp_r2009261319omplex @ F @ G2 )
% 2.62/1.54 = id_complex_complex )
% 2.62/1.54 & ( ( comp_c1884694328l_real @ G2 @ F )
% 2.62/1.54 = id_real ) )
% 2.62/1.54 = ( ! [X: complex > complex] :
% 2.62/1.54 ( ( F @ ( G2 @ X ) )
% 2.62/1.54 = X )
% 2.62/1.54 & ! [X: real] :
% 2.62/1.54 ( ( G2 @ ( F @ X ) )
% 2.62/1.54 = X ) ) ) ).
% 2.62/1.54
% 2.62/1.54 thf(fact_185_isomorphism__expand,axiom,
% 2.62/1.54 ! [F: ( complex > complex ) > complex > complex,G2: ( complex > complex ) > complex > complex] :
% 2.62/1.54 ( ( ( ( comp_c1610621014omplex @ F @ G2 )
% 2.62/1.55 = id_complex_complex )
% 2.62/1.55 & ( ( comp_c1610621014omplex @ G2 @ F )
% 2.62/1.55 = id_complex_complex ) )
% 2.62/1.55 = ( ! [X: complex > complex] :
% 2.62/1.55 ( ( F @ ( G2 @ X ) )
% 2.62/1.55 = X )
% 2.62/1.55 & ! [X: complex > complex] :
% 2.62/1.55 ( ( G2 @ ( F @ X ) )
% 2.62/1.55 = X ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_186_isomorphism__expand,axiom,
% 2.62/1.55 ! [F: complex > complex,G2: complex > complex] :
% 2.62/1.55 ( ( ( ( comp_c130555887omplex @ F @ G2 )
% 2.62/1.55 = id_complex )
% 2.62/1.55 & ( ( comp_c130555887omplex @ G2 @ F )
% 2.62/1.55 = id_complex ) )
% 2.62/1.55 = ( ! [X: complex] :
% 2.62/1.55 ( ( F @ ( G2 @ X ) )
% 2.62/1.55 = X )
% 2.62/1.55 & ! [X: complex] :
% 2.62/1.55 ( ( G2 @ ( F @ X ) )
% 2.62/1.55 = X ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_187_left__right__inverse__eq,axiom,
% 2.62/1.55 ! [F: real > complex,G2: complex > real,H: real > complex] :
% 2.62/1.55 ( ( ( comp_r667767405omplex @ F @ G2 )
% 2.62/1.55 = id_complex )
% 2.62/1.55 => ( ( ( comp_c819638635l_real @ G2 @ H )
% 2.62/1.55 = id_real )
% 2.62/1.55 => ( F = H ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_188_left__right__inverse__eq,axiom,
% 2.62/1.55 ! [F: ( complex > complex ) > complex,G2: complex > complex > complex,H: ( complex > complex ) > complex] :
% 2.62/1.55 ( ( ( comp_c881053372omplex @ F @ G2 )
% 2.62/1.55 = id_complex )
% 2.62/1.55 => ( ( ( comp_c606622857omplex @ G2 @ H )
% 2.62/1.55 = id_complex_complex )
% 2.62/1.55 => ( F = H ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_189_left__right__inverse__eq,axiom,
% 2.62/1.55 ! [F: complex > real,G2: real > complex,H: complex > real] :
% 2.62/1.55 ( ( ( comp_c819638635l_real @ F @ G2 )
% 2.62/1.55 = id_real )
% 2.62/1.55 => ( ( ( comp_r667767405omplex @ G2 @ H )
% 2.62/1.55 = id_complex )
% 2.62/1.55 => ( F = H ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_190_left__right__inverse__eq,axiom,
% 2.62/1.55 ! [F: real > real,G2: real > real,H: real > real] :
% 2.62/1.55 ( ( ( comp_real_real_real @ F @ G2 )
% 2.62/1.55 = id_real )
% 2.62/1.55 => ( ( ( comp_real_real_real @ G2 @ H )
% 2.62/1.55 = id_real )
% 2.62/1.55 => ( F = H ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_191_left__right__inverse__eq,axiom,
% 2.62/1.55 ! [F: ( complex > complex ) > real,G2: real > complex > complex,H: ( complex > complex ) > real] :
% 2.62/1.55 ( ( ( comp_c1884694328l_real @ F @ G2 )
% 2.62/1.55 = id_real )
% 2.62/1.55 => ( ( ( comp_r2009261319omplex @ G2 @ H )
% 2.62/1.55 = id_complex_complex )
% 2.62/1.55 => ( F = H ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_192_left__right__inverse__eq,axiom,
% 2.62/1.55 ! [F: complex > complex > complex,G2: ( complex > complex ) > complex,H: complex > complex > complex] :
% 2.62/1.55 ( ( ( comp_c606622857omplex @ F @ G2 )
% 2.62/1.55 = id_complex_complex )
% 2.62/1.55 => ( ( ( comp_c881053372omplex @ G2 @ H )
% 2.62/1.55 = id_complex )
% 2.62/1.55 => ( F = H ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_193_left__right__inverse__eq,axiom,
% 2.62/1.55 ! [F: real > complex > complex,G2: ( complex > complex ) > real,H: real > complex > complex] :
% 2.62/1.55 ( ( ( comp_r2009261319omplex @ F @ G2 )
% 2.62/1.55 = id_complex_complex )
% 2.62/1.55 => ( ( ( comp_c1884694328l_real @ G2 @ H )
% 2.62/1.55 = id_real )
% 2.62/1.55 => ( F = H ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_194_left__right__inverse__eq,axiom,
% 2.62/1.55 ! [F: ( complex > complex ) > complex > complex,G2: ( complex > complex ) > complex > complex,H: ( complex > complex ) > complex > complex] :
% 2.62/1.55 ( ( ( comp_c1610621014omplex @ F @ G2 )
% 2.62/1.55 = id_complex_complex )
% 2.62/1.55 => ( ( ( comp_c1610621014omplex @ G2 @ H )
% 2.62/1.55 = id_complex_complex )
% 2.62/1.55 => ( F = H ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_195_left__right__inverse__eq,axiom,
% 2.62/1.55 ! [F: complex > complex,G2: complex > complex,H: complex > complex] :
% 2.62/1.55 ( ( ( comp_c130555887omplex @ F @ G2 )
% 2.62/1.55 = id_complex )
% 2.62/1.55 => ( ( ( comp_c130555887omplex @ G2 @ H )
% 2.62/1.55 = id_complex )
% 2.62/1.55 => ( F = H ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_196_rewriteL__comp__comp,axiom,
% 2.62/1.55 ! [F: complex > complex,G2: complex > complex,L: complex > complex,H: complex > complex] :
% 2.62/1.55 ( ( ( comp_c130555887omplex @ F @ G2 )
% 2.62/1.55 = L )
% 2.62/1.55 => ( ( comp_c130555887omplex @ F @ ( comp_c130555887omplex @ G2 @ H ) )
% 2.62/1.55 = ( comp_c130555887omplex @ L @ H ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_197_rewriteR__comp__comp,axiom,
% 2.62/1.55 ! [G2: complex > complex,H: complex > complex,R: complex > complex,F: complex > complex] :
% 2.62/1.55 ( ( ( comp_c130555887omplex @ G2 @ H )
% 2.62/1.55 = R )
% 2.62/1.55 => ( ( comp_c130555887omplex @ ( comp_c130555887omplex @ F @ G2 ) @ H )
% 2.62/1.55 = ( comp_c130555887omplex @ F @ R ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_198_rewriteL__comp__comp2,axiom,
% 2.62/1.55 ! [F: complex > complex,G2: complex > complex,L1: complex > complex,L2: complex > complex,H: complex > complex,R: complex > complex] :
% 2.62/1.55 ( ( ( comp_c130555887omplex @ F @ G2 )
% 2.62/1.55 = ( comp_c130555887omplex @ L1 @ L2 ) )
% 2.62/1.55 => ( ( ( comp_c130555887omplex @ L2 @ H )
% 2.62/1.55 = R )
% 2.62/1.55 => ( ( comp_c130555887omplex @ F @ ( comp_c130555887omplex @ G2 @ H ) )
% 2.62/1.55 = ( comp_c130555887omplex @ L1 @ R ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_199_rewriteR__comp__comp2,axiom,
% 2.62/1.55 ! [G2: complex > complex,H: complex > complex,R1: complex > complex,R2: complex > complex,F: complex > complex,L: complex > complex] :
% 2.62/1.55 ( ( ( comp_c130555887omplex @ G2 @ H )
% 2.62/1.55 = ( comp_c130555887omplex @ R1 @ R2 ) )
% 2.62/1.55 => ( ( ( comp_c130555887omplex @ F @ R1 )
% 2.62/1.55 = L )
% 2.62/1.55 => ( ( comp_c130555887omplex @ ( comp_c130555887omplex @ F @ G2 ) @ H )
% 2.62/1.55 = ( comp_c130555887omplex @ L @ R2 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_200_holomorphic__cong,axiom,
% 2.62/1.55 ! [S2: set_complex,T: set_complex,F: complex > complex,G2: complex > complex] :
% 2.62/1.55 ( ( S2 = T )
% 2.62/1.55 => ( ! [X3: complex] :
% 2.62/1.55 ( ( member_complex @ X3 @ S2 )
% 2.62/1.55 => ( ( F @ X3 )
% 2.62/1.55 = ( G2 @ X3 ) ) )
% 2.62/1.55 => ( ( comple372758642hic_on @ F @ S2 )
% 2.62/1.55 = ( comple372758642hic_on @ G2 @ T ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_201_holomorphic__transform,axiom,
% 2.62/1.55 ! [F: complex > complex,S2: set_complex,G2: complex > complex] :
% 2.62/1.55 ( ( comple372758642hic_on @ F @ S2 )
% 2.62/1.55 => ( ! [X3: complex] :
% 2.62/1.55 ( ( member_complex @ X3 @ S2 )
% 2.62/1.55 => ( ( F @ X3 )
% 2.62/1.55 = ( G2 @ X3 ) ) )
% 2.62/1.55 => ( comple372758642hic_on @ G2 @ S2 ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_202_assms_I5_J,axiom,
% 2.62/1.55 ord_le701908932omplex @ ( image_58037603omplex @ f @ s ) @ t ).
% 2.62/1.55
% 2.62/1.55 thf(fact_203_nat__mult__eq__cancel__disj,axiom,
% 2.62/1.55 ! [K: nat,M: nat,N: nat] :
% 2.62/1.55 ( ( ( times_times_nat @ K @ M )
% 2.62/1.55 = ( times_times_nat @ K @ N ) )
% 2.62/1.55 = ( ( K = zero_zero_nat )
% 2.62/1.55 | ( M = N ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_204_mult__delta__left,axiom,
% 2.62/1.55 ! [B: $o,X2: complex,Y: complex] :
% 2.62/1.55 ( ( B
% 2.62/1.55 => ( ( times_times_complex @ ( if_complex @ B @ X2 @ zero_zero_complex ) @ Y )
% 2.62/1.55 = ( times_times_complex @ X2 @ Y ) ) )
% 2.62/1.55 & ( ~ B
% 2.62/1.55 => ( ( times_times_complex @ ( if_complex @ B @ X2 @ zero_zero_complex ) @ Y )
% 2.62/1.55 = zero_zero_complex ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_205_mult__delta__left,axiom,
% 2.62/1.55 ! [B: $o,X2: nat,Y: nat] :
% 2.62/1.55 ( ( B
% 2.62/1.55 => ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y )
% 2.62/1.55 = ( times_times_nat @ X2 @ Y ) ) )
% 2.62/1.55 & ( ~ B
% 2.62/1.55 => ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y )
% 2.62/1.55 = zero_zero_nat ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_206_mult__delta__left,axiom,
% 2.62/1.55 ! [B: $o,X2: real,Y: real] :
% 2.62/1.55 ( ( B
% 2.62/1.55 => ( ( times_times_real @ ( if_real @ B @ X2 @ zero_zero_real ) @ Y )
% 2.62/1.55 = ( times_times_real @ X2 @ Y ) ) )
% 2.62/1.55 & ( ~ B
% 2.62/1.55 => ( ( times_times_real @ ( if_real @ B @ X2 @ zero_zero_real ) @ Y )
% 2.62/1.55 = zero_zero_real ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_207_mult__delta__right,axiom,
% 2.62/1.55 ! [B: $o,X2: complex,Y: complex] :
% 2.62/1.55 ( ( B
% 2.62/1.55 => ( ( times_times_complex @ X2 @ ( if_complex @ B @ Y @ zero_zero_complex ) )
% 2.62/1.55 = ( times_times_complex @ X2 @ Y ) ) )
% 2.62/1.55 & ( ~ B
% 2.62/1.55 => ( ( times_times_complex @ X2 @ ( if_complex @ B @ Y @ zero_zero_complex ) )
% 2.62/1.55 = zero_zero_complex ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_208_mult__delta__right,axiom,
% 2.62/1.55 ! [B: $o,X2: nat,Y: nat] :
% 2.62/1.55 ( ( B
% 2.62/1.55 => ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y @ zero_zero_nat ) )
% 2.62/1.55 = ( times_times_nat @ X2 @ Y ) ) )
% 2.62/1.55 & ( ~ B
% 2.62/1.55 => ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y @ zero_zero_nat ) )
% 2.62/1.55 = zero_zero_nat ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_209_mult__delta__right,axiom,
% 2.62/1.55 ! [B: $o,X2: real,Y: real] :
% 2.62/1.55 ( ( B
% 2.62/1.55 => ( ( times_times_real @ X2 @ ( if_real @ B @ Y @ zero_zero_real ) )
% 2.62/1.55 = ( times_times_real @ X2 @ Y ) ) )
% 2.62/1.55 & ( ~ B
% 2.62/1.55 => ( ( times_times_real @ X2 @ ( if_real @ B @ Y @ zero_zero_real ) )
% 2.62/1.55 = zero_zero_real ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_210_arcosh__1,axiom,
% 2.62/1.55 ( ( arcosh_real @ one_one_real )
% 2.62/1.55 = zero_zero_real ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_211_arcosh__1,axiom,
% 2.62/1.55 ( ( arcosh_complex @ one_one_complex )
% 2.62/1.55 = zero_zero_complex ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_212_le__zero__eq,axiom,
% 2.62/1.55 ! [N: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.62/1.55 = ( N = zero_zero_nat ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_213_image__id,axiom,
% 2.62/1.55 ( ( image_real_real @ id_real )
% 2.62/1.55 = id_set_real ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_214_image__id,axiom,
% 2.62/1.55 ( ( image_944012797omplex @ id_complex_complex )
% 2.62/1.55 = id_set1618538368omplex ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_215_image__id,axiom,
% 2.62/1.55 ( ( image_58037603omplex @ id_complex )
% 2.62/1.55 = id_set_complex ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_216_set__times__mono2,axiom,
% 2.62/1.55 ! [C2: set_complex,D2: set_complex,E: set_complex,F3: set_complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ C2 @ D2 )
% 2.62/1.55 => ( ( ord_le701908932omplex @ E @ F3 )
% 2.62/1.55 => ( ord_le701908932omplex @ ( times_1316095593omplex @ C2 @ E ) @ ( times_1316095593omplex @ D2 @ F3 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_217_holomorphic__on__compose__gen,axiom,
% 2.62/1.55 ! [F: complex > complex,S2: set_complex,G2: complex > complex,T: set_complex] :
% 2.62/1.55 ( ( comple372758642hic_on @ F @ S2 )
% 2.62/1.55 => ( ( comple372758642hic_on @ G2 @ T )
% 2.62/1.55 => ( ( ord_le701908932omplex @ ( image_58037603omplex @ F @ S2 ) @ T )
% 2.62/1.55 => ( comple372758642hic_on @ ( comp_c130555887omplex @ G2 @ F ) @ S2 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_218_image__comp,axiom,
% 2.62/1.55 ! [F: complex > complex,G2: complex > complex,R: set_complex] :
% 2.62/1.55 ( ( image_58037603omplex @ F @ ( image_58037603omplex @ G2 @ R ) )
% 2.62/1.55 = ( image_58037603omplex @ ( comp_c130555887omplex @ F @ G2 ) @ R ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_219_image__eq__imp__comp,axiom,
% 2.62/1.55 ! [F: complex > complex,A2: set_complex,G2: complex > complex,B3: set_complex,H: complex > complex] :
% 2.62/1.55 ( ( ( image_58037603omplex @ F @ A2 )
% 2.62/1.55 = ( image_58037603omplex @ G2 @ B3 ) )
% 2.62/1.55 => ( ( image_58037603omplex @ ( comp_c130555887omplex @ H @ F ) @ A2 )
% 2.62/1.55 = ( image_58037603omplex @ ( comp_c130555887omplex @ H @ G2 ) @ B3 ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_220_zero__le,axiom,
% 2.62/1.55 ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_221_holomorphic__on__subset,axiom,
% 2.62/1.55 ! [F: complex > complex,S2: set_complex,T: set_complex] :
% 2.62/1.55 ( ( comple372758642hic_on @ F @ S2 )
% 2.62/1.55 => ( ( ord_le701908932omplex @ T @ S2 )
% 2.62/1.55 => ( comple372758642hic_on @ F @ T ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_222_set__times__mono2__b,axiom,
% 2.62/1.55 ! [C2: set_complex,D2: set_complex,E: set_complex,F3: set_complex,X2: complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ C2 @ D2 )
% 2.62/1.55 => ( ( ord_le701908932omplex @ E @ F3 )
% 2.62/1.55 => ( ( member_complex @ X2 @ ( times_1316095593omplex @ C2 @ E ) )
% 2.62/1.55 => ( member_complex @ X2 @ ( times_1316095593omplex @ D2 @ F3 ) ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_223_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 2.62/1.55 ! [A: nat,B: nat,C: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ A @ B )
% 2.62/1.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.62/1.55 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_224_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 2.62/1.55 ! [A: real,B: real,C: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_225_zero__le__mult__iff,axiom,
% 2.62/1.55 ! [A: real,B: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 2.62/1.55 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.62/1.55 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 2.62/1.55 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.62/1.55 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_226_mult__nonneg__nonpos2,axiom,
% 2.62/1.55 ! [A: nat,B: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.62/1.55 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 2.62/1.55 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_227_mult__nonneg__nonpos2,axiom,
% 2.62/1.55 ! [A: real,B: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.62/1.55 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_228_mult__nonpos__nonneg,axiom,
% 2.62/1.55 ! [A: nat,B: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.62/1.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.62/1.55 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_229_mult__nonpos__nonneg,axiom,
% 2.62/1.55 ! [A: real,B: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_230_mult__nonneg__nonpos,axiom,
% 2.62/1.55 ! [A: nat,B: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.62/1.55 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 2.62/1.55 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_231_mult__nonneg__nonpos,axiom,
% 2.62/1.55 ! [A: real,B: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.62/1.55 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_232_mult__nonneg__nonneg,axiom,
% 2.62/1.55 ! [A: nat,B: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.62/1.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.62/1.55 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_233_mult__nonneg__nonneg,axiom,
% 2.62/1.55 ! [A: real,B: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 2.62/1.55 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_234_split__mult__neg__le,axiom,
% 2.62/1.55 ! [A: nat,B: nat] :
% 2.62/1.55 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.62/1.55 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 2.62/1.55 | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.62/1.55 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 2.62/1.55 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_235_split__mult__neg__le,axiom,
% 2.62/1.55 ! [A: real,B: real] :
% 2.62/1.55 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.62/1.55 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 2.62/1.55 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.62/1.55 & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_236_mult__le__0__iff,axiom,
% 2.62/1.55 ! [A: real,B: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 2.62/1.55 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.62/1.55 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 2.62/1.55 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.62/1.55 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_237_mult__right__mono,axiom,
% 2.62/1.55 ! [A: nat,B: nat,C: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ A @ B )
% 2.62/1.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.62/1.55 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_238_mult__right__mono,axiom,
% 2.62/1.55 ! [A: real,B: real,C: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_239_mult__right__mono__neg,axiom,
% 2.62/1.55 ! [B: real,A: real,C: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ B @ A )
% 2.62/1.55 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_240_mult__left__mono,axiom,
% 2.62/1.55 ! [A: nat,B: nat,C: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ A @ B )
% 2.62/1.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.62/1.55 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_241_mult__left__mono,axiom,
% 2.62/1.55 ! [A: real,B: real,C: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_242_mult__nonpos__nonpos,axiom,
% 2.62/1.55 ! [A: real,B: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.62/1.55 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 2.62/1.55 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_243_mult__left__mono__neg,axiom,
% 2.62/1.55 ! [B: real,A: real,C: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ B @ A )
% 2.62/1.55 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_244_split__mult__pos__le,axiom,
% 2.62/1.55 ! [A: real,B: real] :
% 2.62/1.55 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.62/1.55 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 2.62/1.55 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.62/1.55 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 2.62/1.55 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_245_zero__le__square,axiom,
% 2.62/1.55 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_246_mult__mono_H,axiom,
% 2.62/1.55 ! [A: nat,B: nat,C: nat,D: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ A @ B )
% 2.62/1.55 => ( ( ord_less_eq_nat @ C @ D )
% 2.62/1.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.62/1.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.62/1.55 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_247_mult__mono_H,axiom,
% 2.62/1.55 ! [A: real,B: real,C: real,D: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.62/1.55 => ( ( ord_less_eq_real @ C @ D )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_248_mult__mono,axiom,
% 2.62/1.55 ! [A: nat,B: nat,C: nat,D: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ A @ B )
% 2.62/1.55 => ( ( ord_less_eq_nat @ C @ D )
% 2.62/1.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.62/1.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.62/1.55 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_249_mult__mono,axiom,
% 2.62/1.55 ! [A: real,B: real,C: real,D: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.62/1.55 => ( ( ord_less_eq_real @ C @ D )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_250_zero__le__one,axiom,
% 2.62/1.55 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 2.62/1.55
% 2.62/1.55 thf(fact_251_zero__le__one,axiom,
% 2.62/1.55 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 2.62/1.55
% 2.62/1.55 thf(fact_252_not__one__le__zero,axiom,
% 2.62/1.55 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_253_not__one__le__zero,axiom,
% 2.62/1.55 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_254_holomorphic__on__compose,axiom,
% 2.62/1.55 ! [F: complex > complex,S2: set_complex,G2: complex > complex] :
% 2.62/1.55 ( ( comple372758642hic_on @ F @ S2 )
% 2.62/1.55 => ( ( comple372758642hic_on @ G2 @ ( image_58037603omplex @ F @ S2 ) )
% 2.62/1.55 => ( comple372758642hic_on @ ( comp_c130555887omplex @ G2 @ F ) @ S2 ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_255_deriv__left__inverse,axiom,
% 2.62/1.55 ! [F: complex > complex,S: set_complex,G2: complex > complex,T2: set_complex,W2: complex] :
% 2.62/1.55 ( ( comple372758642hic_on @ F @ S )
% 2.62/1.55 => ( ( comple372758642hic_on @ G2 @ T2 )
% 2.62/1.55 => ( ( topolo935673511omplex @ S )
% 2.62/1.55 => ( ( topolo935673511omplex @ T2 )
% 2.62/1.55 => ( ( ord_le701908932omplex @ ( image_58037603omplex @ F @ S ) @ T2 )
% 2.62/1.55 => ( ! [Z3: complex] :
% 2.62/1.55 ( ( member_complex @ Z3 @ S )
% 2.62/1.55 => ( ( G2 @ ( F @ Z3 ) )
% 2.62/1.55 = Z3 ) )
% 2.62/1.55 => ( ( member_complex @ W2 @ S )
% 2.62/1.55 => ( ( times_times_complex @ ( deriv_complex @ F @ W2 ) @ ( deriv_complex @ G2 @ ( F @ W2 ) ) )
% 2.62/1.55 = one_one_complex ) ) ) ) ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_256_mult__left__le,axiom,
% 2.62/1.55 ! [C: nat,A: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 2.62/1.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.62/1.55 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_257_mult__left__le,axiom,
% 2.62/1.55 ! [C: real,A: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ C @ one_one_real )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_258_mult__le__one,axiom,
% 2.62/1.55 ! [A: nat,B: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ A @ one_one_nat )
% 2.62/1.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.62/1.55 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 2.62/1.55 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_259_mult__le__one,axiom,
% 2.62/1.55 ! [A: real,B: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ A @ one_one_real )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 2.62/1.55 => ( ( ord_less_eq_real @ B @ one_one_real )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_260_mult__right__le__one__le,axiom,
% 2.62/1.55 ! [X2: real,Y: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 2.62/1.55 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ X2 @ Y ) @ X2 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_261_mult__left__le__one__le,axiom,
% 2.62/1.55 ! [X2: real,Y: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 2.62/1.55 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ Y @ X2 ) @ X2 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_262_mult__eq__1,axiom,
% 2.62/1.55 ! [A: nat,B: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.62/1.55 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 2.62/1.55 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 2.62/1.55 => ( ( ( times_times_nat @ A @ B )
% 2.62/1.55 = one_one_nat )
% 2.62/1.55 = ( ( A = one_one_nat )
% 2.62/1.55 & ( B = one_one_nat ) ) ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_263_mult__eq__1,axiom,
% 2.62/1.55 ! [A: real,B: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.62/1.55 => ( ( ord_less_eq_real @ A @ one_one_real )
% 2.62/1.55 => ( ( ord_less_eq_real @ B @ one_one_real )
% 2.62/1.55 => ( ( ( times_times_real @ A @ B )
% 2.62/1.55 = one_one_real )
% 2.62/1.55 = ( ( A = one_one_real )
% 2.62/1.55 & ( B = one_one_real ) ) ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_264_artanh__0,axiom,
% 2.62/1.55 ( ( artanh_real @ zero_zero_real )
% 2.62/1.55 = zero_zero_real ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_265_artanh__0,axiom,
% 2.62/1.55 ( ( artanh_complex @ zero_zero_complex )
% 2.62/1.55 = zero_zero_complex ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_266_arsinh__0,axiom,
% 2.62/1.55 ( ( arsinh_real @ zero_zero_real )
% 2.62/1.55 = zero_zero_real ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_267_arsinh__0,axiom,
% 2.62/1.55 ( ( arsinh_complex @ zero_zero_complex )
% 2.62/1.55 = zero_zero_complex ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_268_subset__antisym,axiom,
% 2.62/1.55 ! [A2: set_complex,B3: set_complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ A2 @ B3 )
% 2.62/1.55 => ( ( ord_le701908932omplex @ B3 @ A2 )
% 2.62/1.55 => ( A2 = B3 ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_269_image__eqI,axiom,
% 2.62/1.55 ! [B: complex,F: complex > complex,X2: complex,A2: set_complex] :
% 2.62/1.55 ( ( B
% 2.62/1.55 = ( F @ X2 ) )
% 2.62/1.55 => ( ( member_complex @ X2 @ A2 )
% 2.62/1.55 => ( member_complex @ B @ ( image_58037603omplex @ F @ A2 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_270_subsetI,axiom,
% 2.62/1.55 ! [A2: set_complex,B3: set_complex] :
% 2.62/1.55 ( ! [X3: complex] :
% 2.62/1.55 ( ( member_complex @ X3 @ A2 )
% 2.62/1.55 => ( member_complex @ X3 @ B3 ) )
% 2.62/1.55 => ( ord_le701908932omplex @ A2 @ B3 ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_271_le0,axiom,
% 2.62/1.55 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_272_bot__nat__0_Oextremum,axiom,
% 2.62/1.55 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_273_less__eq__nat_Osimps_I1_J,axiom,
% 2.62/1.55 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_274_le__0__eq,axiom,
% 2.62/1.55 ! [N: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.62/1.55 = ( N = zero_zero_nat ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_275_bot__nat__0_Oextremum__unique,axiom,
% 2.62/1.55 ! [A: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.62/1.55 = ( A = zero_zero_nat ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_276_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.62/1.55 ! [A: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.62/1.55 => ( A = zero_zero_nat ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_277_mult__le__mono2,axiom,
% 2.62/1.55 ! [I: nat,J: nat,K: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ I @ J )
% 2.62/1.55 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_278_mult__le__mono1,axiom,
% 2.62/1.55 ! [I: nat,J: nat,K: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ I @ J )
% 2.62/1.55 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_279_mult__le__mono,axiom,
% 2.62/1.55 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ I @ J )
% 2.62/1.55 => ( ( ord_less_eq_nat @ K @ L )
% 2.62/1.55 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_280_le__square,axiom,
% 2.62/1.55 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_281_le__cube,axiom,
% 2.62/1.55 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_282_imageI,axiom,
% 2.62/1.55 ! [X2: complex,A2: set_complex,F: complex > complex] :
% 2.62/1.55 ( ( member_complex @ X2 @ A2 )
% 2.62/1.55 => ( member_complex @ ( F @ X2 ) @ ( image_58037603omplex @ F @ A2 ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_283_image__iff,axiom,
% 2.62/1.55 ! [Z: complex,F: complex > complex,A2: set_complex] :
% 2.62/1.55 ( ( member_complex @ Z @ ( image_58037603omplex @ F @ A2 ) )
% 2.62/1.55 = ( ? [X: complex] :
% 2.62/1.55 ( ( member_complex @ X @ A2 )
% 2.62/1.55 & ( Z
% 2.62/1.55 = ( F @ X ) ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_284_bex__imageD,axiom,
% 2.62/1.55 ! [F: complex > complex,A2: set_complex,P: complex > $o] :
% 2.62/1.55 ( ? [X4: complex] :
% 2.62/1.55 ( ( member_complex @ X4 @ ( image_58037603omplex @ F @ A2 ) )
% 2.62/1.55 & ( P @ X4 ) )
% 2.62/1.55 => ? [X3: complex] :
% 2.62/1.55 ( ( member_complex @ X3 @ A2 )
% 2.62/1.55 & ( P @ ( F @ X3 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_285_image__cong,axiom,
% 2.62/1.55 ! [M2: set_complex,N2: set_complex,F: complex > complex,G2: complex > complex] :
% 2.62/1.55 ( ( M2 = N2 )
% 2.62/1.55 => ( ! [X3: complex] :
% 2.62/1.55 ( ( member_complex @ X3 @ N2 )
% 2.62/1.55 => ( ( F @ X3 )
% 2.62/1.55 = ( G2 @ X3 ) ) )
% 2.62/1.55 => ( ( image_58037603omplex @ F @ M2 )
% 2.62/1.55 = ( image_58037603omplex @ G2 @ N2 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_286_ball__imageD,axiom,
% 2.62/1.55 ! [F: complex > complex,A2: set_complex,P: complex > $o] :
% 2.62/1.55 ( ! [X3: complex] :
% 2.62/1.55 ( ( member_complex @ X3 @ ( image_58037603omplex @ F @ A2 ) )
% 2.62/1.55 => ( P @ X3 ) )
% 2.62/1.55 => ! [X4: complex] :
% 2.62/1.55 ( ( member_complex @ X4 @ A2 )
% 2.62/1.55 => ( P @ ( F @ X4 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_287_rev__image__eqI,axiom,
% 2.62/1.55 ! [X2: complex,A2: set_complex,B: complex,F: complex > complex] :
% 2.62/1.55 ( ( member_complex @ X2 @ A2 )
% 2.62/1.55 => ( ( B
% 2.62/1.55 = ( F @ X2 ) )
% 2.62/1.55 => ( member_complex @ B @ ( image_58037603omplex @ F @ A2 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_288_in__mono,axiom,
% 2.62/1.55 ! [A2: set_complex,B3: set_complex,X2: complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ A2 @ B3 )
% 2.62/1.55 => ( ( member_complex @ X2 @ A2 )
% 2.62/1.55 => ( member_complex @ X2 @ B3 ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_289_subsetD,axiom,
% 2.62/1.55 ! [A2: set_complex,B3: set_complex,C: complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ A2 @ B3 )
% 2.62/1.55 => ( ( member_complex @ C @ A2 )
% 2.62/1.55 => ( member_complex @ C @ B3 ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_290_equalityE,axiom,
% 2.62/1.55 ! [A2: set_complex,B3: set_complex] :
% 2.62/1.55 ( ( A2 = B3 )
% 2.62/1.55 => ~ ( ( ord_le701908932omplex @ A2 @ B3 )
% 2.62/1.55 => ~ ( ord_le701908932omplex @ B3 @ A2 ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_291_subset__eq,axiom,
% 2.62/1.55 ( ord_le701908932omplex
% 2.62/1.55 = ( ^ [A5: set_complex,B5: set_complex] :
% 2.62/1.55 ! [X: complex] :
% 2.62/1.55 ( ( member_complex @ X @ A5 )
% 2.62/1.55 => ( member_complex @ X @ B5 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_292_equalityD1,axiom,
% 2.62/1.55 ! [A2: set_complex,B3: set_complex] :
% 2.62/1.55 ( ( A2 = B3 )
% 2.62/1.55 => ( ord_le701908932omplex @ A2 @ B3 ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_293_equalityD2,axiom,
% 2.62/1.55 ! [A2: set_complex,B3: set_complex] :
% 2.62/1.55 ( ( A2 = B3 )
% 2.62/1.55 => ( ord_le701908932omplex @ B3 @ A2 ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_294_subset__iff,axiom,
% 2.62/1.55 ( ord_le701908932omplex
% 2.62/1.55 = ( ^ [A5: set_complex,B5: set_complex] :
% 2.62/1.55 ! [T3: complex] :
% 2.62/1.55 ( ( member_complex @ T3 @ A5 )
% 2.62/1.55 => ( member_complex @ T3 @ B5 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_295_subset__refl,axiom,
% 2.62/1.55 ! [A2: set_complex] : ( ord_le701908932omplex @ A2 @ A2 ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_296_Collect__mono,axiom,
% 2.62/1.55 ! [P: complex > $o,Q2: complex > $o] :
% 2.62/1.55 ( ! [X3: complex] :
% 2.62/1.55 ( ( P @ X3 )
% 2.62/1.55 => ( Q2 @ X3 ) )
% 2.62/1.55 => ( ord_le701908932omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q2 ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_297_subset__trans,axiom,
% 2.62/1.55 ! [A2: set_complex,B3: set_complex,C2: set_complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ A2 @ B3 )
% 2.62/1.55 => ( ( ord_le701908932omplex @ B3 @ C2 )
% 2.62/1.55 => ( ord_le701908932omplex @ A2 @ C2 ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_298_set__eq__subset,axiom,
% 2.62/1.55 ( ( ^ [Y2: set_complex,Z4: set_complex] : ( Y2 = Z4 ) )
% 2.62/1.55 = ( ^ [A5: set_complex,B5: set_complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ A5 @ B5 )
% 2.62/1.55 & ( ord_le701908932omplex @ B5 @ A5 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_299_Collect__mono__iff,axiom,
% 2.62/1.55 ! [P: complex > $o,Q2: complex > $o] :
% 2.62/1.55 ( ( ord_le701908932omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q2 ) )
% 2.62/1.55 = ( ! [X: complex] :
% 2.62/1.55 ( ( P @ X )
% 2.62/1.55 => ( Q2 @ X ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_300_image__mono,axiom,
% 2.62/1.55 ! [A2: set_complex,B3: set_complex,F: complex > complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ A2 @ B3 )
% 2.62/1.55 => ( ord_le701908932omplex @ ( image_58037603omplex @ F @ A2 ) @ ( image_58037603omplex @ F @ B3 ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_301_image__subsetI,axiom,
% 2.62/1.55 ! [A2: set_complex,F: complex > complex,B3: set_complex] :
% 2.62/1.55 ( ! [X3: complex] :
% 2.62/1.55 ( ( member_complex @ X3 @ A2 )
% 2.62/1.55 => ( member_complex @ ( F @ X3 ) @ B3 ) )
% 2.62/1.55 => ( ord_le701908932omplex @ ( image_58037603omplex @ F @ A2 ) @ B3 ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_302_subset__imageE,axiom,
% 2.62/1.55 ! [B3: set_complex,F: complex > complex,A2: set_complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ B3 @ ( image_58037603omplex @ F @ A2 ) )
% 2.62/1.55 => ~ ! [C3: set_complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ C3 @ A2 )
% 2.62/1.55 => ( B3
% 2.62/1.55 != ( image_58037603omplex @ F @ C3 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_303_image__subset__iff,axiom,
% 2.62/1.55 ! [F: complex > complex,A2: set_complex,B3: set_complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ ( image_58037603omplex @ F @ A2 ) @ B3 )
% 2.62/1.55 = ( ! [X: complex] :
% 2.62/1.55 ( ( member_complex @ X @ A2 )
% 2.62/1.55 => ( member_complex @ ( F @ X ) @ B3 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_304_subset__image__iff,axiom,
% 2.62/1.55 ! [B3: set_complex,F: complex > complex,A2: set_complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ B3 @ ( image_58037603omplex @ F @ A2 ) )
% 2.62/1.55 = ( ? [AA: set_complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ AA @ A2 )
% 2.62/1.55 & ( B3
% 2.62/1.55 = ( image_58037603omplex @ F @ AA ) ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_305_order__refl,axiom,
% 2.62/1.55 ! [X2: set_complex] : ( ord_le701908932omplex @ X2 @ X2 ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_306_order__refl,axiom,
% 2.62/1.55 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_307_order__refl,axiom,
% 2.62/1.55 ! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_308_kuhn__labelling__lemma_H,axiom,
% 2.62/1.55 ! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q2: nat > $o] :
% 2.62/1.55 ( ! [X3: nat > real] :
% 2.62/1.55 ( ( P @ X3 )
% 2.62/1.55 => ( P @ ( F @ X3 ) ) )
% 2.62/1.55 => ( ! [X3: nat > real] :
% 2.62/1.55 ( ( P @ X3 )
% 2.62/1.55 => ! [I2: nat] :
% 2.62/1.55 ( ( Q2 @ I2 )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I2 ) )
% 2.62/1.55 & ( ord_less_eq_real @ ( X3 @ I2 ) @ one_one_real ) ) ) )
% 2.62/1.55 => ? [L3: ( nat > real ) > nat > nat] :
% 2.62/1.55 ( ! [X4: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L3 @ X4 @ I3 ) @ one_one_nat )
% 2.62/1.55 & ! [X4: nat > real,I3: nat] :
% 2.62/1.55 ( ( ( P @ X4 )
% 2.62/1.55 & ( Q2 @ I3 )
% 2.62/1.55 & ( ( X4 @ I3 )
% 2.62/1.55 = zero_zero_real ) )
% 2.62/1.55 => ( ( L3 @ X4 @ I3 )
% 2.62/1.55 = zero_zero_nat ) )
% 2.62/1.55 & ! [X4: nat > real,I3: nat] :
% 2.62/1.55 ( ( ( P @ X4 )
% 2.62/1.55 & ( Q2 @ I3 )
% 2.62/1.55 & ( ( X4 @ I3 )
% 2.62/1.55 = one_one_real ) )
% 2.62/1.55 => ( ( L3 @ X4 @ I3 )
% 2.62/1.55 = one_one_nat ) )
% 2.62/1.55 & ! [X4: nat > real,I3: nat] :
% 2.62/1.55 ( ( ( P @ X4 )
% 2.62/1.55 & ( Q2 @ I3 )
% 2.62/1.55 & ( ( L3 @ X4 @ I3 )
% 2.62/1.55 = zero_zero_nat ) )
% 2.62/1.55 => ( ord_less_eq_real @ ( X4 @ I3 ) @ ( F @ X4 @ I3 ) ) )
% 2.62/1.55 & ! [X4: nat > real,I3: nat] :
% 2.62/1.55 ( ( ( P @ X4 )
% 2.62/1.55 & ( Q2 @ I3 )
% 2.62/1.55 & ( ( L3 @ X4 @ I3 )
% 2.62/1.55 = one_one_nat ) )
% 2.62/1.55 => ( ord_less_eq_real @ ( F @ X4 @ I3 ) @ ( X4 @ I3 ) ) ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_309_Sup_OSUP__id__eq,axiom,
% 2.62/1.55 ! [Sup: set_real > real,A2: set_real] :
% 2.62/1.55 ( ( Sup @ ( image_real_real @ id_real @ A2 ) )
% 2.62/1.55 = ( Sup @ A2 ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_310_Sup_OSUP__id__eq,axiom,
% 2.62/1.55 ! [Sup: set_complex_complex > complex > complex,A2: set_complex_complex] :
% 2.62/1.55 ( ( Sup @ ( image_944012797omplex @ id_complex_complex @ A2 ) )
% 2.62/1.55 = ( Sup @ A2 ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_311_Sup_OSUP__id__eq,axiom,
% 2.62/1.55 ! [Sup: set_complex > complex,A2: set_complex] :
% 2.62/1.55 ( ( Sup @ ( image_58037603omplex @ id_complex @ A2 ) )
% 2.62/1.55 = ( Sup @ A2 ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_312_le__refl,axiom,
% 2.62/1.55 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_313_le__trans,axiom,
% 2.62/1.55 ! [I: nat,J: nat,K: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ I @ J )
% 2.62/1.55 => ( ( ord_less_eq_nat @ J @ K )
% 2.62/1.55 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_314_eq__imp__le,axiom,
% 2.62/1.55 ! [M: nat,N: nat] :
% 2.62/1.55 ( ( M = N )
% 2.62/1.55 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_315_le__antisym,axiom,
% 2.62/1.55 ! [M: nat,N: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ M @ N )
% 2.62/1.55 => ( ( ord_less_eq_nat @ N @ M )
% 2.62/1.55 => ( M = N ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_316_nat__le__linear,axiom,
% 2.62/1.55 ! [M: nat,N: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ M @ N )
% 2.62/1.55 | ( ord_less_eq_nat @ N @ M ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_317_Nat_Oex__has__greatest__nat,axiom,
% 2.62/1.55 ! [P: nat > $o,K: nat,B: nat] :
% 2.62/1.55 ( ( P @ K )
% 2.62/1.55 => ( ! [Y3: nat] :
% 2.62/1.55 ( ( P @ Y3 )
% 2.62/1.55 => ( ord_less_eq_nat @ Y3 @ B ) )
% 2.62/1.55 => ? [X3: nat] :
% 2.62/1.55 ( ( P @ X3 )
% 2.62/1.55 & ! [Y4: nat] :
% 2.62/1.55 ( ( P @ Y4 )
% 2.62/1.55 => ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_318_dual__order_Oantisym,axiom,
% 2.62/1.55 ! [B: set_complex,A: set_complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ B @ A )
% 2.62/1.55 => ( ( ord_le701908932omplex @ A @ B )
% 2.62/1.55 => ( A = B ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_319_dual__order_Oantisym,axiom,
% 2.62/1.55 ! [B: nat,A: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ B @ A )
% 2.62/1.55 => ( ( ord_less_eq_nat @ A @ B )
% 2.62/1.55 => ( A = B ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_320_dual__order_Oantisym,axiom,
% 2.62/1.55 ! [B: real,A: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ B @ A )
% 2.62/1.55 => ( ( ord_less_eq_real @ A @ B )
% 2.62/1.55 => ( A = B ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_321_dual__order_Oeq__iff,axiom,
% 2.62/1.55 ( ( ^ [Y2: set_complex,Z4: set_complex] : ( Y2 = Z4 ) )
% 2.62/1.55 = ( ^ [A3: set_complex,B2: set_complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ B2 @ A3 )
% 2.62/1.55 & ( ord_le701908932omplex @ A3 @ B2 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_322_dual__order_Oeq__iff,axiom,
% 2.62/1.55 ( ( ^ [Y2: nat,Z4: nat] : ( Y2 = Z4 ) )
% 2.62/1.55 = ( ^ [A3: nat,B2: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ B2 @ A3 )
% 2.62/1.55 & ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_323_dual__order_Oeq__iff,axiom,
% 2.62/1.55 ( ( ^ [Y2: real,Z4: real] : ( Y2 = Z4 ) )
% 2.62/1.55 = ( ^ [A3: real,B2: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ B2 @ A3 )
% 2.62/1.55 & ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_324_dual__order_Otrans,axiom,
% 2.62/1.55 ! [B: set_complex,A: set_complex,C: set_complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ B @ A )
% 2.62/1.55 => ( ( ord_le701908932omplex @ C @ B )
% 2.62/1.55 => ( ord_le701908932omplex @ C @ A ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_325_dual__order_Otrans,axiom,
% 2.62/1.55 ! [B: nat,A: nat,C: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ B @ A )
% 2.62/1.55 => ( ( ord_less_eq_nat @ C @ B )
% 2.62/1.55 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_326_dual__order_Otrans,axiom,
% 2.62/1.55 ! [B: real,A: real,C: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ B @ A )
% 2.62/1.55 => ( ( ord_less_eq_real @ C @ B )
% 2.62/1.55 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_327_linorder__wlog,axiom,
% 2.62/1.55 ! [P: nat > nat > $o,A: nat,B: nat] :
% 2.62/1.55 ( ! [A4: nat,B4: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ A4 @ B4 )
% 2.62/1.55 => ( P @ A4 @ B4 ) )
% 2.62/1.55 => ( ! [A4: nat,B4: nat] :
% 2.62/1.55 ( ( P @ B4 @ A4 )
% 2.62/1.55 => ( P @ A4 @ B4 ) )
% 2.62/1.55 => ( P @ A @ B ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_328_linorder__wlog,axiom,
% 2.62/1.55 ! [P: real > real > $o,A: real,B: real] :
% 2.62/1.55 ( ! [A4: real,B4: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ A4 @ B4 )
% 2.62/1.55 => ( P @ A4 @ B4 ) )
% 2.62/1.55 => ( ! [A4: real,B4: real] :
% 2.62/1.55 ( ( P @ B4 @ A4 )
% 2.62/1.55 => ( P @ A4 @ B4 ) )
% 2.62/1.55 => ( P @ A @ B ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_329_dual__order_Orefl,axiom,
% 2.62/1.55 ! [A: set_complex] : ( ord_le701908932omplex @ A @ A ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_330_dual__order_Orefl,axiom,
% 2.62/1.55 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_331_dual__order_Orefl,axiom,
% 2.62/1.55 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_332_order__trans,axiom,
% 2.62/1.55 ! [X2: set_complex,Y: set_complex,Z: set_complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ X2 @ Y )
% 2.62/1.55 => ( ( ord_le701908932omplex @ Y @ Z )
% 2.62/1.55 => ( ord_le701908932omplex @ X2 @ Z ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_333_order__trans,axiom,
% 2.62/1.55 ! [X2: nat,Y: nat,Z: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ X2 @ Y )
% 2.62/1.55 => ( ( ord_less_eq_nat @ Y @ Z )
% 2.62/1.55 => ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_334_order__trans,axiom,
% 2.62/1.55 ! [X2: real,Y: real,Z: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ X2 @ Y )
% 2.62/1.55 => ( ( ord_less_eq_real @ Y @ Z )
% 2.62/1.55 => ( ord_less_eq_real @ X2 @ Z ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_335_order__class_Oorder_Oantisym,axiom,
% 2.62/1.55 ! [A: set_complex,B: set_complex] :
% 2.62/1.55 ( ( ord_le701908932omplex @ A @ B )
% 2.62/1.55 => ( ( ord_le701908932omplex @ B @ A )
% 2.62/1.55 => ( A = B ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_336_order__class_Oorder_Oantisym,axiom,
% 2.62/1.55 ! [A: nat,B: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ A @ B )
% 2.62/1.55 => ( ( ord_less_eq_nat @ B @ A )
% 2.62/1.55 => ( A = B ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_337_order__class_Oorder_Oantisym,axiom,
% 2.62/1.55 ! [A: real,B: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.62/1.55 => ( ( ord_less_eq_real @ B @ A )
% 2.62/1.55 => ( A = B ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_338_ord__le__eq__trans,axiom,
% 2.62/1.55 ! [A: nat,B: nat,C: nat] :
% 2.62/1.55 ( ( ord_less_eq_nat @ A @ B )
% 2.62/1.55 => ( ( B = C )
% 2.62/1.55 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_339_ord__le__eq__trans,axiom,
% 2.62/1.55 ! [A: real,B: real,C: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.62/1.55 => ( ( B = C )
% 2.62/1.55 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_340_landau__omega_OR__mult__right__mono,axiom,
% 2.62/1.55 ! [B: real,A: real,C: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ B @ A )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ B @ C ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_341_landau__omega_OR__mult__left__mono,axiom,
% 2.62/1.55 ! [B: real,A: real,C: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ B @ A )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ C @ B ) @ ( times_times_real @ C @ A ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_342_landau__omega_OR__linear,axiom,
% 2.62/1.55 ! [Y: real,X2: real] :
% 2.62/1.55 ( ~ ( ord_less_eq_real @ Y @ X2 )
% 2.62/1.55 => ( ord_less_eq_real @ X2 @ Y ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_343_landau__omega_OR__trans,axiom,
% 2.62/1.55 ! [B: real,A: real,C: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ B @ A )
% 2.62/1.55 => ( ( ord_less_eq_real @ C @ B )
% 2.62/1.55 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_344_landau__omega_OR__refl,axiom,
% 2.62/1.55 ! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_345_landau__o_OR__linear,axiom,
% 2.62/1.55 ! [X2: real,Y: real] :
% 2.62/1.55 ( ~ ( ord_less_eq_real @ X2 @ Y )
% 2.62/1.55 => ( ord_less_eq_real @ Y @ X2 ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_346_landau__o_OR__trans,axiom,
% 2.62/1.55 ! [A: real,B: real,C: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.62/1.55 => ( ( ord_less_eq_real @ B @ C )
% 2.62/1.55 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_347_landau__o_OR__mult__left__mono,axiom,
% 2.62/1.55 ! [A: real,B: real,C: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_348_landau__o_OR__mult__right__mono,axiom,
% 2.62/1.55 ! [A: real,B: real,C: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.62/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 2.62/1.55 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_349_ln__ge__zero,axiom,
% 2.62/1.55 ! [X2: real] :
% 2.62/1.55 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 2.62/1.55 => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_350_Ln__1,axiom,
% 2.62/1.55 ( ( ln_ln_complex @ one_one_complex )
% 2.62/1.55 = zero_zero_complex ) ).
% 2.62/1.55
% 2.62/1.55 thf(fact_351_holomorphic__fun__eq__0__on__connected,axiom,
% 2.62/1.55 ! [F: complex > complex,S: set_complex,Z: complex,W2: complex] :
% 2.62/1.55 ( ( comple372758642hic_on @ F @ S )
% 2.62/1.55 => ( ( topolo935673511omplex @ S )
% 2.62/1.55 => ( ( topolo2127351575omplex @ S )
% 2.62/1.55 => ( ! [N3: nat] :
% 2.62/1.55 ( ( compow1098280738omplex @ N3 @ deriv_complex @ F @ Z )
% 2.62/1.55 = zero_zero_complex )
% 2.62/1.55 => ( ( member_complex @ Z @ S )
% 2.62/1.55 => ( ( member_complex @ W2 @ S )
% 2.62/1.55 => ( ( F @ W2 )
% 2.62/1.55 = zero_zero_complex ) ) ) ) ) ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 2.62/1.55 ! [X2: nat,Y: nat] :
% 2.62/1.55 ( ( if_nat @ $false @ X2 @ Y )
% 2.62/1.55 = Y ) ).
% 2.62/1.55
% 2.62/1.55 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 2.62/1.55 ! [X2: nat,Y: nat] :
% 2.62/1.55 ( ( if_nat @ $true @ X2 @ Y )
% 2.62/1.55 = X2 ) ).
% 2.62/1.55
% 2.62/1.55 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 2.62/1.55 ! [X2: real,Y: real] :
% 2.62/1.55 ( ( if_real @ $false @ X2 @ Y )
% 2.62/1.55 = Y ) ).
% 2.62/1.55
% 2.62/1.55 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 2.62/1.55 ! [X2: real,Y: real] :
% 2.62/1.55 ( ( if_real @ $true @ X2 @ Y )
% 2.62/1.55 = X2 ) ).
% 2.62/1.55
% 2.62/1.55 thf(help_If_3_1_If_001t__Complex__Ocomplex_T,axiom,
% 2.62/1.55 ! [P: $o] :
% 2.62/1.55 ( ( P = $true )
% 2.62/1.55 | ( P = $false ) ) ).
% 2.62/1.55
% 2.62/1.55 thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 2.62/1.55 ! [X2: complex,Y: complex] :
% 2.62/1.55 ( ( if_complex @ $false @ X2 @ Y )
% 2.62/1.55 = Y ) ).
% 2.62/1.55
% 2.62/1.55 thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 2.62/1.55 ! [X2: complex,Y: complex] :
% 2.62/1.55 ( ( if_complex @ $true @ X2 @ Y )
% 2.62/1.55 = X2 ) ).
% 2.62/1.55
% 2.62/1.55 thf(conj_0,conjecture,
% 2.62/1.55 ( ( deriv_complex @ id_complex @ w )
% 2.62/1.55 = one_one_complex ) ).
% 2.62/1.55 ------------------------
% 2.62/1.58 % (15307)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/4Mi)
% 2.62/1.58 % (15308)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/27Mi)
% 2.62/1.58 % (15307)Instruction limit reached!
% 2.62/1.58 % (15307)------------------------------
% 2.62/1.58 % (15307)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.58 % (15307)Termination reason: Unknown
% 2.62/1.58 % (15307)Termination phase: shuffling
% 2.62/1.58
% 2.62/1.58 % (15307)Memory used [KB]: 1535
% 2.62/1.58 % (15307)Time elapsed: 0.003 s
% 2.62/1.58 % (15307)Instructions burned: 6 (million)
% 2.62/1.58 % (15307)------------------------------
% 2.62/1.58 % (15307)------------------------------
% 2.62/1.58 % (15306)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/183Mi)
% 2.62/1.58 % (15309)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/2Mi)
% 2.62/1.58 % (15310)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/2Mi)
% 2.62/1.59 % (15311)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/275Mi)
% 2.62/1.59 % (15312)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/18Mi)
% 2.62/1.59 % (15308)Instruction limit reached!
% 2.62/1.59 % (15308)------------------------------
% 2.62/1.59 % (15308)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.59 % (15308)Termination reason: Unknown
% 2.62/1.59 % (15308)Termination phase: Property scanning
% 2.62/1.59
% 2.62/1.59 % (15308)Memory used [KB]: 1791
% 2.62/1.59 % (15308)Time elapsed: 0.010 s
% 2.62/1.59 % (15308)Instructions burned: 27 (million)
% 2.62/1.59 % (15308)------------------------------
% 2.62/1.59 % (15308)------------------------------
% 2.62/1.59 % (15309)Instruction limit reached!
% 2.62/1.59 % (15309)------------------------------
% 2.62/1.59 % (15309)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.59 % (15309)Termination reason: Unknown
% 2.62/1.59 % (15309)Termination phase: shuffling
% 2.62/1.59
% 2.62/1.59 % (15309)Memory used [KB]: 1407
% 2.62/1.59 % (15310)Instruction limit reached!
% 2.62/1.59 % (15310)------------------------------
% 2.62/1.59 % (15310)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.59 % (15310)Termination reason: Unknown
% 2.62/1.59 % (15310)Termination phase: shuffling
% 2.62/1.59
% 2.62/1.59 % (15310)Memory used [KB]: 1407
% 2.62/1.59 % (15310)Time elapsed: 0.004 s
% 2.62/1.59 % (15310)Instructions burned: 3 (million)
% 2.62/1.59 % (15310)------------------------------
% 2.62/1.59 % (15310)------------------------------
% 2.62/1.59 % (15309)Time elapsed: 0.004 s
% 2.62/1.59 % (15309)Instructions burned: 3 (million)
% 2.62/1.59 % (15309)------------------------------
% 2.62/1.59 % (15309)------------------------------
% 2.62/1.59 % (15313)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/3Mi)
% 2.62/1.59 % (15313)Instruction limit reached!
% 2.62/1.59 % (15313)------------------------------
% 2.62/1.59 % (15313)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.59 % (15313)Termination reason: Unknown
% 2.62/1.59 % (15313)Termination phase: shuffling
% 2.62/1.59
% 2.62/1.59 % (15313)Memory used [KB]: 1407
% 2.62/1.59 % (15313)Time elapsed: 0.002 s
% 2.62/1.59 % (15313)Instructions burned: 4 (million)
% 2.62/1.59 % (15313)------------------------------
% 2.62/1.59 % (15313)------------------------------
% 2.67/1.60 % (15312)Instruction limit reached!
% 2.67/1.60 % (15312)------------------------------
% 2.67/1.60 % (15312)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.60 % (15312)Termination reason: Unknown
% 2.67/1.60 % (15312)Termination phase: shuffling
% 2.67/1.60
% 2.67/1.60 % (15312)Memory used [KB]: 1663
% 2.67/1.60 % (15312)Time elapsed: 0.013 s
% 2.67/1.60 % (15312)Instructions burned: 18 (million)
% 2.67/1.60 % (15312)------------------------------
% 2.67/1.60 % (15312)------------------------------
% 2.67/1.60 % (15314)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/37Mi)
% 2.67/1.60 % (15317)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/1041Mi)
% 2.67/1.60 % (15316)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/3Mi)
% 2.67/1.60 % (15315)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/15Mi)
% 2.67/1.60 % (15316)Instruction limit reached!
% 2.67/1.60 % (15316)------------------------------
% 2.67/1.60 % (15316)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.60 % (15316)Termination reason: Unknown
% 2.67/1.60 % (15316)Termination phase: shuffling
% 2.67/1.60
% 2.67/1.60 % (15316)Memory used [KB]: 1407
% 2.67/1.60 % (15316)Time elapsed: 0.003 s
% 2.67/1.60 % (15316)Instructions burned: 3 (million)
% 2.67/1.60 % (15316)------------------------------
% 2.67/1.60 % (15316)------------------------------
% 2.67/1.61 % (15311)Refutation not found, incomplete strategy
% 2.67/1.61 % (15311)------------------------------
% 2.67/1.61 % (15311)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.61 % (15311)Termination reason: Refutation not found, incomplete strategy
% 2.67/1.61
% 2.67/1.61
% 2.67/1.61 % (15311)Memory used [KB]: 6396
% 2.67/1.61 % (15311)Time elapsed: 0.023 s
% 2.67/1.61 % (15311)Instructions burned: 45 (million)
% 2.67/1.61 % (15311)------------------------------
% 2.67/1.61 % (15311)------------------------------
% 2.67/1.61 % (15314)Instruction limit reached!
% 2.67/1.61 % (15314)------------------------------
% 2.67/1.61 % (15314)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.61 % (15314)Termination reason: Unknown
% 2.67/1.61 % (15314)Termination phase: shuffling
% 2.67/1.61
% 2.67/1.61 % (15314)Memory used [KB]: 2174
% 2.67/1.61 % (15314)Time elapsed: 0.013 s
% 2.67/1.61 % (15314)Instructions burned: 39 (million)
% 2.67/1.61 % (15314)------------------------------
% 2.67/1.61 % (15314)------------------------------
% 2.67/1.61 % (15318)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/7Mi)
% 2.67/1.61 % (15315)Instruction limit reached!
% 2.67/1.61 % (15315)------------------------------
% 2.67/1.61 % (15315)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.61 % (15315)Termination reason: Unknown
% 2.67/1.61 % (15315)Termination phase: shuffling
% 2.67/1.61
% 2.67/1.61 % (15315)Memory used [KB]: 1663
% 2.67/1.61 % (15315)Time elapsed: 0.011 s
% 2.67/1.61 % (15315)Instructions burned: 15 (million)
% 2.67/1.61 % (15315)------------------------------
% 2.67/1.61 % (15315)------------------------------
% 2.67/1.62 % (15318)Instruction limit reached!
% 2.67/1.62 % (15318)------------------------------
% 2.67/1.62 % (15318)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.62 % (15318)Termination reason: Unknown
% 2.67/1.62 % (15318)Termination phase: shuffling
% 2.67/1.62
% 2.67/1.62 % (15318)Memory used [KB]: 1535
% 2.67/1.62 % (15318)Time elapsed: 0.004 s
% 2.67/1.62 % (15318)Instructions burned: 8 (million)
% 2.67/1.62 % (15318)------------------------------
% 2.67/1.62 % (15318)------------------------------
% 2.67/1.62 % (15319)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/16Mi)
% 2.67/1.62 % (15320)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/3Mi)
% 2.67/1.62 % (15320)Instruction limit reached!
% 2.67/1.62 % (15320)------------------------------
% 2.67/1.62 % (15320)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.62 % (15320)Termination reason: Unknown
% 2.67/1.62 % (15320)Termination phase: shuffling
% 2.67/1.62
% 2.67/1.62 % (15320)Memory used [KB]: 1407
% 2.67/1.62 % (15320)Time elapsed: 0.002 s
% 2.67/1.62 % (15320)Instructions burned: 3 (million)
% 2.67/1.62 % (15320)------------------------------
% 2.67/1.62 % (15320)------------------------------
% 2.67/1.62 % (15321)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/3Mi)
% 2.67/1.62 % (15321)Instruction limit reached!
% 2.67/1.62 % (15321)------------------------------
% 2.67/1.62 % (15321)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.62 % (15321)Termination reason: Unknown
% 2.67/1.62 % (15321)Termination phase: shuffling
% 2.67/1.62
% 2.67/1.62 % (15321)Memory used [KB]: 1407
% 2.67/1.62 % (15321)Time elapsed: 0.003 s
% 2.67/1.62 % (15321)Instructions burned: 4 (million)
% 2.67/1.62 % (15321)------------------------------
% 2.67/1.62 % (15321)------------------------------
% 2.67/1.62 % (15319)Instruction limit reached!
% 2.67/1.62 % (15319)------------------------------
% 2.67/1.62 % (15319)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.62 % (15319)Termination reason: Unknown
% 2.67/1.62 % (15319)Termination phase: shuffling
% 2.67/1.62
% 2.67/1.62 % (15319)Memory used [KB]: 1791
% 2.67/1.62 % (15319)Time elapsed: 0.007 s
% 2.67/1.62 % (15319)Instructions burned: 17 (million)
% 2.67/1.62 % (15319)------------------------------
% 2.67/1.62 % (15319)------------------------------
% 2.67/1.63 % (15323)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/3Mi)
% 2.67/1.63 % (15323)Instruction limit reached!
% 2.67/1.63 % (15323)------------------------------
% 2.67/1.63 % (15323)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.63 % (15323)Termination reason: Unknown
% 2.67/1.63 % (15323)Termination phase: shuffling
% 2.67/1.63
% 2.67/1.63 % (15323)Memory used [KB]: 1407
% 2.67/1.63 % (15323)Time elapsed: 0.002 s
% 2.67/1.63 % (15323)Instructions burned: 3 (million)
% 2.67/1.63 % (15323)------------------------------
% 2.67/1.63 % (15323)------------------------------
% 2.67/1.63 % (15324)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/4Mi)
% 2.67/1.63 % (15324)Instruction limit reached!
% 2.67/1.63 % (15324)------------------------------
% 2.67/1.63 % (15324)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.63 % (15324)Termination reason: Unknown
% 2.67/1.63 % (15324)Termination phase: shuffling
% 2.67/1.63
% 2.67/1.63 % (15324)Memory used [KB]: 1407
% 2.67/1.63 % (15324)Time elapsed: 0.003 s
% 2.67/1.63 % (15324)Instructions burned: 4 (million)
% 2.67/1.63 % (15324)------------------------------
% 2.67/1.63 % (15324)------------------------------
% 2.67/1.63 % (15325)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/18Mi)
% 2.67/1.63 % (15326)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/710Mi)
% 2.67/1.63 % (15322)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/7Mi)
% 2.67/1.63 % (15322)Instruction limit reached!
% 2.67/1.63 % (15322)------------------------------
% 2.67/1.63 % (15322)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.64 % (15322)Termination reason: Unknown
% 2.67/1.64 % (15322)Termination phase: shuffling
% 2.67/1.64
% 2.67/1.64 % (15322)Memory used [KB]: 1535
% 2.67/1.64 % (15322)Time elapsed: 0.006 s
% 2.67/1.64 % (15322)Instructions burned: 7 (million)
% 2.67/1.64 % (15322)------------------------------
% 2.67/1.64 % (15322)------------------------------
% 2.67/1.64 % (15325)Instruction limit reached!
% 2.67/1.64 % (15325)------------------------------
% 2.67/1.64 % (15325)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.64 % (15325)Termination reason: Unknown
% 2.67/1.64 % (15325)Termination phase: shuffling
% 2.67/1.64
% 2.67/1.64 % (15325)Memory used [KB]: 1791
% 2.67/1.64 % (15325)Time elapsed: 0.008 s
% 2.67/1.64 % (15325)Instructions burned: 18 (million)
% 2.67/1.64 % (15325)------------------------------
% 2.67/1.64 % (15325)------------------------------
% 2.67/1.64 % (15327)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/6Mi)
% 2.67/1.64 % (15327)Instruction limit reached!
% 2.67/1.64 % (15327)------------------------------
% 2.67/1.64 % (15327)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.64 % (15327)Termination reason: Unknown
% 2.67/1.64 % (15327)Termination phase: shuffling
% 2.67/1.64
% 2.67/1.64 % (15327)Memory used [KB]: 1535
% 2.67/1.64 % (15327)Time elapsed: 0.004 s
% 2.67/1.64 % (15327)Instructions burned: 7 (million)
% 2.67/1.64 % (15327)------------------------------
% 2.67/1.64 % (15327)------------------------------
% 2.67/1.64 % (15328)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_15200 for (2999ds/902Mi)
% 2.67/1.64 % (15326)First to succeed.
% 2.67/1.64 % (15326)Refutation found. Thanks to Tanya!
% 2.67/1.64 % SZS status Theorem for DTF2THF_15200
% 2.67/1.64 % SZS output start Proof for DTF2THF_15200
% 2.67/1.64 thf(type_def_5, type, complex: $tType).
% 2.67/1.64 thf(type_def_7, type, set_complex: $tType).
% 2.67/1.64 thf(type_def_8, type, real: $tType).
% 2.67/1.64 thf(type_def_9, type, set_complex_complex: $tType).
% 2.67/1.64 thf(type_def_10, type, set_real: $tType).
% 2.67/1.64 thf(type_def_11, type, nat: $tType).
% 2.67/1.64 thf(type_def_12, type, set_nat: $tType).
% 2.67/1.64 thf(func_def_0, type, set_complex_complex: $tType).
% 2.67/1.64 thf(func_def_1, type, set_complex: $tType).
% 2.67/1.64 thf(func_def_2, type, set_real: $tType).
% 2.67/1.64 thf(func_def_3, type, set_nat: $tType).
% 2.67/1.64 thf(func_def_4, type, complex: $tType).
% 2.67/1.64 thf(func_def_5, type, real: $tType).
% 2.67/1.64 thf(func_def_6, type, nat: $tType).
% 2.67/1.64 thf(func_def_7, type, comple372758642hic_on: (complex > complex) > set_complex > $o).
% 2.67/1.64 thf(func_def_8, type, deriv_complex: (complex > complex) > complex > complex).
% 2.67/1.64 thf(func_def_9, type, deriv_real: (real > real) > real > real).
% 2.67/1.64 thf(func_def_10, type, comp_c1610621014omplex: ((complex > complex) > complex > complex) > ((complex > complex) > complex > complex) > (complex > complex) > complex > complex).
% 2.67/1.64 thf(func_def_11, type, comp_c881053372omplex: ((complex > complex) > complex) > (complex > complex > complex) > complex > complex).
% 2.67/1.64 thf(func_def_12, type, comp_c1884694328l_real: ((complex > complex) > real) > (real > complex > complex) > real > real).
% 2.67/1.64 thf(func_def_13, type, comp_c606622857omplex: (complex > complex > complex) > ((complex > complex) > complex) > (complex > complex) > complex > complex).
% 2.67/1.64 thf(func_def_14, type, comp_c130555887omplex: (complex > complex) > (complex > complex) > complex > complex).
% 2.67/1.64 thf(func_def_15, type, comp_c819638635l_real: (complex > real) > (real > complex) > real > real).
% 2.67/1.64 thf(func_def_16, type, comp_r2009261319omplex: (real > complex > complex) > ((complex > complex) > real) > (complex > complex) > complex > complex).
% 2.67/1.64 thf(func_def_17, type, comp_r667767405omplex: (real > complex) > (complex > real) > complex > complex).
% 2.67/1.64 thf(func_def_18, type, comp_real_real_real: (real > real) > (real > real) > real > real).
% 2.67/1.64 thf(func_def_19, type, id_complex_complex: (complex > complex) > complex > complex).
% 2.67/1.64 thf(func_def_20, type, id_real_real: (real > real) > real > real).
% 2.67/1.64 thf(func_def_21, type, id_complex: complex > complex).
% 2.67/1.64 thf(func_def_22, type, id_real: real > real).
% 2.67/1.64 thf(func_def_23, type, id_set1618538368omplex: set_complex_complex > set_complex_complex).
% 2.67/1.64 thf(func_def_24, type, id_set_complex: set_complex > set_complex).
% 2.67/1.64 thf(func_def_25, type, id_set_real: set_real > set_real).
% 2.67/1.64 thf(func_def_26, type, one_one_complex: complex).
% 2.67/1.64 thf(func_def_27, type, one_one_nat: nat).
% 2.67/1.64 thf(func_def_28, type, one_one_real: real).
% 2.67/1.64 thf(func_def_29, type, times_times_complex: complex > complex > complex).
% 2.67/1.64 thf(func_def_30, type, times_times_nat: nat > nat > nat).
% 2.67/1.64 thf(func_def_31, type, times_times_real: real > real > real).
% 2.67/1.64 thf(func_def_32, type, times_1316095593omplex: set_complex > set_complex > set_complex).
% 2.67/1.64 thf(func_def_33, type, times_times_set_nat: set_nat > set_nat > set_nat).
% 2.67/1.64 thf(func_def_34, type, times_times_set_real: set_real > set_real > set_real).
% 2.67/1.64 thf(func_def_35, type, zero_zero_complex: complex).
% 2.67/1.64 thf(func_def_36, type, zero_zero_nat: nat).
% 2.67/1.64 thf(func_def_37, type, zero_zero_real: real).
% 2.67/1.64 thf(func_def_38, type, if_complex: $o > complex > complex > complex).
% 2.67/1.64 thf(func_def_39, type, if_nat: $o > nat > nat > nat).
% 2.67/1.64 thf(func_def_40, type, if_real: $o > real > real > real).
% 2.67/1.64 thf(func_def_41, type, compow1098280738omplex: nat > ((complex > complex) > complex > complex) > (complex > complex) > complex > complex).
% 2.67/1.64 thf(func_def_42, type, compow1723822618l_real: nat > ((real > real) > real > real) > (real > real) > real > real).
% 2.67/1.64 thf(func_def_43, type, compow1667379464omplex: nat > (complex > complex) > complex > complex).
% 2.67/1.64 thf(func_def_44, type, compow_real_real: nat > (real > real) > real > real).
% 2.67/1.64 thf(func_def_45, type, funpow1854104714omplex: nat > ((complex > complex) > complex > complex) > (complex > complex) > complex > complex).
% 2.67/1.64 thf(func_def_46, type, funpow_real_real: nat > ((real > real) > real > real) > (real > real) > real > real).
% 2.67/1.64 thf(func_def_47, type, funpow_complex: nat > (complex > complex) > complex > complex).
% 2.67/1.64 thf(func_def_48, type, ord_less_eq_nat: nat > nat > $o).
% 2.67/1.64 thf(func_def_49, type, ord_less_eq_real: real > real > $o).
% 2.67/1.64 thf(func_def_50, type, ord_le701908932omplex: set_complex > set_complex > $o).
% 2.67/1.64 thf(func_def_51, type, collect_complex: (complex > $o) > set_complex).
% 2.67/1.64 thf(func_def_52, type, image_944012797omplex: ((complex > complex) > complex > complex) > set_complex_complex > set_complex_complex).
% 2.67/1.64 thf(func_def_53, type, image_58037603omplex: (complex > complex) > set_complex > set_complex).
% 2.67/1.64 thf(func_def_54, type, image_real_real: (real > real) > set_real > set_real).
% 2.67/1.64 thf(func_def_55, type, topolo935673511omplex: set_complex > $o).
% 2.67/1.64 thf(func_def_56, type, topolo2127351575omplex: set_complex > $o).
% 2.67/1.64 thf(func_def_57, type, arcosh_complex: complex > complex).
% 2.67/1.64 thf(func_def_58, type, arcosh_real: real > real).
% 2.67/1.64 thf(func_def_59, type, arsinh_complex: complex > complex).
% 2.67/1.64 thf(func_def_60, type, arsinh_real: real > real).
% 2.67/1.64 thf(func_def_61, type, artanh_complex: complex > complex).
% 2.67/1.64 thf(func_def_62, type, artanh_real: real > real).
% 2.67/1.64 thf(func_def_63, type, ln_ln_complex: complex > complex).
% 2.67/1.64 thf(func_def_64, type, ln_ln_real: real > real).
% 2.67/1.64 thf(func_def_65, type, member_complex: complex > set_complex > $o).
% 2.67/1.64 thf(func_def_66, type, member_nat: nat > set_nat > $o).
% 2.67/1.64 thf(func_def_67, type, member_real: real > set_real > $o).
% 2.67/1.64 thf(func_def_68, type, s: set_complex).
% 2.67/1.64 thf(func_def_69, type, t: set_complex).
% 2.67/1.64 thf(func_def_70, type, f: complex > complex).
% 2.67/1.64 thf(func_def_71, type, g: complex > complex).
% 2.67/1.64 thf(func_def_72, type, w: complex).
% 2.67/1.64 thf(func_def_84, type, sK0: (complex > complex) > complex).
% 2.67/1.64 thf(func_def_86, type, ph2: !>[X0: $tType]:(X0)).
% 2.67/1.64 thf(f896,plain,(
% 2.67/1.64 $false),
% 2.67/1.64 inference(trivial_inequality_removal,[status(thm)],[f895])).
% 2.67/1.64 thf(f895,plain,(
% 2.67/1.64 (one_one_complex != one_one_complex)),
% 2.67/1.64 inference(superposition,[status(thm)],[f872,f887])).
% 2.67/1.64 thf(f887,plain,(
% 2.67/1.64 ( ! [X1 : complex] : ((one_one_complex = (deriv_complex @ id_complex @ X1))) )),
% 2.67/1.64 inference(beta_eta_normalization,[status(thm)],[f886])).
% 2.67/1.64 thf(f886,plain,(
% 2.67/1.64 ( ! [X1 : complex] : (((deriv_complex @ id_complex @ X1) = ((^[Y0 : complex]: (one_one_complex)) @ X1))) )),
% 2.67/1.64 inference(argument_congruence,[status(thm)],[f871])).
% 2.67/1.64 thf(f871,plain,(
% 2.67/1.64 ((deriv_complex @ id_complex) = (^[Y0 : complex]: (one_one_complex)))),
% 2.67/1.64 inference(cnf_transformation,[status(thm)],[f540])).
% 2.67/1.64 thf(f540,plain,(
% 2.67/1.64 ((deriv_complex @ id_complex) = (^[Y0 : complex]: (one_one_complex)))),
% 2.67/1.64 inference(fool_elimination,[status(thm)],[f539])).
% 2.67/1.64 thf(f539,plain,(
% 2.67/1.64 ((deriv_complex @ id_complex) = (^[X0 : complex] : (one_one_complex)))),
% 2.67/1.64 inference(rectify,[status(thm)],[f4])).
% 2.67/1.64 thf(f4,axiom,(
% 2.67/1.64 ((deriv_complex @ id_complex) = (^[X1 : complex] : (one_one_complex)))),
% 2.67/1.64 file('/export/starexec/sandbox/tmp/tmp.gvY0VWrXGK/DTF2THF_15200.p',fact_3_deriv__id)).
% 2.67/1.64 thf(f872,plain,(
% 2.67/1.64 (one_one_complex != (deriv_complex @ id_complex @ w))),
% 2.67/1.64 inference(cnf_transformation,[status(thm)],[f825])).
% 2.67/1.64 thf(f825,plain,(
% 2.67/1.64 (one_one_complex != (deriv_complex @ id_complex @ w))),
% 2.67/1.64 inference(flattening,[status(thm)],[f361])).
% 2.67/1.64 thf(f361,negated_conjecture,(
% 2.67/1.64 ~(one_one_complex = (deriv_complex @ id_complex @ w))),
% 2.67/1.64 inference(negated_conjecture,[status(cth)],[f360])).
% 2.67/1.64 thf(f360,conjecture,(
% 2.67/1.64 (one_one_complex = (deriv_complex @ id_complex @ w))),
% 2.67/1.64 file('/export/starexec/sandbox/tmp/tmp.gvY0VWrXGK/DTF2THF_15200.p',conj_0)).
% 2.67/1.64 % SZS output end Proof for DTF2THF_15200
% 2.67/1.64 % (15326)------------------------------
% 2.67/1.64 % (15326)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.64 % (15326)Termination reason: Refutation
% 2.67/1.64
% 2.67/1.64 % (15326)Memory used [KB]: 6268
% 2.67/1.64 % (15326)Time elapsed: 0.014 s
% 2.67/1.64 % (15326)Instructions burned: 42 (million)
% 2.67/1.64 % (15326)------------------------------
% 2.67/1.64 % (15326)------------------------------
% 2.67/1.64 % (15305)Success in time 0.084 s
%------------------------------------------------------------------------------