TPTP Problem File: SLH0121^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Commuting_Hermitian/0002_Commuting_Hermitian/prob_01332_054602__19473726_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1256 ( 487 unt; 232 typ; 0 def)
% Number of atoms : 2616 (1016 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 8338 ( 292 ~; 76 |; 153 &;6639 @)
% ( 0 <=>;1178 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 961 ( 961 >; 0 *; 0 +; 0 <<)
% Number of symbols : 208 ( 205 usr; 26 con; 0-5 aty)
% Number of variables : 2865 ( 219 ^;2539 !; 107 ?;2865 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:37:30.349
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_J_J,type,
set_Pr6275530937341595591omplex: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
produc1879114310331426279omplex: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_J,type,
produc1634985270395358183omplex: $tType ).
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set_Pr6692490089613684743omplex: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
produc3519440817029397031omplex: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
produc5677646155008957607omplex: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
set_Pr8195022564563857607omplex: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_Pr9093778441882193744at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
produc3969660745209200omplex: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Real__Oreal_J,type,
produc7080596761207171842x_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
produc3259542890344722124omplex: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Nat__Onat_J,type,
produc4941145339993070502ex_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc85711943791777264at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
produc8199716216217303280at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
produc2422161461964618553l_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
produc3741383161447143261al_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
produc7716430852924023517t_real: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
set_mat_complex: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Matrix__Omat_It__Complex__Ocomplex_J,type,
mat_complex: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (205)
thf(sy_c_BNF__Def_OfstOp_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
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bNF_fs1463097484767396021omplex: ( mat_complex > mat_complex > $o ) > ( mat_complex > mat_complex > $o ) > produc352478934956084711omplex > produc352478934956084711omplex ).
thf(sy_c_BNF__Def_OfstOp_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
bNF_fs4645120796853066229omplex: ( mat_complex > mat_complex > $o ) > ( mat_complex > produc352478934956084711omplex > $o ) > produc5677646155008957607omplex > produc352478934956084711omplex ).
thf(sy_c_BNF__Def_OfstOp_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
bNF_fs2272772258120160181omplex: ( mat_complex > mat_complex > $o ) > ( mat_complex > produc5677646155008957607omplex > $o ) > produc1634985270395358183omplex > produc352478934956084711omplex ).
thf(sy_c_BNF__Def_OfstOp_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
bNF_fs2759868622548320629omplex: ( mat_complex > produc352478934956084711omplex > $o ) > ( produc352478934956084711omplex > mat_complex > $o ) > produc352478934956084711omplex > produc5677646155008957607omplex ).
thf(sy_c_BNF__Def_OfstOp_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
bNF_fs4770632251562326709omplex: ( mat_complex > produc352478934956084711omplex > $o ) > ( produc352478934956084711omplex > produc352478934956084711omplex > $o ) > produc5677646155008957607omplex > produc5677646155008957607omplex ).
thf(sy_c_BNF__Def_OfstOp_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
bNF_fs4990717021789639797omplex: ( mat_complex > produc352478934956084711omplex > $o ) > ( produc352478934956084711omplex > produc5677646155008957607omplex > $o ) > produc1634985270395358183omplex > produc5677646155008957607omplex ).
thf(sy_c_BNF__Def_OfstOp_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
bNF_fs3674406735816507829omplex: ( mat_complex > produc5677646155008957607omplex > $o ) > ( produc5677646155008957607omplex > mat_complex > $o ) > produc352478934956084711omplex > produc1634985270395358183omplex ).
thf(sy_c_BNF__Def_OfstOp_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
bNF_fs1977034296766274293omplex: ( mat_complex > produc5677646155008957607omplex > $o ) > ( produc5677646155008957607omplex > produc352478934956084711omplex > $o ) > produc5677646155008957607omplex > produc1634985270395358183omplex ).
thf(sy_c_BNF__Def_OfstOp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_fs6799373675401686564at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_BNF__Def_Opick__middlep_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bNF_pi5684954197124157748at_nat: ( ( nat > nat ) > ( nat > nat ) > $o ) > ( ( nat > nat ) > nat > $o ) > ( nat > nat ) > nat > nat > nat ).
thf(sy_c_BNF__Def_Opick__middlep_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_pi1074727425065620933at_nat: ( ( nat > nat ) > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > nat > nat ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
bNF_pi4007669207202667779omplex: ( mat_complex > mat_complex > $o ) > ( mat_complex > mat_complex > $o ) > mat_complex > mat_complex > mat_complex ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
bNF_pi2386002763760845123omplex: ( mat_complex > mat_complex > $o ) > ( mat_complex > produc352478934956084711omplex > $o ) > mat_complex > produc352478934956084711omplex > mat_complex ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
bNF_pi3374135909036492291omplex: ( mat_complex > mat_complex > $o ) > ( mat_complex > produc5677646155008957607omplex > $o ) > mat_complex > produc5677646155008957607omplex > mat_complex ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
bNF_pi500750589456099523omplex: ( mat_complex > produc352478934956084711omplex > $o ) > ( produc352478934956084711omplex > mat_complex > $o ) > mat_complex > mat_complex > produc352478934956084711omplex ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
bNF_pi5871995902478658819omplex: ( mat_complex > produc352478934956084711omplex > $o ) > ( produc352478934956084711omplex > produc352478934956084711omplex > $o ) > mat_complex > produc352478934956084711omplex > produc352478934956084711omplex ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
bNF_pi5241947103120478659omplex: ( mat_complex > produc352478934956084711omplex > $o ) > ( produc352478934956084711omplex > produc5677646155008957607omplex > $o ) > mat_complex > produc5677646155008957607omplex > produc352478934956084711omplex ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
bNF_pi4775770386732839939omplex: ( mat_complex > produc5677646155008957607omplex > $o ) > ( produc5677646155008957607omplex > mat_complex > $o ) > mat_complex > mat_complex > produc5677646155008957607omplex ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
bNF_pi2228264378097113155omplex: ( mat_complex > produc5677646155008957607omplex > $o ) > ( produc5677646155008957607omplex > produc352478934956084711omplex > $o ) > mat_complex > produc352478934956084711omplex > produc5677646155008957607omplex ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bNF_pi5620451001738351685at_nat: ( nat > ( nat > nat ) > $o ) > ( ( nat > nat ) > nat > $o ) > nat > nat > nat > nat ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_pi7484965678128203350at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > nat ).
thf(sy_c_BNF__Def_OsndOp_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bNF_sn2822976844928563268at_nat: ( ( nat > nat ) > ( nat > nat ) > $o ) > ( ( nat > nat ) > nat > $o ) > produc8199716216217303280at_nat > produc8199716216217303280at_nat ).
thf(sy_c_BNF__Def_OsndOp_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_sn848789568725914837at_nat: ( ( nat > nat ) > nat > $o ) > ( nat > nat > $o ) > produc8199716216217303280at_nat > product_prod_nat_nat ).
thf(sy_c_BNF__Def_OsndOp_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
bNF_sn2463300919646374387omplex: ( mat_complex > mat_complex > $o ) > ( mat_complex > mat_complex > $o ) > produc352478934956084711omplex > produc352478934956084711omplex ).
thf(sy_c_BNF__Def_OsndOp_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
bNF_sn2893073706650928691omplex: ( mat_complex > mat_complex > $o ) > ( mat_complex > produc352478934956084711omplex > $o ) > produc5677646155008957607omplex > produc5677646155008957607omplex ).
thf(sy_c_BNF__Def_OsndOp_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
bNF_sn410462143701312243omplex: ( mat_complex > mat_complex > $o ) > ( mat_complex > produc5677646155008957607omplex > $o ) > produc1634985270395358183omplex > produc1634985270395358183omplex ).
thf(sy_c_BNF__Def_OsndOp_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bNF_sn5394513145398645589at_nat: ( nat > ( nat > nat ) > $o ) > ( ( nat > nat ) > nat > $o ) > product_prod_nat_nat > produc8199716216217303280at_nat ).
thf(sy_c_BNF__Def_OsndOp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_sn8916957246138178918at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Matrix__Omat_It__Complex__Ocomplex_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bNF_Gr4568166622192947294at_nat: set_mat_complex > ( mat_complex > nat > nat ) > ( mat_complex > nat ) > set_Pr9093778441882193744at_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
bNF_Gr7780393699004731405omplex: set_mat_complex > ( mat_complex > mat_complex ) > ( mat_complex > mat_complex ) > set_Pr8195022564563857607omplex ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
bNF_Gr7119546496759113037omplex: set_mat_complex > ( mat_complex > mat_complex ) > ( mat_complex > produc352478934956084711omplex ) > set_Pr6692490089613684743omplex ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
bNF_Gr3087718976987896589omplex: set_mat_complex > ( mat_complex > mat_complex ) > ( mat_complex > produc5677646155008957607omplex ) > set_Pr6275530937341595591omplex ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_Gr5984784631878081519at_nat: set_mat_complex > ( mat_complex > nat ) > ( mat_complex > nat ) > set_Pr1261947904930325089at_nat ).
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produc3802106415914905499omplex: ( mat_complex > mat_complex > set_mat_complex ) > produc352478934956084711omplex > set_mat_complex ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Set__Oset_It__Real__Oreal_J,type,
produc2518032081017119822t_real: ( mat_complex > mat_complex > set_real ) > produc352478934956084711omplex > set_real ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_001_Eo,type,
produc5497593456678237420plex_o: ( mat_complex > produc352478934956084711omplex > $o ) > produc5677646155008957607omplex > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
produc386998054833891163omplex: ( mat_complex > produc352478934956084711omplex > set_mat_complex ) > produc5677646155008957607omplex > set_mat_complex ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_001t__Set__Oset_It__Real__Oreal_J,type,
produc7148884236971998350t_real: ( mat_complex > produc352478934956084711omplex > set_real ) > produc5677646155008957607omplex > set_real ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_001_Eo,type,
produc6770849645695511596plex_o: ( mat_complex > produc5677646155008957607omplex > $o ) > produc1634985270395358183omplex > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
produc8983715372140011163omplex: ( mat_complex > produc5677646155008957607omplex > set_mat_complex ) > produc1634985270395358183omplex > set_mat_complex ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_001t__Set__Oset_It__Real__Oreal_J,type,
produc1354186782250546510t_real: ( mat_complex > produc5677646155008957607omplex > set_real ) > produc1634985270395358183omplex > set_real ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
produc7021537633938252857omplex: ( nat > nat > set_mat_complex ) > product_prod_nat_nat > set_mat_complex ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Real__Oreal_J,type,
produc3668448655016342576t_real: ( nat > nat > set_real ) > product_prod_nat_nat > set_real ).
thf(sy_c_Product__Type_Oprod_Ofst_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc6156676138143019412at_nat: produc8199716216217303280at_nat > nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc9163778666669654339omplex: produc352478934956084711omplex > mat_complex ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Nat__Onat,type,
produc7604368447525475636ex_nat: produc4941145339993070502ex_nat > mat_complex ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
produc2697000228617323907omplex: produc5677646155008957607omplex > mat_complex ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
produc8911724726559533635omplex: produc1634985270395358183omplex > mat_complex ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Real__Oreal,type,
produc7072908607146575376x_real: produc7080596761207171842x_real > mat_complex ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc4700600022864998420at_nat: produc85711943791777264at_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc8687169775924804370omplex: produc3259542890344722124omplex > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Real__Oreal,type,
product_fst_nat_real: produc7716430852924023517t_real > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc811748054312578307omplex: produc3519440817029397031omplex > produc352478934956084711omplex ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc1089987167401105475omplex: produc1879114310331426279omplex > produc5677646155008957607omplex ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Real__Oreal_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc2079669512483292982omplex: produc3969660745209200omplex > real ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Real__Oreal_001t__Nat__Onat,type,
product_fst_real_nat: produc3741383161447143261al_nat > real ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Real__Oreal_001t__Real__Oreal,type,
produc5828954698716094813l_real: produc2422161461964618553l_real > real ).
thf(sy_c_Product__Type_Oprod_Osnd_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc1852801350702243542at_nat: produc8199716216217303280at_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc4897211011226852997omplex: produc352478934956084711omplex > mat_complex ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Nat__Onat,type,
produc8270670584407784050ex_nat: produc4941145339993070502ex_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
produc7343567217041670085omplex: produc5677646155008957607omplex > produc352478934956084711omplex ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
produc943930114779824517omplex: produc1634985270395358183omplex > produc5677646155008957607omplex ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Real__Oreal,type,
produc7625890766153296974x_real: produc7080596761207171842x_real > real ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc396725235424222550at_nat: produc85711943791777264at_nat > nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc130099875952336976omplex: produc3259542890344722124omplex > mat_complex ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Real__Oreal,type,
product_snd_nat_real: produc7716430852924023517t_real > real ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc5458315042736924485omplex: produc3519440817029397031omplex > mat_complex ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc2345564592476172165omplex: produc1879114310331426279omplex > mat_complex ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Real__Oreal_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc2632651671490014580omplex: produc3969660745209200omplex > mat_complex ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Real__Oreal_001t__Nat__Onat,type,
product_snd_real_nat: produc3741383161447143261al_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Real__Oreal_001t__Real__Oreal,type,
produc3484788084999411615l_real: produc2422161461964618553l_real > real ).
thf(sy_c_Product__Type_Oprod_Oswap_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc5003442202721523016at_nat: produc8199716216217303280at_nat > produc85711943791777264at_nat ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc667468844592973047omplex: produc352478934956084711omplex > produc352478934956084711omplex ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
produc3890531975266499383omplex: produc5677646155008957607omplex > produc3519440817029397031omplex ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Matrix__Omat_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
produc8725010470225793527omplex: produc1634985270395358183omplex > produc1879114310331426279omplex ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc3547366087443502024at_nat: produc85711943791777264at_nat > produc8199716216217303280at_nat ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001t__Nat__Onat,type,
product_swap_nat_nat: product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc2005279800961753783omplex: produc3519440817029397031omplex > produc5677646155008957607omplex ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
produc903272911067365367omplex: produc1879114310331426279omplex > produc1634985270395358183omplex ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
collect_mat_complex: ( mat_complex > $o ) > set_mat_complex ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
collec9201399625632817755at_nat: ( produc8199716216217303280at_nat > $o ) > set_Pr9093778441882193744at_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
collec5097175541304549202omplex: ( produc352478934956084711omplex > $o ) > set_Pr8195022564563857607omplex ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
collec2557858944451489170omplex: ( produc5677646155008957607omplex > $o ) > set_Pr6692490089613684743omplex ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_J,type,
collec875000405105826898omplex: ( produc1634985270395358183omplex > $o ) > set_Pr6275530937341595591omplex ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_member_001t__Matrix__Omat_It__Complex__Ocomplex_J,type,
member_mat_complex: mat_complex > set_mat_complex > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member7226740684066999833at_nat: produc8199716216217303280at_nat > set_Pr9093778441882193744at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J,type,
member8347409015010237200omplex: produc352478934956084711omplex > set_Pr8195022564563857607omplex > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J,type,
member74738575112047696omplex: produc5677646155008957607omplex > set_Pr6692490089613684743omplex > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Product____Type__Oprod_It__Matrix__Omat_It__Complex__Ocomplex_J_Mt__Matrix__Omat_It__Complex__Ocomplex_J_J_J_J,type,
member5471586270331035152omplex: produc1634985270395358183omplex > set_Pr6275530937341595591omplex > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_B,type,
b: mat_complex ).
thf(sy_v_B1____,type,
b1: mat_complex ).
thf(sy_v_B2____,type,
b2: mat_complex ).
thf(sy_v_B3____,type,
b3: mat_complex ).
thf(sy_v_B4____,type,
b4: mat_complex ).
thf(sy_v_Ba____,type,
ba: mat_complex ).
thf(sy_v_a____,type,
a: nat ).
thf(sy_v_i,type,
i: nat ).
thf(sy_v_ia____,type,
ia: nat ).
thf(sy_v_j,type,
j: nat ).
thf(sy_v_l,type,
l: list_nat ).
thf(sy_v_la____,type,
la: list_nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_na____,type,
na: nat ).
% Relevant facts (1018)
thf(fact_0_prod_Osel_I1_J,axiom,
! [X1: nat,X2: nat] :
( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
= X1 ) ).
% prod.sel(1)
thf(fact_1_prod_Osel_I1_J,axiom,
! [X1: nat > nat,X2: nat] :
( ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X1 @ X2 ) )
= X1 ) ).
% prod.sel(1)
thf(fact_2_prod_Osel_I1_J,axiom,
! [X1: mat_complex,X2: produc5677646155008957607omplex] :
( ( produc8911724726559533635omplex @ ( produc1901862033385395287omplex @ X1 @ X2 ) )
= X1 ) ).
% prod.sel(1)
thf(fact_3_prod_Osel_I1_J,axiom,
! [X1: mat_complex,X2: produc352478934956084711omplex] :
( ( produc2697000228617323907omplex @ ( produc2861545499953221015omplex @ X1 @ X2 ) )
= X1 ) ).
% prod.sel(1)
thf(fact_4_prod_Osel_I1_J,axiom,
! [X1: mat_complex,X2: mat_complex] :
( ( produc9163778666669654339omplex @ ( produc3658446505030690647omplex @ X1 @ X2 ) )
= X1 ) ).
% prod.sel(1)
thf(fact_5_prod_Osel_I2_J,axiom,
! [X1: nat,X2: nat] :
( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
= X2 ) ).
% prod.sel(2)
thf(fact_6_prod_Osel_I2_J,axiom,
! [X1: nat > nat,X2: nat] :
( ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ X1 @ X2 ) )
= X2 ) ).
% prod.sel(2)
thf(fact_7_prod_Osel_I2_J,axiom,
! [X1: mat_complex,X2: produc5677646155008957607omplex] :
( ( produc943930114779824517omplex @ ( produc1901862033385395287omplex @ X1 @ X2 ) )
= X2 ) ).
% prod.sel(2)
thf(fact_8_prod_Osel_I2_J,axiom,
! [X1: mat_complex,X2: produc352478934956084711omplex] :
( ( produc7343567217041670085omplex @ ( produc2861545499953221015omplex @ X1 @ X2 ) )
= X2 ) ).
% prod.sel(2)
thf(fact_9_prod_Osel_I2_J,axiom,
! [X1: mat_complex,X2: mat_complex] :
( ( produc4897211011226852997omplex @ ( produc3658446505030690647omplex @ X1 @ X2 ) )
= X2 ) ).
% prod.sel(2)
thf(fact_10_B1__def,axiom,
( b1
= ( produc8911724726559533635omplex @ ( split_block_complex @ ba @ a @ a ) ) ) ).
% B1_def
thf(fact_11_B2__def,axiom,
( b2
= ( produc2697000228617323907omplex @ ( produc943930114779824517omplex @ ( split_block_complex @ ba @ a @ a ) ) ) ) ).
% B2_def
thf(fact_12_B3__def,axiom,
( b3
= ( produc9163778666669654339omplex @ ( produc7343567217041670085omplex @ ( produc943930114779824517omplex @ ( split_block_complex @ ba @ a @ a ) ) ) ) ) ).
% B3_def
thf(fact_13_B4__def,axiom,
( b4
= ( produc4897211011226852997omplex @ ( produc7343567217041670085omplex @ ( produc943930114779824517omplex @ ( split_block_complex @ ba @ a @ a ) ) ) ) ) ).
% B4_def
thf(fact_14_prod__cases4,axiom,
! [Y: produc1634985270395358183omplex] :
~ ! [A: mat_complex,B: mat_complex,C: mat_complex,D: mat_complex] :
( Y
!= ( produc1901862033385395287omplex @ A @ ( produc2861545499953221015omplex @ B @ ( produc3658446505030690647omplex @ C @ D ) ) ) ) ).
% prod_cases4
thf(fact_15_prod__induct4,axiom,
! [P: produc1634985270395358183omplex > $o,X: produc1634985270395358183omplex] :
( ! [A: mat_complex,B: mat_complex,C: mat_complex,D: mat_complex] : ( P @ ( produc1901862033385395287omplex @ A @ ( produc2861545499953221015omplex @ B @ ( produc3658446505030690647omplex @ C @ D ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_16_prod_Ocollapse,axiom,
! [Prod: produc1634985270395358183omplex] :
( ( produc1901862033385395287omplex @ ( produc8911724726559533635omplex @ Prod ) @ ( produc943930114779824517omplex @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_17_prod_Ocollapse,axiom,
! [Prod: produc5677646155008957607omplex] :
( ( produc2861545499953221015omplex @ ( produc2697000228617323907omplex @ Prod ) @ ( produc7343567217041670085omplex @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_18_prod_Ocollapse,axiom,
! [Prod: produc352478934956084711omplex] :
( ( produc3658446505030690647omplex @ ( produc9163778666669654339omplex @ Prod ) @ ( produc4897211011226852997omplex @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_19_prod_Ocollapse,axiom,
! [Prod: produc8199716216217303280at_nat] :
( ( produc72220940542539688at_nat @ ( produc6156676138143019412at_nat @ Prod ) @ ( produc1852801350702243542at_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_20_prod_Ocollapse,axiom,
! [Prod: product_prod_nat_nat] :
( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_21_prod_Oexhaust__sel,axiom,
! [Prod: produc1634985270395358183omplex] :
( Prod
= ( produc1901862033385395287omplex @ ( produc8911724726559533635omplex @ Prod ) @ ( produc943930114779824517omplex @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_22_prod_Oexhaust__sel,axiom,
! [Prod: produc5677646155008957607omplex] :
( Prod
= ( produc2861545499953221015omplex @ ( produc2697000228617323907omplex @ Prod ) @ ( produc7343567217041670085omplex @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_23_prod_Oexhaust__sel,axiom,
! [Prod: produc352478934956084711omplex] :
( Prod
= ( produc3658446505030690647omplex @ ( produc9163778666669654339omplex @ Prod ) @ ( produc4897211011226852997omplex @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_24_prod_Oexhaust__sel,axiom,
! [Prod: produc8199716216217303280at_nat] :
( Prod
= ( produc72220940542539688at_nat @ ( produc6156676138143019412at_nat @ Prod ) @ ( produc1852801350702243542at_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_25_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_nat_nat] :
( Prod
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_26_BNF__Def_Osubst__Pair,axiom,
! [P: mat_complex > produc5677646155008957607omplex > $o,X: mat_complex,Y: produc5677646155008957607omplex,A2: produc1634985270395358183omplex] :
( ( P @ X @ Y )
=> ( ( A2
= ( produc1901862033385395287omplex @ X @ Y ) )
=> ( P @ ( produc8911724726559533635omplex @ A2 ) @ ( produc943930114779824517omplex @ A2 ) ) ) ) ).
% BNF_Def.subst_Pair
thf(fact_27_BNF__Def_Osubst__Pair,axiom,
! [P: mat_complex > produc352478934956084711omplex > $o,X: mat_complex,Y: produc352478934956084711omplex,A2: produc5677646155008957607omplex] :
( ( P @ X @ Y )
=> ( ( A2
= ( produc2861545499953221015omplex @ X @ Y ) )
=> ( P @ ( produc2697000228617323907omplex @ A2 ) @ ( produc7343567217041670085omplex @ A2 ) ) ) ) ).
% BNF_Def.subst_Pair
thf(fact_28_BNF__Def_Osubst__Pair,axiom,
! [P: mat_complex > mat_complex > $o,X: mat_complex,Y: mat_complex,A2: produc352478934956084711omplex] :
( ( P @ X @ Y )
=> ( ( A2
= ( produc3658446505030690647omplex @ X @ Y ) )
=> ( P @ ( produc9163778666669654339omplex @ A2 ) @ ( produc4897211011226852997omplex @ A2 ) ) ) ) ).
% BNF_Def.subst_Pair
thf(fact_29_BNF__Def_Osubst__Pair,axiom,
! [P: ( nat > nat ) > nat > $o,X: nat > nat,Y: nat,A2: produc8199716216217303280at_nat] :
( ( P @ X @ Y )
=> ( ( A2
= ( produc72220940542539688at_nat @ X @ Y ) )
=> ( P @ ( produc6156676138143019412at_nat @ A2 ) @ ( produc1852801350702243542at_nat @ A2 ) ) ) ) ).
% BNF_Def.subst_Pair
thf(fact_30_BNF__Def_Osubst__Pair,axiom,
! [P: nat > nat > $o,X: nat,Y: nat,A2: product_prod_nat_nat] :
( ( P @ X @ Y )
=> ( ( A2
= ( product_Pair_nat_nat @ X @ Y ) )
=> ( P @ ( product_fst_nat_nat @ A2 ) @ ( product_snd_nat_nat @ A2 ) ) ) ) ).
% BNF_Def.subst_Pair
thf(fact_31_Pair__inject,axiom,
! [A2: mat_complex,B2: produc5677646155008957607omplex,A3: mat_complex,B3: produc5677646155008957607omplex] :
( ( ( produc1901862033385395287omplex @ A2 @ B2 )
= ( produc1901862033385395287omplex @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_32_Pair__inject,axiom,
! [A2: mat_complex,B2: produc352478934956084711omplex,A3: mat_complex,B3: produc352478934956084711omplex] :
( ( ( produc2861545499953221015omplex @ A2 @ B2 )
= ( produc2861545499953221015omplex @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_33_Pair__inject,axiom,
! [A2: mat_complex,B2: mat_complex,A3: mat_complex,B3: mat_complex] :
( ( ( produc3658446505030690647omplex @ A2 @ B2 )
= ( produc3658446505030690647omplex @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_34_Pair__inject,axiom,
! [A2: nat,B2: nat,A3: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A2 @ B2 )
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_35_Pair__inject,axiom,
! [A2: nat > nat,B2: nat,A3: nat > nat,B3: nat] :
( ( ( produc72220940542539688at_nat @ A2 @ B2 )
= ( produc72220940542539688at_nat @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_36_prod__cases,axiom,
! [P: produc1634985270395358183omplex > $o,P2: produc1634985270395358183omplex] :
( ! [A: mat_complex,B: produc5677646155008957607omplex] : ( P @ ( produc1901862033385395287omplex @ A @ B ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_37_prod__cases,axiom,
! [P: produc5677646155008957607omplex > $o,P2: produc5677646155008957607omplex] :
( ! [A: mat_complex,B: produc352478934956084711omplex] : ( P @ ( produc2861545499953221015omplex @ A @ B ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_38_prod__cases,axiom,
! [P: produc352478934956084711omplex > $o,P2: produc352478934956084711omplex] :
( ! [A: mat_complex,B: mat_complex] : ( P @ ( produc3658446505030690647omplex @ A @ B ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_39_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
( ! [A: nat,B: nat] : ( P @ ( product_Pair_nat_nat @ A @ B ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_40_prod__cases,axiom,
! [P: produc8199716216217303280at_nat > $o,P2: produc8199716216217303280at_nat] :
( ! [A: nat > nat,B: nat] : ( P @ ( produc72220940542539688at_nat @ A @ B ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_41_surj__pair,axiom,
! [P2: produc1634985270395358183omplex] :
? [X3: mat_complex,Y2: produc5677646155008957607omplex] :
( P2
= ( produc1901862033385395287omplex @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_42_surj__pair,axiom,
! [P2: produc5677646155008957607omplex] :
? [X3: mat_complex,Y2: produc352478934956084711omplex] :
( P2
= ( produc2861545499953221015omplex @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_43_surj__pair,axiom,
! [P2: produc352478934956084711omplex] :
? [X3: mat_complex,Y2: mat_complex] :
( P2
= ( produc3658446505030690647omplex @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_44_surj__pair,axiom,
! [P2: product_prod_nat_nat] :
? [X3: nat,Y2: nat] :
( P2
= ( product_Pair_nat_nat @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_45_surj__pair,axiom,
! [P2: produc8199716216217303280at_nat] :
? [X3: nat > nat,Y2: nat] :
( P2
= ( produc72220940542539688at_nat @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_46_gcd_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [A: nat,B: nat] :
( X
!= ( product_Pair_nat_nat @ A @ B ) ) ).
% gcd.cases
thf(fact_47_old_Oprod_Oinducts,axiom,
! [P: produc1634985270395358183omplex > $o,Prod: produc1634985270395358183omplex] :
( ! [A: mat_complex,B: produc5677646155008957607omplex] : ( P @ ( produc1901862033385395287omplex @ A @ B ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_48_old_Oprod_Oinducts,axiom,
! [P: produc5677646155008957607omplex > $o,Prod: produc5677646155008957607omplex] :
( ! [A: mat_complex,B: produc352478934956084711omplex] : ( P @ ( produc2861545499953221015omplex @ A @ B ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_49_old_Oprod_Oinducts,axiom,
! [P: produc352478934956084711omplex > $o,Prod: produc352478934956084711omplex] :
( ! [A: mat_complex,B: mat_complex] : ( P @ ( produc3658446505030690647omplex @ A @ B ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_50_old_Oprod_Oinducts,axiom,
! [P: product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ! [A: nat,B: nat] : ( P @ ( product_Pair_nat_nat @ A @ B ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_51_old_Oprod_Oinducts,axiom,
! [P: produc8199716216217303280at_nat > $o,Prod: produc8199716216217303280at_nat] :
( ! [A: nat > nat,B: nat] : ( P @ ( produc72220940542539688at_nat @ A @ B ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_52_old_Oprod_Oexhaust,axiom,
! [Y: produc1634985270395358183omplex] :
~ ! [A: mat_complex,B: produc5677646155008957607omplex] :
( Y
!= ( produc1901862033385395287omplex @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_53_old_Oprod_Oexhaust,axiom,
! [Y: produc5677646155008957607omplex] :
~ ! [A: mat_complex,B: produc352478934956084711omplex] :
( Y
!= ( produc2861545499953221015omplex @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_54_old_Oprod_Oexhaust,axiom,
! [Y: produc352478934956084711omplex] :
~ ! [A: mat_complex,B: mat_complex] :
( Y
!= ( produc3658446505030690647omplex @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_55_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A: nat,B: nat] :
( Y
!= ( product_Pair_nat_nat @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_56_old_Oprod_Oexhaust,axiom,
! [Y: produc8199716216217303280at_nat] :
~ ! [A: nat > nat,B: nat] :
( Y
!= ( produc72220940542539688at_nat @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_57_old_Oprod_Oinject,axiom,
! [A2: mat_complex,B2: produc5677646155008957607omplex,A3: mat_complex,B3: produc5677646155008957607omplex] :
( ( ( produc1901862033385395287omplex @ A2 @ B2 )
= ( produc1901862033385395287omplex @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_58_old_Oprod_Oinject,axiom,
! [A2: mat_complex,B2: produc352478934956084711omplex,A3: mat_complex,B3: produc352478934956084711omplex] :
( ( ( produc2861545499953221015omplex @ A2 @ B2 )
= ( produc2861545499953221015omplex @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_59_old_Oprod_Oinject,axiom,
! [A2: mat_complex,B2: mat_complex,A3: mat_complex,B3: mat_complex] :
( ( ( produc3658446505030690647omplex @ A2 @ B2 )
= ( produc3658446505030690647omplex @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_60_old_Oprod_Oinject,axiom,
! [A2: nat,B2: nat,A3: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A2 @ B2 )
= ( product_Pair_nat_nat @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_61_old_Oprod_Oinject,axiom,
! [A2: nat > nat,B2: nat,A3: nat > nat,B3: nat] :
( ( ( produc72220940542539688at_nat @ A2 @ B2 )
= ( produc72220940542539688at_nat @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_62_prod_Oinject,axiom,
! [X1: mat_complex,X2: produc5677646155008957607omplex,Y1: mat_complex,Y22: produc5677646155008957607omplex] :
( ( ( produc1901862033385395287omplex @ X1 @ X2 )
= ( produc1901862033385395287omplex @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X2 = Y22 ) ) ) ).
% prod.inject
thf(fact_63_prod_Oinject,axiom,
! [X1: mat_complex,X2: produc352478934956084711omplex,Y1: mat_complex,Y22: produc352478934956084711omplex] :
( ( ( produc2861545499953221015omplex @ X1 @ X2 )
= ( produc2861545499953221015omplex @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X2 = Y22 ) ) ) ).
% prod.inject
thf(fact_64_prod_Oinject,axiom,
! [X1: mat_complex,X2: mat_complex,Y1: mat_complex,Y22: mat_complex] :
( ( ( produc3658446505030690647omplex @ X1 @ X2 )
= ( produc3658446505030690647omplex @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X2 = Y22 ) ) ) ).
% prod.inject
thf(fact_65_prod_Oinject,axiom,
! [X1: nat,X2: nat,Y1: nat,Y22: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X2 )
= ( product_Pair_nat_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X2 = Y22 ) ) ) ).
% prod.inject
thf(fact_66_prod_Oinject,axiom,
! [X1: nat > nat,X2: nat,Y1: nat > nat,Y22: nat] :
( ( ( produc72220940542539688at_nat @ X1 @ X2 )
= ( produc72220940542539688at_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X2 = Y22 ) ) ) ).
% prod.inject
thf(fact_67_fst__eqD,axiom,
! [X: mat_complex,Y: produc5677646155008957607omplex,A2: mat_complex] :
( ( ( produc8911724726559533635omplex @ ( produc1901862033385395287omplex @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_68_fst__eqD,axiom,
! [X: mat_complex,Y: produc352478934956084711omplex,A2: mat_complex] :
( ( ( produc2697000228617323907omplex @ ( produc2861545499953221015omplex @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_69_fst__eqD,axiom,
! [X: mat_complex,Y: mat_complex,A2: mat_complex] :
( ( ( produc9163778666669654339omplex @ ( produc3658446505030690647omplex @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_70_fst__eqD,axiom,
! [X: nat > nat,Y: nat,A2: nat > nat] :
( ( ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_71_fst__eqD,axiom,
! [X: nat,Y: nat,A2: nat] :
( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_72_fstI,axiom,
! [X: produc1634985270395358183omplex,Y: mat_complex,Z: produc5677646155008957607omplex] :
( ( X
= ( produc1901862033385395287omplex @ Y @ Z ) )
=> ( ( produc8911724726559533635omplex @ X )
= Y ) ) ).
% fstI
thf(fact_73_fstI,axiom,
! [X: produc5677646155008957607omplex,Y: mat_complex,Z: produc352478934956084711omplex] :
( ( X
= ( produc2861545499953221015omplex @ Y @ Z ) )
=> ( ( produc2697000228617323907omplex @ X )
= Y ) ) ).
% fstI
thf(fact_74_fstI,axiom,
! [X: produc352478934956084711omplex,Y: mat_complex,Z: mat_complex] :
( ( X
= ( produc3658446505030690647omplex @ Y @ Z ) )
=> ( ( produc9163778666669654339omplex @ X )
= Y ) ) ).
% fstI
thf(fact_75_fstI,axiom,
! [X: produc8199716216217303280at_nat,Y: nat > nat,Z: nat] :
( ( X
= ( produc72220940542539688at_nat @ Y @ Z ) )
=> ( ( produc6156676138143019412at_nat @ X )
= Y ) ) ).
% fstI
thf(fact_76_fstI,axiom,
! [X: product_prod_nat_nat,Y: nat,Z: nat] :
( ( X
= ( product_Pair_nat_nat @ Y @ Z ) )
=> ( ( product_fst_nat_nat @ X )
= Y ) ) ).
% fstI
thf(fact_77_snd__eqD,axiom,
! [X: mat_complex,Y: produc5677646155008957607omplex,A2: produc5677646155008957607omplex] :
( ( ( produc943930114779824517omplex @ ( produc1901862033385395287omplex @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_78_snd__eqD,axiom,
! [X: mat_complex,Y: produc352478934956084711omplex,A2: produc352478934956084711omplex] :
( ( ( produc7343567217041670085omplex @ ( produc2861545499953221015omplex @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_79_snd__eqD,axiom,
! [X: mat_complex,Y: mat_complex,A2: mat_complex] :
( ( ( produc4897211011226852997omplex @ ( produc3658446505030690647omplex @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_80_snd__eqD,axiom,
! [X: nat > nat,Y: nat,A2: nat] :
( ( ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_81_snd__eqD,axiom,
! [X: nat,Y: nat,A2: nat] :
( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_82_sndI,axiom,
! [X: produc1634985270395358183omplex,Y: mat_complex,Z: produc5677646155008957607omplex] :
( ( X
= ( produc1901862033385395287omplex @ Y @ Z ) )
=> ( ( produc943930114779824517omplex @ X )
= Z ) ) ).
% sndI
thf(fact_83_sndI,axiom,
! [X: produc5677646155008957607omplex,Y: mat_complex,Z: produc352478934956084711omplex] :
( ( X
= ( produc2861545499953221015omplex @ Y @ Z ) )
=> ( ( produc7343567217041670085omplex @ X )
= Z ) ) ).
% sndI
thf(fact_84_sndI,axiom,
! [X: produc352478934956084711omplex,Y: mat_complex,Z: mat_complex] :
( ( X
= ( produc3658446505030690647omplex @ Y @ Z ) )
=> ( ( produc4897211011226852997omplex @ X )
= Z ) ) ).
% sndI
thf(fact_85_sndI,axiom,
! [X: produc8199716216217303280at_nat,Y: nat > nat,Z: nat] :
( ( X
= ( produc72220940542539688at_nat @ Y @ Z ) )
=> ( ( produc1852801350702243542at_nat @ X )
= Z ) ) ).
% sndI
thf(fact_86_sndI,axiom,
! [X: product_prod_nat_nat,Y: nat,Z: nat] :
( ( X
= ( product_Pair_nat_nat @ Y @ Z ) )
=> ( ( product_snd_nat_nat @ X )
= Z ) ) ).
% sndI
thf(fact_87_prod__induct3,axiom,
! [P: produc1634985270395358183omplex > $o,X: produc1634985270395358183omplex] :
( ! [A: mat_complex,B: mat_complex,C: produc352478934956084711omplex] : ( P @ ( produc1901862033385395287omplex @ A @ ( produc2861545499953221015omplex @ B @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_88_prod__induct3,axiom,
! [P: produc5677646155008957607omplex > $o,X: produc5677646155008957607omplex] :
( ! [A: mat_complex,B: mat_complex,C: mat_complex] : ( P @ ( produc2861545499953221015omplex @ A @ ( produc3658446505030690647omplex @ B @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_89_prod__cases3,axiom,
! [Y: produc1634985270395358183omplex] :
~ ! [A: mat_complex,B: mat_complex,C: produc352478934956084711omplex] :
( Y
!= ( produc1901862033385395287omplex @ A @ ( produc2861545499953221015omplex @ B @ C ) ) ) ).
% prod_cases3
thf(fact_90_prod__cases3,axiom,
! [Y: produc5677646155008957607omplex] :
~ ! [A: mat_complex,B: mat_complex,C: mat_complex] :
( Y
!= ( produc2861545499953221015omplex @ A @ ( produc3658446505030690647omplex @ B @ C ) ) ) ).
% prod_cases3
thf(fact_91_prod__eq__iff,axiom,
( ( ^ [Y3: produc1634985270395358183omplex,Z2: produc1634985270395358183omplex] : ( Y3 = Z2 ) )
= ( ^ [S: produc1634985270395358183omplex,T: produc1634985270395358183omplex] :
( ( ( produc8911724726559533635omplex @ S )
= ( produc8911724726559533635omplex @ T ) )
& ( ( produc943930114779824517omplex @ S )
= ( produc943930114779824517omplex @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_92_prod__eq__iff,axiom,
( ( ^ [Y3: produc5677646155008957607omplex,Z2: produc5677646155008957607omplex] : ( Y3 = Z2 ) )
= ( ^ [S: produc5677646155008957607omplex,T: produc5677646155008957607omplex] :
( ( ( produc2697000228617323907omplex @ S )
= ( produc2697000228617323907omplex @ T ) )
& ( ( produc7343567217041670085omplex @ S )
= ( produc7343567217041670085omplex @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_93_prod__eq__iff,axiom,
( ( ^ [Y3: produc352478934956084711omplex,Z2: produc352478934956084711omplex] : ( Y3 = Z2 ) )
= ( ^ [S: produc352478934956084711omplex,T: produc352478934956084711omplex] :
( ( ( produc9163778666669654339omplex @ S )
= ( produc9163778666669654339omplex @ T ) )
& ( ( produc4897211011226852997omplex @ S )
= ( produc4897211011226852997omplex @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_94_prod__eq__iff,axiom,
( ( ^ [Y3: produc8199716216217303280at_nat,Z2: produc8199716216217303280at_nat] : ( Y3 = Z2 ) )
= ( ^ [S: produc8199716216217303280at_nat,T: produc8199716216217303280at_nat] :
( ( ( produc6156676138143019412at_nat @ S )
= ( produc6156676138143019412at_nat @ T ) )
& ( ( produc1852801350702243542at_nat @ S )
= ( produc1852801350702243542at_nat @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_95_prod__eq__iff,axiom,
( ( ^ [Y3: product_prod_nat_nat,Z2: product_prod_nat_nat] : ( Y3 = Z2 ) )
= ( ^ [S: product_prod_nat_nat,T: product_prod_nat_nat] :
( ( ( product_fst_nat_nat @ S )
= ( product_fst_nat_nat @ T ) )
& ( ( product_snd_nat_nat @ S )
= ( product_snd_nat_nat @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_96_prod__eqI,axiom,
! [P2: produc1634985270395358183omplex,Q: produc1634985270395358183omplex] :
( ( ( produc8911724726559533635omplex @ P2 )
= ( produc8911724726559533635omplex @ Q ) )
=> ( ( ( produc943930114779824517omplex @ P2 )
= ( produc943930114779824517omplex @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_97_prod__eqI,axiom,
! [P2: produc5677646155008957607omplex,Q: produc5677646155008957607omplex] :
( ( ( produc2697000228617323907omplex @ P2 )
= ( produc2697000228617323907omplex @ Q ) )
=> ( ( ( produc7343567217041670085omplex @ P2 )
= ( produc7343567217041670085omplex @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_98_prod__eqI,axiom,
! [P2: produc352478934956084711omplex,Q: produc352478934956084711omplex] :
( ( ( produc9163778666669654339omplex @ P2 )
= ( produc9163778666669654339omplex @ Q ) )
=> ( ( ( produc4897211011226852997omplex @ P2 )
= ( produc4897211011226852997omplex @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_99_prod__eqI,axiom,
! [P2: produc8199716216217303280at_nat,Q: produc8199716216217303280at_nat] :
( ( ( produc6156676138143019412at_nat @ P2 )
= ( produc6156676138143019412at_nat @ Q ) )
=> ( ( ( produc1852801350702243542at_nat @ P2 )
= ( produc1852801350702243542at_nat @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_100_prod__eqI,axiom,
! [P2: product_prod_nat_nat,Q: product_prod_nat_nat] :
( ( ( product_fst_nat_nat @ P2 )
= ( product_fst_nat_nat @ Q ) )
=> ( ( ( product_snd_nat_nat @ P2 )
= ( product_snd_nat_nat @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_101_prod_Oexpand,axiom,
! [Prod: produc1634985270395358183omplex,Prod2: produc1634985270395358183omplex] :
( ( ( ( produc8911724726559533635omplex @ Prod )
= ( produc8911724726559533635omplex @ Prod2 ) )
& ( ( produc943930114779824517omplex @ Prod )
= ( produc943930114779824517omplex @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_102_prod_Oexpand,axiom,
! [Prod: produc5677646155008957607omplex,Prod2: produc5677646155008957607omplex] :
( ( ( ( produc2697000228617323907omplex @ Prod )
= ( produc2697000228617323907omplex @ Prod2 ) )
& ( ( produc7343567217041670085omplex @ Prod )
= ( produc7343567217041670085omplex @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_103_prod_Oexpand,axiom,
! [Prod: produc352478934956084711omplex,Prod2: produc352478934956084711omplex] :
( ( ( ( produc9163778666669654339omplex @ Prod )
= ( produc9163778666669654339omplex @ Prod2 ) )
& ( ( produc4897211011226852997omplex @ Prod )
= ( produc4897211011226852997omplex @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_104_prod_Oexpand,axiom,
! [Prod: produc8199716216217303280at_nat,Prod2: produc8199716216217303280at_nat] :
( ( ( ( produc6156676138143019412at_nat @ Prod )
= ( produc6156676138143019412at_nat @ Prod2 ) )
& ( ( produc1852801350702243542at_nat @ Prod )
= ( produc1852801350702243542at_nat @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_105_prod_Oexpand,axiom,
! [Prod: product_prod_nat_nat,Prod2: product_prod_nat_nat] :
( ( ( ( product_fst_nat_nat @ Prod )
= ( product_fst_nat_nat @ Prod2 ) )
& ( ( product_snd_nat_nat @ Prod )
= ( product_snd_nat_nat @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_106_surjective__pairing,axiom,
! [T2: produc1634985270395358183omplex] :
( T2
= ( produc1901862033385395287omplex @ ( produc8911724726559533635omplex @ T2 ) @ ( produc943930114779824517omplex @ T2 ) ) ) ).
% surjective_pairing
thf(fact_107_surjective__pairing,axiom,
! [T2: produc5677646155008957607omplex] :
( T2
= ( produc2861545499953221015omplex @ ( produc2697000228617323907omplex @ T2 ) @ ( produc7343567217041670085omplex @ T2 ) ) ) ).
% surjective_pairing
thf(fact_108_surjective__pairing,axiom,
! [T2: produc352478934956084711omplex] :
( T2
= ( produc3658446505030690647omplex @ ( produc9163778666669654339omplex @ T2 ) @ ( produc4897211011226852997omplex @ T2 ) ) ) ).
% surjective_pairing
thf(fact_109_surjective__pairing,axiom,
! [T2: produc8199716216217303280at_nat] :
( T2
= ( produc72220940542539688at_nat @ ( produc6156676138143019412at_nat @ T2 ) @ ( produc1852801350702243542at_nat @ T2 ) ) ) ).
% surjective_pairing
thf(fact_110_surjective__pairing,axiom,
! [T2: product_prod_nat_nat] :
( T2
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T2 ) @ ( product_snd_nat_nat @ T2 ) ) ) ).
% surjective_pairing
thf(fact_111_mem__Collect__eq,axiom,
! [A2: mat_complex,P: mat_complex > $o] :
( ( member_mat_complex @ A2 @ ( collect_mat_complex @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_112_mem__Collect__eq,axiom,
! [A2: real,P: real > $o] :
( ( member_real @ A2 @ ( collect_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_113_Collect__mem__eq,axiom,
! [A4: set_mat_complex] :
( ( collect_mat_complex
@ ^ [X4: mat_complex] : ( member_mat_complex @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
! [A4: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_115_conjI__realizer,axiom,
! [P: mat_complex > $o,P2: mat_complex,Q2: produc5677646155008957607omplex > $o,Q: produc5677646155008957607omplex] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc8911724726559533635omplex @ ( produc1901862033385395287omplex @ P2 @ Q ) ) )
& ( Q2 @ ( produc943930114779824517omplex @ ( produc1901862033385395287omplex @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_116_conjI__realizer,axiom,
! [P: mat_complex > $o,P2: mat_complex,Q2: produc352478934956084711omplex > $o,Q: produc352478934956084711omplex] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc2697000228617323907omplex @ ( produc2861545499953221015omplex @ P2 @ Q ) ) )
& ( Q2 @ ( produc7343567217041670085omplex @ ( produc2861545499953221015omplex @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_117_conjI__realizer,axiom,
! [P: mat_complex > $o,P2: mat_complex,Q2: mat_complex > $o,Q: mat_complex] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc9163778666669654339omplex @ ( produc3658446505030690647omplex @ P2 @ Q ) ) )
& ( Q2 @ ( produc4897211011226852997omplex @ ( produc3658446505030690647omplex @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_118_conjI__realizer,axiom,
! [P: ( nat > nat ) > $o,P2: nat > nat,Q2: nat > $o,Q: nat] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ P2 @ Q ) ) )
& ( Q2 @ ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_119_conjI__realizer,axiom,
! [P: nat > $o,P2: nat,Q2: nat > $o,Q: nat] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q ) ) )
& ( Q2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_120_split__pairs,axiom,
! [A4: mat_complex,B4: produc5677646155008957607omplex,X5: produc1634985270395358183omplex] :
( ( ( produc1901862033385395287omplex @ A4 @ B4 )
= X5 )
= ( ( ( produc8911724726559533635omplex @ X5 )
= A4 )
& ( ( produc943930114779824517omplex @ X5 )
= B4 ) ) ) ).
% split_pairs
thf(fact_121_split__pairs,axiom,
! [A4: mat_complex,B4: produc352478934956084711omplex,X5: produc5677646155008957607omplex] :
( ( ( produc2861545499953221015omplex @ A4 @ B4 )
= X5 )
= ( ( ( produc2697000228617323907omplex @ X5 )
= A4 )
& ( ( produc7343567217041670085omplex @ X5 )
= B4 ) ) ) ).
% split_pairs
thf(fact_122_split__pairs,axiom,
! [A4: mat_complex,B4: mat_complex,X5: produc352478934956084711omplex] :
( ( ( produc3658446505030690647omplex @ A4 @ B4 )
= X5 )
= ( ( ( produc9163778666669654339omplex @ X5 )
= A4 )
& ( ( produc4897211011226852997omplex @ X5 )
= B4 ) ) ) ).
% split_pairs
thf(fact_123_split__pairs,axiom,
! [A4: nat > nat,B4: nat,X5: produc8199716216217303280at_nat] :
( ( ( produc72220940542539688at_nat @ A4 @ B4 )
= X5 )
= ( ( ( produc6156676138143019412at_nat @ X5 )
= A4 )
& ( ( produc1852801350702243542at_nat @ X5 )
= B4 ) ) ) ).
% split_pairs
thf(fact_124_split__pairs,axiom,
! [A4: nat,B4: nat,X5: product_prod_nat_nat] :
( ( ( product_Pair_nat_nat @ A4 @ B4 )
= X5 )
= ( ( ( product_fst_nat_nat @ X5 )
= A4 )
& ( ( product_snd_nat_nat @ X5 )
= B4 ) ) ) ).
% split_pairs
thf(fact_125_exI__realizer,axiom,
! [P: produc5677646155008957607omplex > mat_complex > $o,Y: produc5677646155008957607omplex,X: mat_complex] :
( ( P @ Y @ X )
=> ( P @ ( produc943930114779824517omplex @ ( produc1901862033385395287omplex @ X @ Y ) ) @ ( produc8911724726559533635omplex @ ( produc1901862033385395287omplex @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_126_exI__realizer,axiom,
! [P: produc352478934956084711omplex > mat_complex > $o,Y: produc352478934956084711omplex,X: mat_complex] :
( ( P @ Y @ X )
=> ( P @ ( produc7343567217041670085omplex @ ( produc2861545499953221015omplex @ X @ Y ) ) @ ( produc2697000228617323907omplex @ ( produc2861545499953221015omplex @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_127_exI__realizer,axiom,
! [P: mat_complex > mat_complex > $o,Y: mat_complex,X: mat_complex] :
( ( P @ Y @ X )
=> ( P @ ( produc4897211011226852997omplex @ ( produc3658446505030690647omplex @ X @ Y ) ) @ ( produc9163778666669654339omplex @ ( produc3658446505030690647omplex @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_128_exI__realizer,axiom,
! [P: nat > ( nat > nat ) > $o,Y: nat,X: nat > nat] :
( ( P @ Y @ X )
=> ( P @ ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ X @ Y ) ) @ ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_129_exI__realizer,axiom,
! [P: nat > nat > $o,Y: nat,X: nat] :
( ( P @ Y @ X )
=> ( P @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_130_exE__realizer_H,axiom,
! [P: produc5677646155008957607omplex > mat_complex > $o,P2: produc1634985270395358183omplex] :
( ( P @ ( produc943930114779824517omplex @ P2 ) @ ( produc8911724726559533635omplex @ P2 ) )
=> ~ ! [X3: mat_complex,Y2: produc5677646155008957607omplex] :
~ ( P @ Y2 @ X3 ) ) ).
% exE_realizer'
thf(fact_131_exE__realizer_H,axiom,
! [P: produc352478934956084711omplex > mat_complex > $o,P2: produc5677646155008957607omplex] :
( ( P @ ( produc7343567217041670085omplex @ P2 ) @ ( produc2697000228617323907omplex @ P2 ) )
=> ~ ! [X3: mat_complex,Y2: produc352478934956084711omplex] :
~ ( P @ Y2 @ X3 ) ) ).
% exE_realizer'
thf(fact_132_exE__realizer_H,axiom,
! [P: mat_complex > mat_complex > $o,P2: produc352478934956084711omplex] :
( ( P @ ( produc4897211011226852997omplex @ P2 ) @ ( produc9163778666669654339omplex @ P2 ) )
=> ~ ! [X3: mat_complex,Y2: mat_complex] :
~ ( P @ Y2 @ X3 ) ) ).
% exE_realizer'
thf(fact_133_exE__realizer_H,axiom,
! [P: nat > ( nat > nat ) > $o,P2: produc8199716216217303280at_nat] :
( ( P @ ( produc1852801350702243542at_nat @ P2 ) @ ( produc6156676138143019412at_nat @ P2 ) )
=> ~ ! [X3: nat > nat,Y2: nat] :
~ ( P @ Y2 @ X3 ) ) ).
% exE_realizer'
thf(fact_134_exE__realizer_H,axiom,
! [P: nat > nat > $o,P2: product_prod_nat_nat] :
( ( P @ ( product_snd_nat_nat @ P2 ) @ ( product_fst_nat_nat @ P2 ) )
=> ~ ! [X3: nat,Y2: nat] :
~ ( P @ Y2 @ X3 ) ) ).
% exE_realizer'
thf(fact_135_eq__snd__iff,axiom,
! [B2: produc5677646155008957607omplex,P2: produc1634985270395358183omplex] :
( ( B2
= ( produc943930114779824517omplex @ P2 ) )
= ( ? [A5: mat_complex] :
( P2
= ( produc1901862033385395287omplex @ A5 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_136_eq__snd__iff,axiom,
! [B2: produc352478934956084711omplex,P2: produc5677646155008957607omplex] :
( ( B2
= ( produc7343567217041670085omplex @ P2 ) )
= ( ? [A5: mat_complex] :
( P2
= ( produc2861545499953221015omplex @ A5 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_137_eq__snd__iff,axiom,
! [B2: mat_complex,P2: produc352478934956084711omplex] :
( ( B2
= ( produc4897211011226852997omplex @ P2 ) )
= ( ? [A5: mat_complex] :
( P2
= ( produc3658446505030690647omplex @ A5 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_138_eq__snd__iff,axiom,
! [B2: nat,P2: produc8199716216217303280at_nat] :
( ( B2
= ( produc1852801350702243542at_nat @ P2 ) )
= ( ? [A5: nat > nat] :
( P2
= ( produc72220940542539688at_nat @ A5 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_139_eq__snd__iff,axiom,
! [B2: nat,P2: product_prod_nat_nat] :
( ( B2
= ( product_snd_nat_nat @ P2 ) )
= ( ? [A5: nat] :
( P2
= ( product_Pair_nat_nat @ A5 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_140_eq__fst__iff,axiom,
! [A2: mat_complex,P2: produc1634985270395358183omplex] :
( ( A2
= ( produc8911724726559533635omplex @ P2 ) )
= ( ? [B5: produc5677646155008957607omplex] :
( P2
= ( produc1901862033385395287omplex @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_141_eq__fst__iff,axiom,
! [A2: mat_complex,P2: produc5677646155008957607omplex] :
( ( A2
= ( produc2697000228617323907omplex @ P2 ) )
= ( ? [B5: produc352478934956084711omplex] :
( P2
= ( produc2861545499953221015omplex @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_142_eq__fst__iff,axiom,
! [A2: mat_complex,P2: produc352478934956084711omplex] :
( ( A2
= ( produc9163778666669654339omplex @ P2 ) )
= ( ? [B5: mat_complex] :
( P2
= ( produc3658446505030690647omplex @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_143_eq__fst__iff,axiom,
! [A2: nat > nat,P2: produc8199716216217303280at_nat] :
( ( A2
= ( produc6156676138143019412at_nat @ P2 ) )
= ( ? [B5: nat] :
( P2
= ( produc72220940542539688at_nat @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_144_eq__fst__iff,axiom,
! [A2: nat,P2: product_prod_nat_nat] :
( ( A2
= ( product_fst_nat_nat @ P2 ) )
= ( ? [B5: nat] :
( P2
= ( product_Pair_nat_nat @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_145_prod_Oswap__def,axiom,
( produc903272911067365367omplex
= ( ^ [P3: produc1879114310331426279omplex] : ( produc1901862033385395287omplex @ ( produc2345564592476172165omplex @ P3 ) @ ( produc1089987167401105475omplex @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_146_prod_Oswap__def,axiom,
( produc2005279800961753783omplex
= ( ^ [P3: produc3519440817029397031omplex] : ( produc2861545499953221015omplex @ ( produc5458315042736924485omplex @ P3 ) @ ( produc811748054312578307omplex @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_147_prod_Oswap__def,axiom,
( produc3547366087443502024at_nat
= ( ^ [P3: produc85711943791777264at_nat] : ( produc72220940542539688at_nat @ ( produc396725235424222550at_nat @ P3 ) @ ( produc4700600022864998420at_nat @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_148_prod_Oswap__def,axiom,
( produc8725010470225793527omplex
= ( ^ [P3: produc1634985270395358183omplex] : ( produc3303496511081742935omplex @ ( produc943930114779824517omplex @ P3 ) @ ( produc8911724726559533635omplex @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_149_prod_Oswap__def,axiom,
( produc3890531975266499383omplex
= ( ^ [P3: produc5677646155008957607omplex] : ( produc976293325648475415omplex @ ( produc7343567217041670085omplex @ P3 ) @ ( produc2697000228617323907omplex @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_150_prod_Oswap__def,axiom,
( produc667468844592973047omplex
= ( ^ [P3: produc352478934956084711omplex] : ( produc3658446505030690647omplex @ ( produc4897211011226852997omplex @ P3 ) @ ( produc9163778666669654339omplex @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_151_prod_Oswap__def,axiom,
( produc5003442202721523016at_nat
= ( ^ [P3: produc8199716216217303280at_nat] : ( produc7839516862119294504at_nat @ ( produc1852801350702243542at_nat @ P3 ) @ ( produc6156676138143019412at_nat @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_152_prod_Oswap__def,axiom,
( product_swap_nat_nat
= ( ^ [P3: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( product_snd_nat_nat @ P3 ) @ ( product_fst_nat_nat @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_153_Suc_Oprems_I1_J,axiom,
member_mat_complex @ ba @ ( carrier_mat_complex @ na @ na ) ).
% Suc.prems(1)
thf(fact_154_assms_I1_J,axiom,
member_mat_complex @ b @ ( carrier_mat_complex @ n @ n ) ).
% assms(1)
thf(fact_155_swap__simp,axiom,
! [X: produc5677646155008957607omplex,Y: mat_complex] :
( ( produc903272911067365367omplex @ ( produc3303496511081742935omplex @ X @ Y ) )
= ( produc1901862033385395287omplex @ Y @ X ) ) ).
% swap_simp
thf(fact_156_swap__simp,axiom,
! [X: produc352478934956084711omplex,Y: mat_complex] :
( ( produc2005279800961753783omplex @ ( produc976293325648475415omplex @ X @ Y ) )
= ( produc2861545499953221015omplex @ Y @ X ) ) ).
% swap_simp
thf(fact_157_swap__simp,axiom,
! [X: nat,Y: nat > nat] :
( ( produc3547366087443502024at_nat @ ( produc7839516862119294504at_nat @ X @ Y ) )
= ( produc72220940542539688at_nat @ Y @ X ) ) ).
% swap_simp
thf(fact_158_swap__simp,axiom,
! [X: mat_complex,Y: produc5677646155008957607omplex] :
( ( produc8725010470225793527omplex @ ( produc1901862033385395287omplex @ X @ Y ) )
= ( produc3303496511081742935omplex @ Y @ X ) ) ).
% swap_simp
thf(fact_159_swap__simp,axiom,
! [X: mat_complex,Y: produc352478934956084711omplex] :
( ( produc3890531975266499383omplex @ ( produc2861545499953221015omplex @ X @ Y ) )
= ( produc976293325648475415omplex @ Y @ X ) ) ).
% swap_simp
thf(fact_160_swap__simp,axiom,
! [X: mat_complex,Y: mat_complex] :
( ( produc667468844592973047omplex @ ( produc3658446505030690647omplex @ X @ Y ) )
= ( produc3658446505030690647omplex @ Y @ X ) ) ).
% swap_simp
thf(fact_161_swap__simp,axiom,
! [X: nat,Y: nat] :
( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
= ( product_Pair_nat_nat @ Y @ X ) ) ).
% swap_simp
thf(fact_162_swap__simp,axiom,
! [X: nat > nat,Y: nat] :
( ( produc5003442202721523016at_nat @ ( produc72220940542539688at_nat @ X @ Y ) )
= ( produc7839516862119294504at_nat @ Y @ X ) ) ).
% swap_simp
thf(fact_163_fst__swap,axiom,
! [X: produc1879114310331426279omplex] :
( ( produc8911724726559533635omplex @ ( produc903272911067365367omplex @ X ) )
= ( produc2345564592476172165omplex @ X ) ) ).
% fst_swap
thf(fact_164_fst__swap,axiom,
! [X: produc3519440817029397031omplex] :
( ( produc2697000228617323907omplex @ ( produc2005279800961753783omplex @ X ) )
= ( produc5458315042736924485omplex @ X ) ) ).
% fst_swap
thf(fact_165_fst__swap,axiom,
! [X: produc85711943791777264at_nat] :
( ( produc6156676138143019412at_nat @ ( produc3547366087443502024at_nat @ X ) )
= ( produc396725235424222550at_nat @ X ) ) ).
% fst_swap
thf(fact_166_fst__swap,axiom,
! [X: produc1634985270395358183omplex] :
( ( produc1089987167401105475omplex @ ( produc8725010470225793527omplex @ X ) )
= ( produc943930114779824517omplex @ X ) ) ).
% fst_swap
thf(fact_167_fst__swap,axiom,
! [X: produc5677646155008957607omplex] :
( ( produc811748054312578307omplex @ ( produc3890531975266499383omplex @ X ) )
= ( produc7343567217041670085omplex @ X ) ) ).
% fst_swap
thf(fact_168_fst__swap,axiom,
! [X: produc352478934956084711omplex] :
( ( produc9163778666669654339omplex @ ( produc667468844592973047omplex @ X ) )
= ( produc4897211011226852997omplex @ X ) ) ).
% fst_swap
thf(fact_169_fst__swap,axiom,
! [X: produc8199716216217303280at_nat] :
( ( produc4700600022864998420at_nat @ ( produc5003442202721523016at_nat @ X ) )
= ( produc1852801350702243542at_nat @ X ) ) ).
% fst_swap
thf(fact_170_fst__swap,axiom,
! [X: product_prod_nat_nat] :
( ( product_fst_nat_nat @ ( product_swap_nat_nat @ X ) )
= ( product_snd_nat_nat @ X ) ) ).
% fst_swap
thf(fact_171_snd__swap,axiom,
! [X: produc1634985270395358183omplex] :
( ( produc2345564592476172165omplex @ ( produc8725010470225793527omplex @ X ) )
= ( produc8911724726559533635omplex @ X ) ) ).
% snd_swap
thf(fact_172_snd__swap,axiom,
! [X: produc5677646155008957607omplex] :
( ( produc5458315042736924485omplex @ ( produc3890531975266499383omplex @ X ) )
= ( produc2697000228617323907omplex @ X ) ) ).
% snd_swap
thf(fact_173_snd__swap,axiom,
! [X: produc8199716216217303280at_nat] :
( ( produc396725235424222550at_nat @ ( produc5003442202721523016at_nat @ X ) )
= ( produc6156676138143019412at_nat @ X ) ) ).
% snd_swap
thf(fact_174_snd__swap,axiom,
! [X: produc1879114310331426279omplex] :
( ( produc943930114779824517omplex @ ( produc903272911067365367omplex @ X ) )
= ( produc1089987167401105475omplex @ X ) ) ).
% snd_swap
thf(fact_175_snd__swap,axiom,
! [X: produc3519440817029397031omplex] :
( ( produc7343567217041670085omplex @ ( produc2005279800961753783omplex @ X ) )
= ( produc811748054312578307omplex @ X ) ) ).
% snd_swap
thf(fact_176_snd__swap,axiom,
! [X: produc352478934956084711omplex] :
( ( produc4897211011226852997omplex @ ( produc667468844592973047omplex @ X ) )
= ( produc9163778666669654339omplex @ X ) ) ).
% snd_swap
thf(fact_177_snd__swap,axiom,
! [X: produc85711943791777264at_nat] :
( ( produc1852801350702243542at_nat @ ( produc3547366087443502024at_nat @ X ) )
= ( produc4700600022864998420at_nat @ X ) ) ).
% snd_swap
thf(fact_178_snd__swap,axiom,
! [X: product_prod_nat_nat] :
( ( product_snd_nat_nat @ ( product_swap_nat_nat @ X ) )
= ( product_fst_nat_nat @ X ) ) ).
% snd_swap
thf(fact_179__092_060open_062a_A_092_060le_062_An_092_060close_062,axiom,
ord_less_eq_nat @ a @ na ).
% \<open>a \<le> n\<close>
thf(fact_180_split__block__diag__carrier_I1_J,axiom,
! [D2: mat_complex,N: nat,A2: nat,D1: mat_complex,D22: mat_complex,D3: mat_complex,D4: mat_complex] :
( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_eq_nat @ A2 @ N )
=> ( ( ( split_block_complex @ D2 @ A2 @ A2 )
= ( produc1901862033385395287omplex @ D1 @ ( produc2861545499953221015omplex @ D22 @ ( produc3658446505030690647omplex @ D3 @ D4 ) ) ) )
=> ( member_mat_complex @ D1 @ ( carrier_mat_complex @ A2 @ A2 ) ) ) ) ) ).
% split_block_diag_carrier(1)
thf(fact_181_fstOp__def,axiom,
( bNF_fs6799373675401686564at_nat
= ( ^ [P4: nat > nat > $o,Q3: nat > nat > $o,Ac: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Ac ) @ ( bNF_pi7484965678128203350at_nat @ P4 @ Q3 @ ( product_fst_nat_nat @ Ac ) @ ( product_snd_nat_nat @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_182_fstOp__def,axiom,
( bNF_fs6984906391148982419at_nat
= ( ^ [P4: ( nat > nat ) > nat > $o,Q3: nat > nat > $o,Ac: produc8199716216217303280at_nat] : ( produc72220940542539688at_nat @ ( produc6156676138143019412at_nat @ Ac ) @ ( bNF_pi1074727425065620933at_nat @ P4 @ Q3 @ ( produc6156676138143019412at_nat @ Ac ) @ ( produc1852801350702243542at_nat @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_183_fstOp__def,axiom,
( bNF_fs1463097484767396021omplex
= ( ^ [P4: mat_complex > mat_complex > $o,Q3: mat_complex > mat_complex > $o,Ac: produc352478934956084711omplex] : ( produc3658446505030690647omplex @ ( produc9163778666669654339omplex @ Ac ) @ ( bNF_pi4007669207202667779omplex @ P4 @ Q3 @ ( produc9163778666669654339omplex @ Ac ) @ ( produc4897211011226852997omplex @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_184_fstOp__def,axiom,
( bNF_fs4645120796853066229omplex
= ( ^ [P4: mat_complex > mat_complex > $o,Q3: mat_complex > produc352478934956084711omplex > $o,Ac: produc5677646155008957607omplex] : ( produc3658446505030690647omplex @ ( produc2697000228617323907omplex @ Ac ) @ ( bNF_pi2386002763760845123omplex @ P4 @ Q3 @ ( produc2697000228617323907omplex @ Ac ) @ ( produc7343567217041670085omplex @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_185_fstOp__def,axiom,
( bNF_fs2759868622548320629omplex
= ( ^ [P4: mat_complex > produc352478934956084711omplex > $o,Q3: produc352478934956084711omplex > mat_complex > $o,Ac: produc352478934956084711omplex] : ( produc2861545499953221015omplex @ ( produc9163778666669654339omplex @ Ac ) @ ( bNF_pi500750589456099523omplex @ P4 @ Q3 @ ( produc9163778666669654339omplex @ Ac ) @ ( produc4897211011226852997omplex @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_186_fstOp__def,axiom,
( bNF_fs2272772258120160181omplex
= ( ^ [P4: mat_complex > mat_complex > $o,Q3: mat_complex > produc5677646155008957607omplex > $o,Ac: produc1634985270395358183omplex] : ( produc3658446505030690647omplex @ ( produc8911724726559533635omplex @ Ac ) @ ( bNF_pi3374135909036492291omplex @ P4 @ Q3 @ ( produc8911724726559533635omplex @ Ac ) @ ( produc943930114779824517omplex @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_187_fstOp__def,axiom,
( bNF_fs4770632251562326709omplex
= ( ^ [P4: mat_complex > produc352478934956084711omplex > $o,Q3: produc352478934956084711omplex > produc352478934956084711omplex > $o,Ac: produc5677646155008957607omplex] : ( produc2861545499953221015omplex @ ( produc2697000228617323907omplex @ Ac ) @ ( bNF_pi5871995902478658819omplex @ P4 @ Q3 @ ( produc2697000228617323907omplex @ Ac ) @ ( produc7343567217041670085omplex @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_188_fstOp__def,axiom,
( bNF_fs3674406735816507829omplex
= ( ^ [P4: mat_complex > produc5677646155008957607omplex > $o,Q3: produc5677646155008957607omplex > mat_complex > $o,Ac: produc352478934956084711omplex] : ( produc1901862033385395287omplex @ ( produc9163778666669654339omplex @ Ac ) @ ( bNF_pi4775770386732839939omplex @ P4 @ Q3 @ ( produc9163778666669654339omplex @ Ac ) @ ( produc4897211011226852997omplex @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_189_fstOp__def,axiom,
( bNF_fs4990717021789639797omplex
= ( ^ [P4: mat_complex > produc352478934956084711omplex > $o,Q3: produc352478934956084711omplex > produc5677646155008957607omplex > $o,Ac: produc1634985270395358183omplex] : ( produc2861545499953221015omplex @ ( produc8911724726559533635omplex @ Ac ) @ ( bNF_pi5241947103120478659omplex @ P4 @ Q3 @ ( produc8911724726559533635omplex @ Ac ) @ ( produc943930114779824517omplex @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_190_fstOp__def,axiom,
( bNF_fs1977034296766274293omplex
= ( ^ [P4: mat_complex > produc5677646155008957607omplex > $o,Q3: produc5677646155008957607omplex > produc352478934956084711omplex > $o,Ac: produc5677646155008957607omplex] : ( produc1901862033385395287omplex @ ( produc2697000228617323907omplex @ Ac ) @ ( bNF_pi2228264378097113155omplex @ P4 @ Q3 @ ( produc2697000228617323907omplex @ Ac ) @ ( produc7343567217041670085omplex @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_191_sndOp__def,axiom,
( bNF_sn410462143701312243omplex
= ( ^ [P4: mat_complex > mat_complex > $o,Q3: mat_complex > produc5677646155008957607omplex > $o,Ac: produc1634985270395358183omplex] : ( produc1901862033385395287omplex @ ( bNF_pi3374135909036492291omplex @ P4 @ Q3 @ ( produc8911724726559533635omplex @ Ac ) @ ( produc943930114779824517omplex @ Ac ) ) @ ( produc943930114779824517omplex @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_192_sndOp__def,axiom,
( bNF_sn2893073706650928691omplex
= ( ^ [P4: mat_complex > mat_complex > $o,Q3: mat_complex > produc352478934956084711omplex > $o,Ac: produc5677646155008957607omplex] : ( produc2861545499953221015omplex @ ( bNF_pi2386002763760845123omplex @ P4 @ Q3 @ ( produc2697000228617323907omplex @ Ac ) @ ( produc7343567217041670085omplex @ Ac ) ) @ ( produc7343567217041670085omplex @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_193_sndOp__def,axiom,
( bNF_sn2463300919646374387omplex
= ( ^ [P4: mat_complex > mat_complex > $o,Q3: mat_complex > mat_complex > $o,Ac: produc352478934956084711omplex] : ( produc3658446505030690647omplex @ ( bNF_pi4007669207202667779omplex @ P4 @ Q3 @ ( produc9163778666669654339omplex @ Ac ) @ ( produc4897211011226852997omplex @ Ac ) ) @ ( produc4897211011226852997omplex @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_194_sndOp__def,axiom,
( bNF_sn848789568725914837at_nat
= ( ^ [P4: ( nat > nat ) > nat > $o,Q3: nat > nat > $o,Ac: produc8199716216217303280at_nat] : ( product_Pair_nat_nat @ ( bNF_pi1074727425065620933at_nat @ P4 @ Q3 @ ( produc6156676138143019412at_nat @ Ac ) @ ( produc1852801350702243542at_nat @ Ac ) ) @ ( produc1852801350702243542at_nat @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_195_sndOp__def,axiom,
( bNF_sn2822976844928563268at_nat
= ( ^ [P4: ( nat > nat ) > ( nat > nat ) > $o,Q3: ( nat > nat ) > nat > $o,Ac: produc8199716216217303280at_nat] : ( produc72220940542539688at_nat @ ( bNF_pi5684954197124157748at_nat @ P4 @ Q3 @ ( produc6156676138143019412at_nat @ Ac ) @ ( produc1852801350702243542at_nat @ Ac ) ) @ ( produc1852801350702243542at_nat @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_196_sndOp__def,axiom,
( bNF_sn8916957246138178918at_nat
= ( ^ [P4: nat > nat > $o,Q3: nat > nat > $o,Ac: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( bNF_pi7484965678128203350at_nat @ P4 @ Q3 @ ( product_fst_nat_nat @ Ac ) @ ( product_snd_nat_nat @ Ac ) ) @ ( product_snd_nat_nat @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_197_sndOp__def,axiom,
( bNF_sn5394513145398645589at_nat
= ( ^ [P4: nat > ( nat > nat ) > $o,Q3: ( nat > nat ) > nat > $o,Ac: product_prod_nat_nat] : ( produc72220940542539688at_nat @ ( bNF_pi5620451001738351685at_nat @ P4 @ Q3 @ ( product_fst_nat_nat @ Ac ) @ ( product_snd_nat_nat @ Ac ) ) @ ( product_snd_nat_nat @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_198_image2__eqI,axiom,
! [B2: mat_complex,F: mat_complex > mat_complex,X: mat_complex,C2: produc5677646155008957607omplex,G: mat_complex > produc5677646155008957607omplex,A4: set_mat_complex] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member_mat_complex @ X @ A4 )
=> ( member5471586270331035152omplex @ ( produc1901862033385395287omplex @ B2 @ C2 ) @ ( bNF_Gr3087718976987896589omplex @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_199_image2__eqI,axiom,
! [B2: mat_complex,F: real > mat_complex,X: real,C2: produc5677646155008957607omplex,G: real > produc5677646155008957607omplex,A4: set_real] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member_real @ X @ A4 )
=> ( member5471586270331035152omplex @ ( produc1901862033385395287omplex @ B2 @ C2 ) @ ( bNF_Gr474249821954933062omplex @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_200_image2__eqI,axiom,
! [B2: mat_complex,F: mat_complex > mat_complex,X: mat_complex,C2: produc352478934956084711omplex,G: mat_complex > produc352478934956084711omplex,A4: set_mat_complex] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member_mat_complex @ X @ A4 )
=> ( member74738575112047696omplex @ ( produc2861545499953221015omplex @ B2 @ C2 ) @ ( bNF_Gr7119546496759113037omplex @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_201_image2__eqI,axiom,
! [B2: mat_complex,F: real > mat_complex,X: real,C2: produc352478934956084711omplex,G: real > produc352478934956084711omplex,A4: set_real] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member_real @ X @ A4 )
=> ( member74738575112047696omplex @ ( produc2861545499953221015omplex @ B2 @ C2 ) @ ( bNF_Gr8775370692967121158omplex @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_202_image2__eqI,axiom,
! [B2: mat_complex,F: mat_complex > mat_complex,X: mat_complex,C2: mat_complex,G: mat_complex > mat_complex,A4: set_mat_complex] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member_mat_complex @ X @ A4 )
=> ( member8347409015010237200omplex @ ( produc3658446505030690647omplex @ B2 @ C2 ) @ ( bNF_Gr7780393699004731405omplex @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_203_image2__eqI,axiom,
! [B2: mat_complex,F: real > mat_complex,X: real,C2: mat_complex,G: real > mat_complex,A4: set_real] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member_real @ X @ A4 )
=> ( member8347409015010237200omplex @ ( produc3658446505030690647omplex @ B2 @ C2 ) @ ( bNF_Gr1157811550679916870omplex @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_204_image2__eqI,axiom,
! [B2: nat,F: mat_complex > nat,X: mat_complex,C2: nat,G: mat_complex > nat,A4: set_mat_complex] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member_mat_complex @ X @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C2 ) @ ( bNF_Gr5984784631878081519at_nat @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_205_image2__eqI,axiom,
! [B2: nat,F: real > nat,X: real,C2: nat,G: real > nat,A4: set_real] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member_real @ X @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C2 ) @ ( bNF_Gr3333497931924356776at_nat @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_206_image2__eqI,axiom,
! [B2: nat > nat,F: mat_complex > nat > nat,X: mat_complex,C2: nat,G: mat_complex > nat,A4: set_mat_complex] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member_mat_complex @ X @ A4 )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B2 @ C2 ) @ ( bNF_Gr4568166622192947294at_nat @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_207_image2__eqI,axiom,
! [B2: nat > nat,F: real > nat > nat,X: real,C2: nat,G: real > nat,A4: set_real] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member_real @ X @ A4 )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B2 @ C2 ) @ ( bNF_Gr78910094189767063at_nat @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_208_Suc_Oprems_I2_J,axiom,
ord_less_nat @ zero_zero_nat @ na ).
% Suc.prems(2)
thf(fact_209_ssubst__Pair__rhs,axiom,
! [R: mat_complex,S2: produc5677646155008957607omplex,R2: set_Pr6275530937341595591omplex,S3: produc5677646155008957607omplex] :
( ( member5471586270331035152omplex @ ( produc1901862033385395287omplex @ R @ S2 ) @ R2 )
=> ( ( S3 = S2 )
=> ( member5471586270331035152omplex @ ( produc1901862033385395287omplex @ R @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_210_ssubst__Pair__rhs,axiom,
! [R: mat_complex,S2: produc352478934956084711omplex,R2: set_Pr6692490089613684743omplex,S3: produc352478934956084711omplex] :
( ( member74738575112047696omplex @ ( produc2861545499953221015omplex @ R @ S2 ) @ R2 )
=> ( ( S3 = S2 )
=> ( member74738575112047696omplex @ ( produc2861545499953221015omplex @ R @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_211_ssubst__Pair__rhs,axiom,
! [R: mat_complex,S2: mat_complex,R2: set_Pr8195022564563857607omplex,S3: mat_complex] :
( ( member8347409015010237200omplex @ ( produc3658446505030690647omplex @ R @ S2 ) @ R2 )
=> ( ( S3 = S2 )
=> ( member8347409015010237200omplex @ ( produc3658446505030690647omplex @ R @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_212_ssubst__Pair__rhs,axiom,
! [R: nat,S2: nat,R2: set_Pr1261947904930325089at_nat,S3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S2 ) @ R2 )
=> ( ( S3 = S2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_213_ssubst__Pair__rhs,axiom,
! [R: nat > nat,S2: nat,R2: set_Pr9093778441882193744at_nat,S3: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ R @ S2 ) @ R2 )
=> ( ( S3 = S2 )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ R @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_214_split__block__diagonal,axiom,
! [D2: mat_complex,N: nat,A2: nat,D1: mat_complex,D22: mat_complex,D3: mat_complex,D4: mat_complex] :
( ( diagonal_mat_complex @ D2 )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_eq_nat @ A2 @ N )
=> ( ( ( split_block_complex @ D2 @ A2 @ A2 )
= ( produc1901862033385395287omplex @ D1 @ ( produc2861545499953221015omplex @ D22 @ ( produc3658446505030690647omplex @ D3 @ D4 ) ) ) )
=> ( ( diagonal_mat_complex @ D1 )
& ( diagonal_mat_complex @ D4 ) ) ) ) ) ) ).
% split_block_diagonal
thf(fact_215_split__block__diag__carrier_I2_J,axiom,
! [D2: mat_complex,N: nat,A2: nat,D1: mat_complex,D22: mat_complex,D3: mat_complex,D4: mat_complex] :
( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_eq_nat @ A2 @ N )
=> ( ( ( split_block_complex @ D2 @ A2 @ A2 )
= ( produc1901862033385395287omplex @ D1 @ ( produc2861545499953221015omplex @ D22 @ ( produc3658446505030690647omplex @ D3 @ D4 ) ) ) )
=> ( member_mat_complex @ D4 @ ( carrier_mat_complex @ ( minus_minus_nat @ N @ A2 ) @ ( minus_minus_nat @ N @ A2 ) ) ) ) ) ) ).
% split_block_diag_carrier(2)
thf(fact_216_assms_I2_J,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% assms(2)
thf(fact_217_mem__case__prodE,axiom,
! [Z: mat_complex,C2: mat_complex > produc5677646155008957607omplex > set_mat_complex,P2: produc1634985270395358183omplex] :
( ( member_mat_complex @ Z @ ( produc8983715372140011163omplex @ C2 @ P2 ) )
=> ~ ! [X3: mat_complex,Y2: produc5677646155008957607omplex] :
( ( P2
= ( produc1901862033385395287omplex @ X3 @ Y2 ) )
=> ~ ( member_mat_complex @ Z @ ( C2 @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_218_mem__case__prodE,axiom,
! [Z: real,C2: mat_complex > produc5677646155008957607omplex > set_real,P2: produc1634985270395358183omplex] :
( ( member_real @ Z @ ( produc1354186782250546510t_real @ C2 @ P2 ) )
=> ~ ! [X3: mat_complex,Y2: produc5677646155008957607omplex] :
( ( P2
= ( produc1901862033385395287omplex @ X3 @ Y2 ) )
=> ~ ( member_real @ Z @ ( C2 @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_219_mem__case__prodE,axiom,
! [Z: mat_complex,C2: mat_complex > produc352478934956084711omplex > set_mat_complex,P2: produc5677646155008957607omplex] :
( ( member_mat_complex @ Z @ ( produc386998054833891163omplex @ C2 @ P2 ) )
=> ~ ! [X3: mat_complex,Y2: produc352478934956084711omplex] :
( ( P2
= ( produc2861545499953221015omplex @ X3 @ Y2 ) )
=> ~ ( member_mat_complex @ Z @ ( C2 @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_220_mem__case__prodE,axiom,
! [Z: real,C2: mat_complex > produc352478934956084711omplex > set_real,P2: produc5677646155008957607omplex] :
( ( member_real @ Z @ ( produc7148884236971998350t_real @ C2 @ P2 ) )
=> ~ ! [X3: mat_complex,Y2: produc352478934956084711omplex] :
( ( P2
= ( produc2861545499953221015omplex @ X3 @ Y2 ) )
=> ~ ( member_real @ Z @ ( C2 @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_221_mem__case__prodE,axiom,
! [Z: mat_complex,C2: mat_complex > mat_complex > set_mat_complex,P2: produc352478934956084711omplex] :
( ( member_mat_complex @ Z @ ( produc3802106415914905499omplex @ C2 @ P2 ) )
=> ~ ! [X3: mat_complex,Y2: mat_complex] :
( ( P2
= ( produc3658446505030690647omplex @ X3 @ Y2 ) )
=> ~ ( member_mat_complex @ Z @ ( C2 @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_222_mem__case__prodE,axiom,
! [Z: real,C2: mat_complex > mat_complex > set_real,P2: produc352478934956084711omplex] :
( ( member_real @ Z @ ( produc2518032081017119822t_real @ C2 @ P2 ) )
=> ~ ! [X3: mat_complex,Y2: mat_complex] :
( ( P2
= ( produc3658446505030690647omplex @ X3 @ Y2 ) )
=> ~ ( member_real @ Z @ ( C2 @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_223_mem__case__prodE,axiom,
! [Z: mat_complex,C2: nat > nat > set_mat_complex,P2: product_prod_nat_nat] :
( ( member_mat_complex @ Z @ ( produc7021537633938252857omplex @ C2 @ P2 ) )
=> ~ ! [X3: nat,Y2: nat] :
( ( P2
= ( product_Pair_nat_nat @ X3 @ Y2 ) )
=> ~ ( member_mat_complex @ Z @ ( C2 @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_224_mem__case__prodE,axiom,
! [Z: real,C2: nat > nat > set_real,P2: product_prod_nat_nat] :
( ( member_real @ Z @ ( produc3668448655016342576t_real @ C2 @ P2 ) )
=> ~ ! [X3: nat,Y2: nat] :
( ( P2
= ( product_Pair_nat_nat @ X3 @ Y2 ) )
=> ~ ( member_real @ Z @ ( C2 @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_225_mem__case__prodE,axiom,
! [Z: mat_complex,C2: ( nat > nat ) > nat > set_mat_complex,P2: produc8199716216217303280at_nat] :
( ( member_mat_complex @ Z @ ( produc5815470908825352714omplex @ C2 @ P2 ) )
=> ~ ! [X3: nat > nat,Y2: nat] :
( ( P2
= ( produc72220940542539688at_nat @ X3 @ Y2 ) )
=> ~ ( member_mat_complex @ Z @ ( C2 @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_226_mem__case__prodE,axiom,
! [Z: real,C2: ( nat > nat ) > nat > set_real,P2: produc8199716216217303280at_nat] :
( ( member_real @ Z @ ( produc5041090368443872287t_real @ C2 @ P2 ) )
=> ~ ! [X3: nat > nat,Y2: nat] :
( ( P2
= ( produc72220940542539688at_nat @ X3 @ Y2 ) )
=> ~ ( member_real @ Z @ ( C2 @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_227_Product__Type_OCollect__case__prodD,axiom,
! [X: produc1634985270395358183omplex,A4: mat_complex > produc5677646155008957607omplex > $o] :
( ( member5471586270331035152omplex @ X @ ( collec875000405105826898omplex @ ( produc6770849645695511596plex_o @ A4 ) ) )
=> ( A4 @ ( produc8911724726559533635omplex @ X ) @ ( produc943930114779824517omplex @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_228_Product__Type_OCollect__case__prodD,axiom,
! [X: produc5677646155008957607omplex,A4: mat_complex > produc352478934956084711omplex > $o] :
( ( member74738575112047696omplex @ X @ ( collec2557858944451489170omplex @ ( produc5497593456678237420plex_o @ A4 ) ) )
=> ( A4 @ ( produc2697000228617323907omplex @ X ) @ ( produc7343567217041670085omplex @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_229_Product__Type_OCollect__case__prodD,axiom,
! [X: produc352478934956084711omplex,A4: mat_complex > mat_complex > $o] :
( ( member8347409015010237200omplex @ X @ ( collec5097175541304549202omplex @ ( produc3878269250153479468plex_o @ A4 ) ) )
=> ( A4 @ ( produc9163778666669654339omplex @ X ) @ ( produc4897211011226852997omplex @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_230_Product__Type_OCollect__case__prodD,axiom,
! [X: produc8199716216217303280at_nat,A4: ( nat > nat ) > nat > $o] :
( ( member7226740684066999833at_nat @ X @ ( collec9201399625632817755at_nat @ ( produc1191635805960096411_nat_o @ A4 ) ) )
=> ( A4 @ ( produc6156676138143019412at_nat @ X ) @ ( produc1852801350702243542at_nat @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_231_Product__Type_OCollect__case__prodD,axiom,
! [X: product_prod_nat_nat,A4: nat > nat > $o] :
( ( member8440522571783428010at_nat @ X @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A4 ) ) )
=> ( A4 @ ( product_fst_nat_nat @ X ) @ ( product_snd_nat_nat @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_232_minus__prod__def,axiom,
( minus_4365393887724441320at_nat
= ( ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( minus_minus_nat @ ( product_fst_nat_nat @ X4 ) @ ( product_fst_nat_nat @ Y4 ) ) @ ( minus_minus_nat @ ( product_snd_nat_nat @ X4 ) @ ( product_snd_nat_nat @ Y4 ) ) ) ) ) ).
% minus_prod_def
thf(fact_233_minus__prod__def,axiom,
( minus_5557628854490389828t_real
= ( ^ [X4: produc7716430852924023517t_real,Y4: produc7716430852924023517t_real] : ( produc7837566107596912789t_real @ ( minus_minus_nat @ ( product_fst_nat_real @ X4 ) @ ( product_fst_nat_real @ Y4 ) ) @ ( minus_minus_real @ ( product_snd_nat_real @ X4 ) @ ( product_snd_nat_real @ Y4 ) ) ) ) ) ).
% minus_prod_def
thf(fact_234_minus__prod__def,axiom,
( minus_1582581163013509572al_nat
= ( ^ [X4: produc3741383161447143261al_nat,Y4: produc3741383161447143261al_nat] : ( produc3181502643871035669al_nat @ ( minus_minus_real @ ( product_fst_real_nat @ X4 ) @ ( product_fst_real_nat @ Y4 ) ) @ ( minus_minus_nat @ ( product_snd_real_nat @ X4 ) @ ( product_snd_real_nat @ Y4 ) ) ) ) ) ).
% minus_prod_def
thf(fact_235_minus__prod__def,axiom,
( minus_885040589139849760l_real
= ( ^ [X4: produc2422161461964618553l_real,Y4: produc2422161461964618553l_real] : ( produc4511245868158468465l_real @ ( minus_minus_real @ ( produc5828954698716094813l_real @ X4 ) @ ( produc5828954698716094813l_real @ Y4 ) ) @ ( minus_minus_real @ ( produc3484788084999411615l_real @ X4 ) @ ( produc3484788084999411615l_real @ Y4 ) ) ) ) ) ).
% minus_prod_def
thf(fact_236_minus__prod__def,axiom,
( minus_9125208095613564965omplex
= ( ^ [X4: produc3259542890344722124omplex,Y4: produc3259542890344722124omplex] : ( produc4998868960714853886omplex @ ( minus_minus_nat @ ( produc8687169775924804370omplex @ X4 ) @ ( produc8687169775924804370omplex @ Y4 ) ) @ ( minus_2412168080157227406omplex @ ( produc130099875952336976omplex @ X4 ) @ ( produc130099875952336976omplex @ Y4 ) ) ) ) ) ).
% minus_prod_def
thf(fact_237_minus__prod__def,axiom,
( minus_1583438508407137535ex_nat
= ( ^ [X4: produc4941145339993070502ex_nat,Y4: produc4941145339993070502ex_nat] : ( produc3916067632315525152ex_nat @ ( minus_2412168080157227406omplex @ ( produc7604368447525475636ex_nat @ X4 ) @ ( produc7604368447525475636ex_nat @ Y4 ) ) @ ( minus_minus_nat @ ( produc8270670584407784050ex_nat @ X4 ) @ ( produc8270670584407784050ex_nat @ Y4 ) ) ) ) ) ).
% minus_prod_def
thf(fact_238_minus__prod__def,axiom,
( minus_5460563211077212123x_real
= ( ^ [X4: produc7080596761207171842x_real,Y4: produc7080596761207171842x_real] : ( produc8172131712939336444x_real @ ( minus_2412168080157227406omplex @ ( produc7072908607146575376x_real @ X4 ) @ ( produc7072908607146575376x_real @ Y4 ) ) @ ( minus_minus_real @ ( produc7625890766153296974x_real @ X4 ) @ ( produc7625890766153296974x_real @ Y4 ) ) ) ) ) ).
% minus_prod_def
thf(fact_239_minus__prod__def,axiom,
( minus_7607308147470025289omplex
= ( ^ [X4: produc3969660745209200omplex,Y4: produc3969660745209200omplex] : ( produc3178892618276054050omplex @ ( minus_minus_real @ ( produc2079669512483292982omplex @ X4 ) @ ( produc2079669512483292982omplex @ Y4 ) ) @ ( minus_2412168080157227406omplex @ ( produc2632651671490014580omplex @ X4 ) @ ( produc2632651671490014580omplex @ Y4 ) ) ) ) ) ).
% minus_prod_def
thf(fact_240_minus__prod__def,axiom,
( minus_9067931446424981591at_nat
= ( ^ [X4: produc8199716216217303280at_nat,Y4: produc8199716216217303280at_nat] : ( produc72220940542539688at_nat @ ( minus_minus_nat_nat @ ( produc6156676138143019412at_nat @ X4 ) @ ( produc6156676138143019412at_nat @ Y4 ) ) @ ( minus_minus_nat @ ( produc1852801350702243542at_nat @ X4 ) @ ( produc1852801350702243542at_nat @ Y4 ) ) ) ) ) ).
% minus_prod_def
thf(fact_241_minus__prod__def,axiom,
( minus_2734116836287720782omplex
= ( ^ [X4: produc352478934956084711omplex,Y4: produc352478934956084711omplex] : ( produc3658446505030690647omplex @ ( minus_2412168080157227406omplex @ ( produc9163778666669654339omplex @ X4 ) @ ( produc9163778666669654339omplex @ Y4 ) ) @ ( minus_2412168080157227406omplex @ ( produc4897211011226852997omplex @ X4 ) @ ( produc4897211011226852997omplex @ Y4 ) ) ) ) ) ).
% minus_prod_def
thf(fact_242_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_243_diff__less__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_minus_nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_244_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_245_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_246_less__eq__prod__def,axiom,
( ord_le2083499100524348295omplex
= ( ^ [X4: produc1634985270395358183omplex,Y4: produc1634985270395358183omplex] :
( ( ord_le1403324449407493959omplex @ ( produc8911724726559533635omplex @ X4 ) @ ( produc8911724726559533635omplex @ Y4 ) )
& ( ord_le5571771215207171143omplex @ ( produc943930114779824517omplex @ X4 ) @ ( produc943930114779824517omplex @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_247_less__eq__prod__def,axiom,
( ord_le5571771215207171143omplex
= ( ^ [X4: produc5677646155008957607omplex,Y4: produc5677646155008957607omplex] :
( ( ord_le1403324449407493959omplex @ ( produc2697000228617323907omplex @ X4 ) @ ( produc2697000228617323907omplex @ Y4 ) )
& ( ord_le6144634291999778183omplex @ ( produc7343567217041670085omplex @ X4 ) @ ( produc7343567217041670085omplex @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_248_less__eq__prod__def,axiom,
( ord_le6144634291999778183omplex
= ( ^ [X4: produc352478934956084711omplex,Y4: produc352478934956084711omplex] :
( ( ord_le1403324449407493959omplex @ ( produc9163778666669654339omplex @ X4 ) @ ( produc9163778666669654339omplex @ Y4 ) )
& ( ord_le1403324449407493959omplex @ ( produc4897211011226852997omplex @ X4 ) @ ( produc4897211011226852997omplex @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_249_less__eq__prod__def,axiom,
( ord_le2819838839419867280at_nat
= ( ^ [X4: produc8199716216217303280at_nat,Y4: produc8199716216217303280at_nat] :
( ( ord_less_eq_nat_nat @ ( produc6156676138143019412at_nat @ X4 ) @ ( produc6156676138143019412at_nat @ Y4 ) )
& ( ord_less_eq_nat @ ( produc1852801350702243542at_nat @ X4 ) @ ( produc1852801350702243542at_nat @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_250_less__eq__prod__def,axiom,
( ord_le8460144461188290721at_nat
= ( ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
( ( ord_less_eq_nat @ ( product_fst_nat_nat @ X4 ) @ ( product_fst_nat_nat @ Y4 ) )
& ( ord_less_eq_nat @ ( product_snd_nat_nat @ X4 ) @ ( product_snd_nat_nat @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_251_less__eq__prod__def,axiom,
( ord_le8710666929947597437t_real
= ( ^ [X4: produc7716430852924023517t_real,Y4: produc7716430852924023517t_real] :
( ( ord_less_eq_nat @ ( product_fst_nat_real @ X4 ) @ ( product_fst_nat_real @ Y4 ) )
& ( ord_less_eq_real @ ( product_snd_nat_real @ X4 ) @ ( product_snd_nat_real @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_252_less__eq__prod__def,axiom,
( ord_le4735619238470717181al_nat
= ( ^ [X4: produc3741383161447143261al_nat,Y4: produc3741383161447143261al_nat] :
( ( ord_less_eq_real @ ( product_fst_real_nat @ X4 ) @ ( product_fst_real_nat @ Y4 ) )
& ( ord_less_eq_nat @ ( product_snd_real_nat @ X4 ) @ ( product_snd_real_nat @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_253_less__eq__prod__def,axiom,
( ord_le1075799226346578649l_real
= ( ^ [X4: produc2422161461964618553l_real,Y4: produc2422161461964618553l_real] :
( ( ord_less_eq_real @ ( produc5828954698716094813l_real @ X4 ) @ ( produc5828954698716094813l_real @ Y4 ) )
& ( ord_less_eq_real @ ( produc3484788084999411615l_real @ X4 ) @ ( produc3484788084999411615l_real @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_254_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_255_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_256_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ K2 )
=> ~ ( P @ I ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_257_diff__gt__0__iff__gt,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B2 ) )
= ( ord_less_real @ B2 @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_258_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_259_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_260_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C2 ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_261_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: real,C2: real,B2: real] :
( ( minus_minus_real @ ( minus_minus_real @ A2 @ C2 ) @ B2 )
= ( minus_minus_real @ ( minus_minus_real @ A2 @ B2 ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_262_diff__eq__diff__eq,axiom,
! [A2: real,B2: real,C2: real,D5: real] :
( ( ( minus_minus_real @ A2 @ B2 )
= ( minus_minus_real @ C2 @ D5 ) )
=> ( ( A2 = B2 )
= ( C2 = D5 ) ) ) ).
% diff_eq_diff_eq
thf(fact_263_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_264_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_265_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_266_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_267_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_268_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_269_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_270_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_271_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_272_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_273_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_274_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_275_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_276_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_277_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_278_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_279_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_280_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_281_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_282_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_283_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_284_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_285_zero__prod__def,axiom,
( zero_z3979849011205770936at_nat
= ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ).
% zero_prod_def
thf(fact_286_zero__prod__def,axiom,
( zero_z738777567634093332t_real
= ( produc7837566107596912789t_real @ zero_zero_nat @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_287_zero__prod__def,axiom,
( zero_z5987101913011988884al_nat
= ( produc3181502643871035669al_nat @ zero_zero_real @ zero_zero_nat ) ) ).
% zero_prod_def
thf(fact_288_zero__prod__def,axiom,
( zero_z1365759597461889520l_real
= ( produc4511245868158468465l_real @ zero_zero_real @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_289_Pair__le,axiom,
! [A2: mat_complex,B2: produc5677646155008957607omplex,C2: mat_complex,D5: produc5677646155008957607omplex] :
( ( ord_le2083499100524348295omplex @ ( produc1901862033385395287omplex @ A2 @ B2 ) @ ( produc1901862033385395287omplex @ C2 @ D5 ) )
= ( ( ord_le1403324449407493959omplex @ A2 @ C2 )
& ( ord_le5571771215207171143omplex @ B2 @ D5 ) ) ) ).
% Pair_le
thf(fact_290_Pair__le,axiom,
! [A2: mat_complex,B2: produc352478934956084711omplex,C2: mat_complex,D5: produc352478934956084711omplex] :
( ( ord_le5571771215207171143omplex @ ( produc2861545499953221015omplex @ A2 @ B2 ) @ ( produc2861545499953221015omplex @ C2 @ D5 ) )
= ( ( ord_le1403324449407493959omplex @ A2 @ C2 )
& ( ord_le6144634291999778183omplex @ B2 @ D5 ) ) ) ).
% Pair_le
thf(fact_291_Pair__le,axiom,
! [A2: mat_complex,B2: mat_complex,C2: mat_complex,D5: mat_complex] :
( ( ord_le6144634291999778183omplex @ ( produc3658446505030690647omplex @ A2 @ B2 ) @ ( produc3658446505030690647omplex @ C2 @ D5 ) )
= ( ( ord_le1403324449407493959omplex @ A2 @ C2 )
& ( ord_le1403324449407493959omplex @ B2 @ D5 ) ) ) ).
% Pair_le
thf(fact_292_Pair__le,axiom,
! [A2: nat > nat,B2: nat,C2: nat > nat,D5: nat] :
( ( ord_le2819838839419867280at_nat @ ( produc72220940542539688at_nat @ A2 @ B2 ) @ ( produc72220940542539688at_nat @ C2 @ D5 ) )
= ( ( ord_less_eq_nat_nat @ A2 @ C2 )
& ( ord_less_eq_nat @ B2 @ D5 ) ) ) ).
% Pair_le
thf(fact_293_Pair__le,axiom,
! [A2: nat,B2: nat,C2: nat,D5: nat] :
( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( product_Pair_nat_nat @ C2 @ D5 ) )
= ( ( ord_less_eq_nat @ A2 @ C2 )
& ( ord_less_eq_nat @ B2 @ D5 ) ) ) ).
% Pair_le
thf(fact_294_Pair__le,axiom,
! [A2: nat,B2: real,C2: nat,D5: real] :
( ( ord_le8710666929947597437t_real @ ( produc7837566107596912789t_real @ A2 @ B2 ) @ ( produc7837566107596912789t_real @ C2 @ D5 ) )
= ( ( ord_less_eq_nat @ A2 @ C2 )
& ( ord_less_eq_real @ B2 @ D5 ) ) ) ).
% Pair_le
thf(fact_295_Pair__le,axiom,
! [A2: real,B2: nat,C2: real,D5: nat] :
( ( ord_le4735619238470717181al_nat @ ( produc3181502643871035669al_nat @ A2 @ B2 ) @ ( produc3181502643871035669al_nat @ C2 @ D5 ) )
= ( ( ord_less_eq_real @ A2 @ C2 )
& ( ord_less_eq_nat @ B2 @ D5 ) ) ) ).
% Pair_le
thf(fact_296_Pair__le,axiom,
! [A2: real,B2: real,C2: real,D5: real] :
( ( ord_le1075799226346578649l_real @ ( produc4511245868158468465l_real @ A2 @ B2 ) @ ( produc4511245868158468465l_real @ C2 @ D5 ) )
= ( ( ord_less_eq_real @ A2 @ C2 )
& ( ord_less_eq_real @ B2 @ D5 ) ) ) ).
% Pair_le
thf(fact_297_Pair__mono,axiom,
! [X: mat_complex,X6: mat_complex,Y: produc5677646155008957607omplex,Y6: produc5677646155008957607omplex] :
( ( ord_le1403324449407493959omplex @ X @ X6 )
=> ( ( ord_le5571771215207171143omplex @ Y @ Y6 )
=> ( ord_le2083499100524348295omplex @ ( produc1901862033385395287omplex @ X @ Y ) @ ( produc1901862033385395287omplex @ X6 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_298_Pair__mono,axiom,
! [X: mat_complex,X6: mat_complex,Y: produc352478934956084711omplex,Y6: produc352478934956084711omplex] :
( ( ord_le1403324449407493959omplex @ X @ X6 )
=> ( ( ord_le6144634291999778183omplex @ Y @ Y6 )
=> ( ord_le5571771215207171143omplex @ ( produc2861545499953221015omplex @ X @ Y ) @ ( produc2861545499953221015omplex @ X6 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_299_Pair__mono,axiom,
! [X: mat_complex,X6: mat_complex,Y: mat_complex,Y6: mat_complex] :
( ( ord_le1403324449407493959omplex @ X @ X6 )
=> ( ( ord_le1403324449407493959omplex @ Y @ Y6 )
=> ( ord_le6144634291999778183omplex @ ( produc3658446505030690647omplex @ X @ Y ) @ ( produc3658446505030690647omplex @ X6 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_300_Pair__mono,axiom,
! [X: nat > nat,X6: nat > nat,Y: nat,Y6: nat] :
( ( ord_less_eq_nat_nat @ X @ X6 )
=> ( ( ord_less_eq_nat @ Y @ Y6 )
=> ( ord_le2819838839419867280at_nat @ ( produc72220940542539688at_nat @ X @ Y ) @ ( produc72220940542539688at_nat @ X6 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_301_Pair__mono,axiom,
! [X: nat,X6: nat,Y: nat,Y6: nat] :
( ( ord_less_eq_nat @ X @ X6 )
=> ( ( ord_less_eq_nat @ Y @ Y6 )
=> ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X6 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_302_Pair__mono,axiom,
! [X: nat,X6: nat,Y: real,Y6: real] :
( ( ord_less_eq_nat @ X @ X6 )
=> ( ( ord_less_eq_real @ Y @ Y6 )
=> ( ord_le8710666929947597437t_real @ ( produc7837566107596912789t_real @ X @ Y ) @ ( produc7837566107596912789t_real @ X6 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_303_Pair__mono,axiom,
! [X: real,X6: real,Y: nat,Y6: nat] :
( ( ord_less_eq_real @ X @ X6 )
=> ( ( ord_less_eq_nat @ Y @ Y6 )
=> ( ord_le4735619238470717181al_nat @ ( produc3181502643871035669al_nat @ X @ Y ) @ ( produc3181502643871035669al_nat @ X6 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_304_Pair__mono,axiom,
! [X: real,X6: real,Y: real,Y6: real] :
( ( ord_less_eq_real @ X @ X6 )
=> ( ( ord_less_eq_real @ Y @ Y6 )
=> ( ord_le1075799226346578649l_real @ ( produc4511245868158468465l_real @ X @ Y ) @ ( produc4511245868158468465l_real @ X6 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_305_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_306_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_307_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_308_diff__zero,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_zero
thf(fact_309_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A5: real,B5: real] :
( ( minus_minus_real @ A5 @ B5 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_310_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_311_diff__0__right,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_0_right
thf(fact_312_diff__self,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% diff_self
thf(fact_313_diff__eq__diff__less__eq,axiom,
! [A2: real,B2: real,C2: real,D5: real] :
( ( ( minus_minus_real @ A2 @ B2 )
= ( minus_minus_real @ C2 @ D5 ) )
=> ( ( ord_less_eq_real @ A2 @ B2 )
= ( ord_less_eq_real @ C2 @ D5 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_314_diff__right__mono,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C2 ) @ ( minus_minus_real @ B2 @ C2 ) ) ) ).
% diff_right_mono
thf(fact_315_diff__left__mono,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ C2 @ A2 ) @ ( minus_minus_real @ C2 @ B2 ) ) ) ).
% diff_left_mono
thf(fact_316_diff__mono,axiom,
! [A2: real,B2: real,D5: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ D5 @ C2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C2 ) @ ( minus_minus_real @ B2 @ D5 ) ) ) ) ).
% diff_mono
thf(fact_317_diff__strict__right__mono,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ord_less_real @ ( minus_minus_real @ A2 @ C2 ) @ ( minus_minus_real @ B2 @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_318_diff__strict__left__mono,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ord_less_real @ ( minus_minus_real @ C2 @ A2 ) @ ( minus_minus_real @ C2 @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_319_diff__eq__diff__less,axiom,
! [A2: real,B2: real,C2: real,D5: real] :
( ( ( minus_minus_real @ A2 @ B2 )
= ( minus_minus_real @ C2 @ D5 ) )
=> ( ( ord_less_real @ A2 @ B2 )
= ( ord_less_real @ C2 @ D5 ) ) ) ).
% diff_eq_diff_less
thf(fact_320_diff__strict__mono,axiom,
! [A2: real,B2: real,D5: real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ D5 @ C2 )
=> ( ord_less_real @ ( minus_minus_real @ A2 @ C2 ) @ ( minus_minus_real @ B2 @ D5 ) ) ) ) ).
% diff_strict_mono
thf(fact_321_diff__Pair,axiom,
! [A2: nat,B2: nat,C2: nat,D5: nat] :
( ( minus_4365393887724441320at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( product_Pair_nat_nat @ C2 @ D5 ) )
= ( product_Pair_nat_nat @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_minus_nat @ B2 @ D5 ) ) ) ).
% diff_Pair
thf(fact_322_diff__Pair,axiom,
! [A2: nat,B2: real,C2: nat,D5: real] :
( ( minus_5557628854490389828t_real @ ( produc7837566107596912789t_real @ A2 @ B2 ) @ ( produc7837566107596912789t_real @ C2 @ D5 ) )
= ( produc7837566107596912789t_real @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_minus_real @ B2 @ D5 ) ) ) ).
% diff_Pair
thf(fact_323_diff__Pair,axiom,
! [A2: real,B2: nat,C2: real,D5: nat] :
( ( minus_1582581163013509572al_nat @ ( produc3181502643871035669al_nat @ A2 @ B2 ) @ ( produc3181502643871035669al_nat @ C2 @ D5 ) )
= ( produc3181502643871035669al_nat @ ( minus_minus_real @ A2 @ C2 ) @ ( minus_minus_nat @ B2 @ D5 ) ) ) ).
% diff_Pair
thf(fact_324_diff__Pair,axiom,
! [A2: real,B2: real,C2: real,D5: real] :
( ( minus_885040589139849760l_real @ ( produc4511245868158468465l_real @ A2 @ B2 ) @ ( produc4511245868158468465l_real @ C2 @ D5 ) )
= ( produc4511245868158468465l_real @ ( minus_minus_real @ A2 @ C2 ) @ ( minus_minus_real @ B2 @ D5 ) ) ) ).
% diff_Pair
thf(fact_325_diff__Pair,axiom,
! [A2: nat,B2: mat_complex,C2: nat,D5: mat_complex] :
( ( minus_9125208095613564965omplex @ ( produc4998868960714853886omplex @ A2 @ B2 ) @ ( produc4998868960714853886omplex @ C2 @ D5 ) )
= ( produc4998868960714853886omplex @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_2412168080157227406omplex @ B2 @ D5 ) ) ) ).
% diff_Pair
thf(fact_326_diff__Pair,axiom,
! [A2: mat_complex,B2: nat,C2: mat_complex,D5: nat] :
( ( minus_1583438508407137535ex_nat @ ( produc3916067632315525152ex_nat @ A2 @ B2 ) @ ( produc3916067632315525152ex_nat @ C2 @ D5 ) )
= ( produc3916067632315525152ex_nat @ ( minus_2412168080157227406omplex @ A2 @ C2 ) @ ( minus_minus_nat @ B2 @ D5 ) ) ) ).
% diff_Pair
thf(fact_327_diff__Pair,axiom,
! [A2: mat_complex,B2: real,C2: mat_complex,D5: real] :
( ( minus_5460563211077212123x_real @ ( produc8172131712939336444x_real @ A2 @ B2 ) @ ( produc8172131712939336444x_real @ C2 @ D5 ) )
= ( produc8172131712939336444x_real @ ( minus_2412168080157227406omplex @ A2 @ C2 ) @ ( minus_minus_real @ B2 @ D5 ) ) ) ).
% diff_Pair
thf(fact_328_diff__Pair,axiom,
! [A2: real,B2: mat_complex,C2: real,D5: mat_complex] :
( ( minus_7607308147470025289omplex @ ( produc3178892618276054050omplex @ A2 @ B2 ) @ ( produc3178892618276054050omplex @ C2 @ D5 ) )
= ( produc3178892618276054050omplex @ ( minus_minus_real @ A2 @ C2 ) @ ( minus_2412168080157227406omplex @ B2 @ D5 ) ) ) ).
% diff_Pair
thf(fact_329_diff__Pair,axiom,
! [A2: nat > nat,B2: nat,C2: nat > nat,D5: nat] :
( ( minus_9067931446424981591at_nat @ ( produc72220940542539688at_nat @ A2 @ B2 ) @ ( produc72220940542539688at_nat @ C2 @ D5 ) )
= ( produc72220940542539688at_nat @ ( minus_minus_nat_nat @ A2 @ C2 ) @ ( minus_minus_nat @ B2 @ D5 ) ) ) ).
% diff_Pair
thf(fact_330_diff__Pair,axiom,
! [A2: mat_complex,B2: mat_complex,C2: mat_complex,D5: mat_complex] :
( ( minus_2734116836287720782omplex @ ( produc3658446505030690647omplex @ A2 @ B2 ) @ ( produc3658446505030690647omplex @ C2 @ D5 ) )
= ( produc3658446505030690647omplex @ ( minus_2412168080157227406omplex @ A2 @ C2 ) @ ( minus_2412168080157227406omplex @ B2 @ D5 ) ) ) ).
% diff_Pair
thf(fact_331_fst__zero,axiom,
( ( product_fst_nat_nat @ zero_z3979849011205770936at_nat )
= zero_zero_nat ) ).
% fst_zero
thf(fact_332_fst__mono,axiom,
! [X: produc1634985270395358183omplex,Y: produc1634985270395358183omplex] :
( ( ord_le2083499100524348295omplex @ X @ Y )
=> ( ord_le1403324449407493959omplex @ ( produc8911724726559533635omplex @ X ) @ ( produc8911724726559533635omplex @ Y ) ) ) ).
% fst_mono
thf(fact_333_fst__mono,axiom,
! [X: produc5677646155008957607omplex,Y: produc5677646155008957607omplex] :
( ( ord_le5571771215207171143omplex @ X @ Y )
=> ( ord_le1403324449407493959omplex @ ( produc2697000228617323907omplex @ X ) @ ( produc2697000228617323907omplex @ Y ) ) ) ).
% fst_mono
thf(fact_334_fst__mono,axiom,
! [X: produc352478934956084711omplex,Y: produc352478934956084711omplex] :
( ( ord_le6144634291999778183omplex @ X @ Y )
=> ( ord_le1403324449407493959omplex @ ( produc9163778666669654339omplex @ X ) @ ( produc9163778666669654339omplex @ Y ) ) ) ).
% fst_mono
thf(fact_335_fst__mono,axiom,
! [X: produc8199716216217303280at_nat,Y: produc8199716216217303280at_nat] :
( ( ord_le2819838839419867280at_nat @ X @ Y )
=> ( ord_less_eq_nat_nat @ ( produc6156676138143019412at_nat @ X ) @ ( produc6156676138143019412at_nat @ Y ) ) ) ).
% fst_mono
thf(fact_336_fst__mono,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( ord_le8460144461188290721at_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( product_fst_nat_nat @ X ) @ ( product_fst_nat_nat @ Y ) ) ) ).
% fst_mono
thf(fact_337_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_338_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_339_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_340_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_341_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_342_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_343_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_344_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_345_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_346_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_347_snd__zero,axiom,
( ( product_snd_nat_nat @ zero_z3979849011205770936at_nat )
= zero_zero_nat ) ).
% snd_zero
thf(fact_348_snd__mono,axiom,
! [X: produc1634985270395358183omplex,Y: produc1634985270395358183omplex] :
( ( ord_le2083499100524348295omplex @ X @ Y )
=> ( ord_le5571771215207171143omplex @ ( produc943930114779824517omplex @ X ) @ ( produc943930114779824517omplex @ Y ) ) ) ).
% snd_mono
thf(fact_349_snd__mono,axiom,
! [X: produc5677646155008957607omplex,Y: produc5677646155008957607omplex] :
( ( ord_le5571771215207171143omplex @ X @ Y )
=> ( ord_le6144634291999778183omplex @ ( produc7343567217041670085omplex @ X ) @ ( produc7343567217041670085omplex @ Y ) ) ) ).
% snd_mono
thf(fact_350_snd__mono,axiom,
! [X: produc352478934956084711omplex,Y: produc352478934956084711omplex] :
( ( ord_le6144634291999778183omplex @ X @ Y )
=> ( ord_le1403324449407493959omplex @ ( produc4897211011226852997omplex @ X ) @ ( produc4897211011226852997omplex @ Y ) ) ) ).
% snd_mono
thf(fact_351_snd__mono,axiom,
! [X: produc8199716216217303280at_nat,Y: produc8199716216217303280at_nat] :
( ( ord_le2819838839419867280at_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( produc1852801350702243542at_nat @ X ) @ ( produc1852801350702243542at_nat @ Y ) ) ) ).
% snd_mono
thf(fact_352_snd__mono,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( ord_le8460144461188290721at_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( product_snd_nat_nat @ X ) @ ( product_snd_nat_nat @ Y ) ) ) ).
% snd_mono
thf(fact_353_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_354_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_355_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_356_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_357_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_358_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_359_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_360_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_361_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_362_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_363_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_364_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_365_fst__diff,axiom,
! [X: produc8199716216217303280at_nat,Y: produc8199716216217303280at_nat] :
( ( produc6156676138143019412at_nat @ ( minus_9067931446424981591at_nat @ X @ Y ) )
= ( minus_minus_nat_nat @ ( produc6156676138143019412at_nat @ X ) @ ( produc6156676138143019412at_nat @ Y ) ) ) ).
% fst_diff
thf(fact_366_fst__diff,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( product_fst_nat_nat @ ( minus_4365393887724441320at_nat @ X @ Y ) )
= ( minus_minus_nat @ ( product_fst_nat_nat @ X ) @ ( product_fst_nat_nat @ Y ) ) ) ).
% fst_diff
thf(fact_367_fst__diff,axiom,
! [X: produc1634985270395358183omplex,Y: produc1634985270395358183omplex] :
( ( produc8911724726559533635omplex @ ( minus_5093045068546291278omplex @ X @ Y ) )
= ( minus_2412168080157227406omplex @ ( produc8911724726559533635omplex @ X ) @ ( produc8911724726559533635omplex @ Y ) ) ) ).
% fst_diff
thf(fact_368_fst__diff,axiom,
! [X: produc5677646155008957607omplex,Y: produc5677646155008957607omplex] :
( ( produc2697000228617323907omplex @ ( minus_4882995375997288846omplex @ X @ Y ) )
= ( minus_2412168080157227406omplex @ ( produc2697000228617323907omplex @ X ) @ ( produc2697000228617323907omplex @ Y ) ) ) ).
% fst_diff
thf(fact_369_fst__diff,axiom,
! [X: produc352478934956084711omplex,Y: produc352478934956084711omplex] :
( ( produc9163778666669654339omplex @ ( minus_2734116836287720782omplex @ X @ Y ) )
= ( minus_2412168080157227406omplex @ ( produc9163778666669654339omplex @ X ) @ ( produc9163778666669654339omplex @ Y ) ) ) ).
% fst_diff
thf(fact_370_snd__diff,axiom,
! [X: produc1634985270395358183omplex,Y: produc1634985270395358183omplex] :
( ( produc943930114779824517omplex @ ( minus_5093045068546291278omplex @ X @ Y ) )
= ( minus_4882995375997288846omplex @ ( produc943930114779824517omplex @ X ) @ ( produc943930114779824517omplex @ Y ) ) ) ).
% snd_diff
thf(fact_371_snd__diff,axiom,
! [X: produc5677646155008957607omplex,Y: produc5677646155008957607omplex] :
( ( produc7343567217041670085omplex @ ( minus_4882995375997288846omplex @ X @ Y ) )
= ( minus_2734116836287720782omplex @ ( produc7343567217041670085omplex @ X ) @ ( produc7343567217041670085omplex @ Y ) ) ) ).
% snd_diff
thf(fact_372_snd__diff,axiom,
! [X: produc8199716216217303280at_nat,Y: produc8199716216217303280at_nat] :
( ( produc1852801350702243542at_nat @ ( minus_9067931446424981591at_nat @ X @ Y ) )
= ( minus_minus_nat @ ( produc1852801350702243542at_nat @ X ) @ ( produc1852801350702243542at_nat @ Y ) ) ) ).
% snd_diff
thf(fact_373_snd__diff,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( product_snd_nat_nat @ ( minus_4365393887724441320at_nat @ X @ Y ) )
= ( minus_minus_nat @ ( product_snd_nat_nat @ X ) @ ( product_snd_nat_nat @ Y ) ) ) ).
% snd_diff
thf(fact_374_snd__diff,axiom,
! [X: produc352478934956084711omplex,Y: produc352478934956084711omplex] :
( ( produc4897211011226852997omplex @ ( minus_2734116836287720782omplex @ X @ Y ) )
= ( minus_2412168080157227406omplex @ ( produc4897211011226852997omplex @ X ) @ ( produc4897211011226852997omplex @ Y ) ) ) ).
% snd_diff
thf(fact_375_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_376_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_377_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_378_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_379_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_380_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_381_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_382_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_383_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A2 ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_384_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_385_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_386_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_387_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_388_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_389_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B5: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B5 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_390_diff__ge__0__iff__ge,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B2 ) )
= ( ord_less_eq_real @ B2 @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_391_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A5: real,B5: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B5 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_392_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A5: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A5 @ B5 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_393_split__block__commute__subblock,axiom,
! [D2: mat_complex,N: nat,B4: mat_complex,A2: nat,B1: mat_complex,B22: mat_complex,B32: mat_complex,B42: mat_complex,D1: mat_complex,D22: mat_complex,D3: mat_complex,D4: mat_complex] :
( ( diagonal_mat_complex @ D2 )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( ord_less_eq_nat @ A2 @ N )
=> ( ( ( split_block_complex @ B4 @ A2 @ A2 )
= ( produc1901862033385395287omplex @ B1 @ ( produc2861545499953221015omplex @ B22 @ ( produc3658446505030690647omplex @ B32 @ B42 ) ) ) )
=> ( ( ( split_block_complex @ D2 @ A2 @ A2 )
= ( produc1901862033385395287omplex @ D1 @ ( produc2861545499953221015omplex @ D22 @ ( produc3658446505030690647omplex @ D3 @ D4 ) ) ) )
=> ( ( ( times_8009071140041733218omplex @ B4 @ D2 )
= ( times_8009071140041733218omplex @ D2 @ B4 ) )
=> ( ( times_8009071140041733218omplex @ B42 @ D4 )
= ( times_8009071140041733218omplex @ D4 @ B42 ) ) ) ) ) ) ) ) ) ).
% split_block_commute_subblock
thf(fact_394_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K2: nat] :
? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( F @ K2 @ I ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ! [K3: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
& ( F @ K4 @ I3 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_395_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I: nat] :
( ( ord_less_nat @ K2 @ I )
=> ( P @ I ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_396_mult_Oleft__commute,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( times_times_nat @ B2 @ ( times_times_nat @ A2 @ C2 ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_397_mult_Oleft__commute,axiom,
! [B2: real,A2: real,C2: real] :
( ( times_times_real @ B2 @ ( times_times_real @ A2 @ C2 ) )
= ( times_times_real @ A2 @ ( times_times_real @ B2 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_398_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A5: nat,B5: nat] : ( times_times_nat @ B5 @ A5 ) ) ) ).
% mult.commute
thf(fact_399_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A5: real,B5: real] : ( times_times_real @ B5 @ A5 ) ) ) ).
% mult.commute
thf(fact_400_semigroup__mult__class_Omult_Oassoc,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
= ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) ) ).
% semigroup_mult_class.mult.assoc
thf(fact_401_semigroup__mult__class_Omult_Oassoc,axiom,
! [A2: real,B2: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B2 ) @ C2 )
= ( times_times_real @ A2 @ ( times_times_real @ B2 @ C2 ) ) ) ).
% semigroup_mult_class.mult.assoc
thf(fact_402_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
= ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_403_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: real,B2: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B2 ) @ C2 )
= ( times_times_real @ A2 @ ( times_times_real @ B2 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_404_arithmetic__simps_I63_J,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% arithmetic_simps(63)
thf(fact_405_arithmetic__simps_I63_J,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% arithmetic_simps(63)
thf(fact_406_arithmetic__simps_I62_J,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% arithmetic_simps(62)
thf(fact_407_arithmetic__simps_I62_J,axiom,
! [A2: real] :
( ( times_times_real @ zero_zero_real @ A2 )
= zero_zero_real ) ).
% arithmetic_simps(62)
thf(fact_408_verit__eq__simplify_I6_J,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_409_verit__eq__simplify_I6_J,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% verit_eq_simplify(6)
thf(fact_410_verit__comp__simplify_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify(2)
thf(fact_411_verit__comp__simplify_I2_J,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% verit_comp_simplify(2)
thf(fact_412_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_413_verit__la__disequality,axiom,
! [A2: real,B2: real] :
( ( A2 = B2 )
| ~ ( ord_less_eq_real @ A2 @ B2 )
| ~ ( ord_less_eq_real @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_414_verit__comp__simplify_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify(1)
thf(fact_415_verit__comp__simplify_I1_J,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% verit_comp_simplify(1)
thf(fact_416_semiring__norm_I113_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% semiring_norm(113)
thf(fact_417_semiring__norm_I113_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% semiring_norm(113)
thf(fact_418_semiring__norm_I137_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% semiring_norm(137)
thf(fact_419_semiring__norm_I137_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% semiring_norm(137)
thf(fact_420_verit__comp__simplify_I3_J,axiom,
! [B3: nat,A3: nat] :
( ( ~ ( ord_less_eq_nat @ B3 @ A3 ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% verit_comp_simplify(3)
thf(fact_421_verit__comp__simplify_I3_J,axiom,
! [B3: real,A3: real] :
( ( ~ ( ord_less_eq_real @ B3 @ A3 ) )
= ( ord_less_real @ A3 @ B3 ) ) ).
% verit_comp_simplify(3)
thf(fact_422_mult__le__cancel__left,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A2 @ B2 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_423_mult__le__cancel__right,axiom,
! [A2: real,C2: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A2 @ B2 ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_424_mult__left__less__imp__less,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_425_mult__left__less__imp__less,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_426_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A2: nat,B2: nat,C2: nat,D5: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C2 @ D5 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D5 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_427_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A2: real,B2: real,C2: real,D5: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ C2 @ D5 )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D5 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_428_mult__less__cancel__left,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A2 @ B2 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_429_mult__right__less__imp__less,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_430_mult__right__less__imp__less,axiom,
! [A2: real,C2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_431_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A2: nat,B2: nat,C2: nat,D5: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C2 @ D5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D5 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_432_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A2: real,B2: real,C2: real,D5: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ C2 @ D5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D5 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_433_mult__less__cancel__right,axiom,
! [A2: real,C2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A2 @ B2 ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_434_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_435_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_436_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_437_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_438_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_439_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_440_mult__le__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_441_mult__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_442_mult__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_443_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_444_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_445_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_446_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_447_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_448_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_449_mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel1
thf(fact_450_mult__less__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_451_mult__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_452_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_453_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_454_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_455_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_456_mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel1
thf(fact_457_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_458_mult__right__cancel,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C2 )
= ( times_times_nat @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_459_mult__right__cancel,axiom,
! [C2: real,A2: real,B2: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ C2 )
= ( times_times_real @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_460_mult__cancel__right,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ C2 )
= ( times_times_nat @ B2 @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_461_mult__cancel__right,axiom,
! [A2: real,C2: real,B2: real] :
( ( ( times_times_real @ A2 @ C2 )
= ( times_times_real @ B2 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_462_mult__left__cancel,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A2 )
= ( times_times_nat @ C2 @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_463_mult__left__cancel,axiom,
! [C2: real,A2: real,B2: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A2 )
= ( times_times_real @ C2 @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_464_mult__cancel__left,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ( times_times_nat @ C2 @ A2 )
= ( times_times_nat @ C2 @ B2 ) )
= ( ( C2 = zero_zero_nat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_465_mult__cancel__left,axiom,
! [C2: real,A2: real,B2: real] :
( ( ( times_times_real @ C2 @ A2 )
= ( times_times_real @ C2 @ B2 ) )
= ( ( C2 = zero_zero_real )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_466_no__zero__divisors,axiom,
! [A2: nat,B2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B2 != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B2 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_467_no__zero__divisors,axiom,
! [A2: real,B2: real] :
( ( A2 != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( ( times_times_real @ A2 @ B2 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_468_mult__eq__0__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_469_mult__eq__0__iff,axiom,
! [A2: real,B2: real] :
( ( ( times_times_real @ A2 @ B2 )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_470_divisors__zero,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_471_divisors__zero,axiom,
! [A2: real,B2: real] :
( ( ( times_times_real @ A2 @ B2 )
= zero_zero_real )
=> ( ( A2 = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_472_mult__not__zero,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B2 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_473_mult__not__zero,axiom,
! [A2: real,B2: real] :
( ( ( times_times_real @ A2 @ B2 )
!= zero_zero_real )
=> ( ( A2 != zero_zero_real )
& ( B2 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_474_right__diff__distrib_H,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( times_times_nat @ A2 @ ( minus_minus_nat @ B2 @ C2 ) )
= ( minus_minus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_475_right__diff__distrib_H,axiom,
! [A2: real,B2: real,C2: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ B2 @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ B2 ) @ ( times_times_real @ A2 @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_476_left__diff__distrib_H,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B2 @ C2 ) @ A2 )
= ( minus_minus_nat @ ( times_times_nat @ B2 @ A2 ) @ ( times_times_nat @ C2 @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_477_left__diff__distrib_H,axiom,
! [B2: real,C2: real,A2: real] :
( ( times_times_real @ ( minus_minus_real @ B2 @ C2 ) @ A2 )
= ( minus_minus_real @ ( times_times_real @ B2 @ A2 ) @ ( times_times_real @ C2 @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_478_right__diff__distrib,axiom,
! [A2: real,B2: real,C2: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ B2 @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ B2 ) @ ( times_times_real @ A2 @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_479_left__diff__distrib,axiom,
! [A2: real,B2: real,C2: real] :
( ( times_times_real @ ( minus_minus_real @ A2 @ B2 ) @ C2 )
= ( minus_minus_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_480_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_481_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_482_zero__le__mult__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_483_mult__nonneg__nonpos2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_484_mult__nonneg__nonpos2,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B2 @ A2 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_485_mult__nonpos__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_486_mult__nonpos__nonneg,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_487_mult__nonneg__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_488_mult__nonneg__nonpos,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_489_mult__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_490_mult__nonneg__nonneg,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_491_split__mult__neg__le,axiom,
! [A2: nat,B2: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_492_split__mult__neg__le,axiom,
! [A2: real,B2: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_493_mult__le__0__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_494_mult__right__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_495_mult__right__mono,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_496_mult__right__mono__neg,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_497_mult__left__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_498_mult__left__mono,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_499_mult__nonpos__nonpos,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_500_mult__left__mono__neg,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_501_split__mult__pos__le,axiom,
! [A2: real,B2: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_502_zero__le__square,axiom,
! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_503_mult__mono_H,axiom,
! [A2: nat,B2: nat,C2: nat,D5: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D5 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_504_mult__mono_H,axiom,
! [A2: real,B2: real,C2: real,D5: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ C2 @ D5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D5 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_505_mult__mono,axiom,
! [A2: nat,B2: nat,C2: nat,D5: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D5 ) ) ) ) ) ) ).
% mult_mono
thf(fact_506_mult__mono,axiom,
! [A2: real,B2: real,C2: real,D5: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ C2 @ D5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D5 ) ) ) ) ) ) ).
% mult_mono
thf(fact_507_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_508_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_509_mult__less__cancel__right__disj,axiom,
! [A2: real,C2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A2 @ B2 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_510_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_511_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_512_mult__strict__right__mono__neg,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_513_mult__less__cancel__left__disj,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A2 @ B2 ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_514_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_515_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_516_mult__strict__left__mono__neg,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_517_mult__less__cancel__left__pos,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ord_less_real @ A2 @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_518_mult__less__cancel__left__neg,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ord_less_real @ B2 @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_519_zero__less__mult__pos2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_520_zero__less__mult__pos2,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B2 @ A2 ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_521_zero__less__mult__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_522_zero__less__mult__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_523_zero__less__mult__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_524_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_525_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B2 @ A2 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_526_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_527_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_528_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_529_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_530_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_531_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_532_mult__less__0__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_533_not__square__less__zero,axiom,
! [A2: real] :
~ ( ord_less_real @ ( times_times_real @ A2 @ A2 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_534_mult__neg__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_535_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A2: nat,B2: nat,C2: nat,D5: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ D5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D5 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_536_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A2: real,B2: real,C2: real,D5: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ C2 @ D5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D5 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_537_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A2: nat,B2: nat,C2: nat,D5: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C2 @ D5 )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D5 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_538_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A2: real,B2: real,C2: real,D5: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ C2 @ D5 )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D5 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_539_mult__right__le__imp__le,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_540_mult__right__le__imp__le,axiom,
! [A2: real,C2: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_541_mult__left__le__imp__le,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_542_mult__left__le__imp__le,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_543_mult__le__cancel__left__pos,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ord_less_eq_real @ A2 @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_544_mult__le__cancel__left__neg,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
= ( ord_less_eq_real @ B2 @ A2 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_545_mult__le__cancel__iff2,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_546_mult__le__cancel__iff1,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_547_diagonal__mat__commute,axiom,
! [A4: mat_complex,N: nat,B4: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( diagonal_mat_complex @ A4 )
=> ( ( diagonal_mat_complex @ B4 )
=> ( ( times_8009071140041733218omplex @ A4 @ B4 )
= ( times_8009071140041733218omplex @ B4 @ A4 ) ) ) ) ) ) ).
% diagonal_mat_commute
thf(fact_548_mult__less__iff1,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_549_diagonal__mat__times__diag,axiom,
! [A4: mat_complex,N: nat,B4: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( diagonal_mat_complex @ A4 )
=> ( ( diagonal_mat_complex @ B4 )
=> ( diagonal_mat_complex @ ( times_8009071140041733218omplex @ A4 @ B4 ) ) ) ) ) ) ).
% diagonal_mat_times_diag
thf(fact_550_diagonal__mat__sq__diag,axiom,
! [B4: mat_complex,N: nat] :
( ( diagonal_mat_complex @ B4 )
=> ( ( member_mat_complex @ B4 @ ( carrier_mat_complex @ N @ N ) )
=> ( diagonal_mat_complex @ ( times_8009071140041733218omplex @ B4 @ B4 ) ) ) ) ).
% diagonal_mat_sq_diag
thf(fact_551_poly__cancel__eq__conv,axiom,
! [X: real,A2: real,Y: real,B2: real] :
( ( X = zero_zero_real )
=> ( ( A2 != zero_zero_real )
=> ( ( Y = zero_zero_real )
= ( ( minus_minus_real @ ( times_times_real @ A2 @ Y ) @ ( times_times_real @ B2 @ X ) )
= zero_zero_real ) ) ) ) ).
% poly_cancel_eq_conv
thf(fact_552_less__eq__fract__respect,axiom,
! [B2: real,B3: real,D5: real,D6: real,A2: real,A3: real,C2: real,C3: real] :
( ( B2 != zero_zero_real )
=> ( ( B3 != zero_zero_real )
=> ( ( D5 != zero_zero_real )
=> ( ( D6 != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ B3 )
= ( times_times_real @ A3 @ B2 ) )
=> ( ( ( times_times_real @ C2 @ D6 )
= ( times_times_real @ C3 @ D5 ) )
=> ( ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A2 @ D5 ) @ ( times_times_real @ B2 @ D5 ) ) @ ( times_times_real @ ( times_times_real @ C2 @ B2 ) @ ( times_times_real @ B2 @ D5 ) ) )
= ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A3 @ D6 ) @ ( times_times_real @ B3 @ D6 ) ) @ ( times_times_real @ ( times_times_real @ C3 @ B3 ) @ ( times_times_real @ B3 @ D6 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_553_mult__minus__distrib__mat,axiom,
! [A4: mat_complex,Nr: nat,N: nat,B4: mat_complex,Nc: nat,C4: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B4 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( member_mat_complex @ C4 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A4 @ ( minus_2412168080157227406omplex @ B4 @ C4 ) )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ A4 @ B4 ) @ ( times_8009071140041733218omplex @ A4 @ C4 ) ) ) ) ) ) ).
% mult_minus_distrib_mat
thf(fact_554_minus__mult__distrib__mat,axiom,
! [A4: mat_complex,Nr: nat,N: nat,B4: mat_complex,C4: mat_complex,Nc: nat] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B4 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ C4 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( minus_2412168080157227406omplex @ A4 @ B4 ) @ C4 )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ A4 @ C4 ) @ ( times_8009071140041733218omplex @ B4 @ C4 ) ) ) ) ) ) ).
% minus_mult_distrib_mat
thf(fact_555_mult__carrier__mat,axiom,
! [A4: mat_complex,Nr: nat,N: nat,B4: mat_complex,Nc: nat] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ Nr @ N ) )
=> ( ( member_mat_complex @ B4 @ ( carrier_mat_complex @ N @ Nc ) )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A4 @ B4 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_556_assoc__mult__mat,axiom,
! [A4: mat_complex,N_1: nat,N_2: nat,B4: mat_complex,N_3: nat,C4: mat_complex,N_4: nat] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N_1 @ N_2 ) )
=> ( ( member_mat_complex @ B4 @ ( carrier_mat_complex @ N_2 @ N_3 ) )
=> ( ( member_mat_complex @ C4 @ ( carrier_mat_complex @ N_3 @ N_4 ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A4 @ B4 ) @ C4 )
= ( times_8009071140041733218omplex @ A4 @ ( times_8009071140041733218omplex @ B4 @ C4 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_557_minus__carrier__mat,axiom,
! [B4: mat_complex,Nr: nat,Nc: nat,A4: mat_complex] :
( ( member_mat_complex @ B4 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( minus_2412168080157227406omplex @ A4 @ B4 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% minus_carrier_mat
thf(fact_558_minus__carrier__mat_H,axiom,
! [A4: mat_complex,Nr: nat,Nc: nat,B4: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B4 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( minus_2412168080157227406omplex @ A4 @ B4 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% minus_carrier_mat'
thf(fact_559_subrelI,axiom,
! [R: set_Pr6275530937341595591omplex,S2: set_Pr6275530937341595591omplex] :
( ! [X3: mat_complex,Y2: produc5677646155008957607omplex] :
( ( member5471586270331035152omplex @ ( produc1901862033385395287omplex @ X3 @ Y2 ) @ R )
=> ( member5471586270331035152omplex @ ( produc1901862033385395287omplex @ X3 @ Y2 ) @ S2 ) )
=> ( ord_le2282573146336397159omplex @ R @ S2 ) ) ).
% subrelI
thf(fact_560_subrelI,axiom,
! [R: set_Pr6692490089613684743omplex,S2: set_Pr6692490089613684743omplex] :
( ! [X3: mat_complex,Y2: produc352478934956084711omplex] :
( ( member74738575112047696omplex @ ( produc2861545499953221015omplex @ X3 @ Y2 ) @ R )
=> ( member74738575112047696omplex @ ( produc2861545499953221015omplex @ X3 @ Y2 ) @ S2 ) )
=> ( ord_le3058207763710613415omplex @ R @ S2 ) ) ).
% subrelI
thf(fact_561_subrelI,axiom,
! [R: set_Pr8195022564563857607omplex,S2: set_Pr8195022564563857607omplex] :
( ! [X3: mat_complex,Y2: mat_complex] :
( ( member8347409015010237200omplex @ ( produc3658446505030690647omplex @ X3 @ Y2 ) @ R )
=> ( member8347409015010237200omplex @ ( produc3658446505030690647omplex @ X3 @ Y2 ) @ S2 ) )
=> ( ord_le3181491735367534695omplex @ R @ S2 ) ) ).
% subrelI
thf(fact_562_subrelI,axiom,
! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ! [X3: nat,Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ S2 ) )
=> ( ord_le3146513528884898305at_nat @ R @ S2 ) ) ).
% subrelI
thf(fact_563_subrelI,axiom,
! [R: set_Pr9093778441882193744at_nat,S2: set_Pr9093778441882193744at_nat] :
( ! [X3: nat > nat,Y2: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y2 ) @ R )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y2 ) @ S2 ) )
=> ( ord_le3678578370064672496at_nat @ R @ S2 ) ) ).
% subrelI
thf(fact_564_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_565_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_566_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_567_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_568_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_569_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_570_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_571_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_572_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_573_ord__eq__le__subst,axiom,
! [A2: real,F: nat > real,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_574_ord__eq__le__subst,axiom,
! [A2: nat,F: real > nat,B2: real,C2: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_575_ord__eq__le__subst,axiom,
! [A2: real,F: real > real,B2: real,C2: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_576_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_577_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_578_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_579_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_580_order__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_581_order__subst2,axiom,
! [A2: real,B2: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_582_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_583_order__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C2: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_584_order__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_585_order__subst1,axiom,
! [A2: real,F: real > real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_586_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_587_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ( ord_less_eq_real @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_588_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_589_antisym,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_590_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_591_dual__order_Otrans,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C2 @ B2 )
=> ( ord_less_eq_real @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_592_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_593_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_594_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_595_dual__order_Oantisym,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_596_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_597_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ B5 @ A5 )
& ( ord_less_eq_real @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_598_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( P @ A @ B ) )
=> ( ! [A: nat,B: nat] :
( ( P @ B @ A )
=> ( P @ A @ B ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_599_linorder__wlog,axiom,
! [P: real > real > $o,A2: real,B2: real] :
( ! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( P @ A @ B ) )
=> ( ! [A: real,B: real] :
( ( P @ B @ A )
=> ( P @ A @ B ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_600_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_601_order__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X @ Z ) ) ) ).
% order_trans
thf(fact_602_order_Otrans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_603_order_Otrans,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ord_less_eq_real @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_604_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_605_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_606_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_607_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_608_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_609_ord__le__eq__trans,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_real @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_610_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_611_ord__eq__le__trans,axiom,
! [A2: real,B2: real,C2: real] :
( ( A2 = B2 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ord_less_eq_real @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_612_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_613_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_614_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_615_le__cases3,axiom,
! [X: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z ) )
=> ( ( ( ord_less_eq_real @ X @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X ) )
=> ~ ( ( ord_less_eq_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_616_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_617_nle__le,axiom,
! [A2: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_618_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_619_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_620_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_621_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_622_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_623_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_624_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_625_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_626_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_627_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_628_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_629_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_630_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_631_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_632_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_633_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_634_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_635_order__less__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C2: real] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_636_order__less__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C2: nat] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_637_order__less__subst1,axiom,
! [A2: real,F: real > real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_638_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_639_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_640_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_641_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_642_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_643_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_644_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_645_ord__eq__less__subst,axiom,
! [A2: real,F: nat > real,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_646_ord__eq__less__subst,axiom,
! [A2: nat,F: real > nat,B2: real,C2: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_647_ord__eq__less__subst,axiom,
! [A2: real,F: real > real,B2: real,C2: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_648_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_649_order__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_650_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_651_order__less__asym_H,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ~ ( ord_less_real @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_652_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_653_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_654_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_655_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_656_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_657_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_658_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_659_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_660_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_661_order_Ostrict__implies__not__eq,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_662_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_663_dual__order_Ostrict__trans,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_real @ C2 @ B2 )
=> ( ord_less_real @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_664_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_665_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_666_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_667_order_Ostrict__trans,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_668_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( P @ A @ B ) )
=> ( ! [A: nat] : ( P @ A @ A )
=> ( ! [A: nat,B: nat] :
( ( P @ B @ A )
=> ( P @ A @ B ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_669_linorder__less__wlog,axiom,
! [P: real > real > $o,A2: real,B2: real] :
( ! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( P @ A @ B ) )
=> ( ! [A: real] : ( P @ A @ A )
=> ( ! [A: real,B: real] :
( ( P @ B @ A )
=> ( P @ A @ B ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_670_exists__least__iff,axiom,
( ( ^ [P5: nat > $o] :
? [X7: nat] : ( P5 @ X7 ) )
= ( ^ [P4: nat > $o] :
? [N3: nat] :
( ( P4 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P4 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_671_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_672_dual__order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_673_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_674_dual__order_Oasym,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ~ ( ord_less_real @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_675_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_676_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_677_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_678_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_679_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_680_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_681_ord__less__eq__trans,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_682_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_683_ord__eq__less__trans,axiom,
! [A2: real,B2: real,C2: real] :
( ( A2 = B2 )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_684_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_685_order_Oasym,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ~ ( ord_less_real @ B2 @ A2 ) ) ).
% order.asym
thf(fact_686_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_687_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_688_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y ) ) ) ).
% dense
thf(fact_689_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_690_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_691_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_692_mat__assoc__test_I9_J,axiom,
! [A4: mat_complex,N: nat,B4: mat_complex,C4: mat_complex,D2: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A4 @ ( minus_2412168080157227406omplex @ B4 @ C4 ) ) @ D2 )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A4 @ B4 ) @ D2 ) @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A4 @ C4 ) @ D2 ) ) ) ) ) ) ) ).
% mat_assoc_test(9)
thf(fact_693_mat__assoc__test_I1_J,axiom,
! [A4: mat_complex,N: nat,B4: mat_complex,C4: mat_complex,D2: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A4 @ B4 ) @ ( times_8009071140041733218omplex @ C4 @ D2 ) )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A4 @ B4 ) @ C4 ) @ D2 ) ) ) ) ) ) ).
% mat_assoc_test(1)
thf(fact_694_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_695_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_696_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_697_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_698_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_699_order__less__le__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_700_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_701_order__less__le__subst2,axiom,
! [A2: real,B2: real,F: real > real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_702_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_703_order__less__le__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C2: nat] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_704_order__less__le__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C2: real] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_705_order__less__le__subst1,axiom,
! [A2: real,F: real > real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_706_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_707_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_708_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_709_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_710_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_711_order__le__less__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C2: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_712_order__le__less__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C2: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_713_order__le__less__subst1,axiom,
! [A2: real,F: real > real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_714_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_715_order__less__le__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_716_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_717_order__le__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_718_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_719_order__neq__le__trans,axiom,
! [A2: real,B2: real] :
( ( A2 != B2 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_720_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_721_order__le__neq__trans,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_722_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_723_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_724_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_725_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_726_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_727_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_728_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_729_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_730_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_731_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_732_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_733_dual__order_Ostrict__implies__order,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ord_less_eq_real @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_734_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_735_order_Ostrict__implies__order,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_736_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_737_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B5: real,A5: real] :
( ( ord_less_eq_real @ B5 @ A5 )
& ~ ( ord_less_eq_real @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_738_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_739_dual__order_Ostrict__trans2,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C2 @ B2 )
=> ( ord_less_real @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_740_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_741_dual__order_Ostrict__trans1,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_real @ C2 @ B2 )
=> ( ord_less_real @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_742_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_743_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B5: real,A5: real] :
( ( ord_less_eq_real @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_744_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_745_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A5: real] :
( ( ord_less_real @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_746_dense__le__bounded,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_747_dense__ge__bounded,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_748_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_749_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ~ ( ord_less_eq_real @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_750_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_751_order_Ostrict__trans2,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_752_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_753_order_Ostrict__trans1,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_754_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_755_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_756_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_757_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B5: real] :
( ( ord_less_real @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_758_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_759_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_760_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_761_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_762_dense__le,axiom,
! [Y: real,Z: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_le
thf(fact_763_dense__ge,axiom,
! [Z: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_ge
thf(fact_764_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_765_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_766_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_767_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_768_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_769_nless__le,axiom,
! [A2: real,B2: real] :
( ( ~ ( ord_less_real @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_real @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_770_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_771_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_772_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_773_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_774_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
! [A2: real,X: real,Y: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ X @ Y ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ X ) @ ( times_times_real @ A2 @ Y ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib
thf(fact_775_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
! [A2: real,B2: real,X: real] :
( ( times_times_real @ ( minus_minus_real @ A2 @ B2 ) @ X )
= ( minus_minus_real @ ( times_times_real @ A2 @ X ) @ ( times_times_real @ B2 @ X ) ) ) ).
% vector_space_over_itself.scale_left_diff_distrib
thf(fact_776_inf__period_I1_J,axiom,
! [P: real > $o,D2: real,Q2: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D2 ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D2 ) ) ) )
=> ! [X8: real,K3: real] :
( ( ( P @ X8 )
& ( Q2 @ X8 ) )
= ( ( P @ ( minus_minus_real @ X8 @ ( times_times_real @ K3 @ D2 ) ) )
& ( Q2 @ ( minus_minus_real @ X8 @ ( times_times_real @ K3 @ D2 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_777_vector__space__over__itself_Ovector__space__assms_I3_J,axiom,
! [A2: real,B2: real,X: real] :
( ( times_times_real @ A2 @ ( times_times_real @ B2 @ X ) )
= ( times_times_real @ ( times_times_real @ A2 @ B2 ) @ X ) ) ).
% vector_space_over_itself.vector_space_assms(3)
thf(fact_778_vector__space__over__itself_Oscale__left__commute,axiom,
! [A2: real,B2: real,X: real] :
( ( times_times_real @ A2 @ ( times_times_real @ B2 @ X ) )
= ( times_times_real @ B2 @ ( times_times_real @ A2 @ X ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_779_minf_I7_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z3 )
=> ~ ( ord_less_nat @ T2 @ X8 ) ) ).
% minf(7)
thf(fact_780_minf_I7_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z3 )
=> ~ ( ord_less_real @ T2 @ X8 ) ) ).
% minf(7)
thf(fact_781_minf_I5_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z3 )
=> ( ord_less_nat @ X8 @ T2 ) ) ).
% minf(5)
thf(fact_782_minf_I5_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z3 )
=> ( ord_less_real @ X8 @ T2 ) ) ).
% minf(5)
thf(fact_783_minf_I4_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z3 )
=> ( X8 != T2 ) ) ).
% minf(4)
thf(fact_784_minf_I4_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z3 )
=> ( X8 != T2 ) ) ).
% minf(4)
thf(fact_785_minf_I3_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z3 )
=> ( X8 != T2 ) ) ).
% minf(3)
thf(fact_786_minf_I3_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z3 )
=> ( X8 != T2 ) ) ).
% minf(3)
thf(fact_787_minf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q2: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z3 )
=> ( ( ( P @ X8 )
| ( Q2 @ X8 ) )
= ( ( P6 @ X8 )
| ( Q4 @ X8 ) ) ) ) ) ) ).
% minf(2)
thf(fact_788_minf_I2_J,axiom,
! [P: real > $o,P6: real > $o,Q2: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z3 )
=> ( ( ( P @ X8 )
| ( Q2 @ X8 ) )
= ( ( P6 @ X8 )
| ( Q4 @ X8 ) ) ) ) ) ) ).
% minf(2)
thf(fact_789_minf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q2: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z3 )
=> ( ( ( P @ X8 )
& ( Q2 @ X8 ) )
= ( ( P6 @ X8 )
& ( Q4 @ X8 ) ) ) ) ) ) ).
% minf(1)
thf(fact_790_minf_I1_J,axiom,
! [P: real > $o,P6: real > $o,Q2: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z3 )
=> ( ( ( P @ X8 )
& ( Q2 @ X8 ) )
= ( ( P6 @ X8 )
& ( Q4 @ X8 ) ) ) ) ) ) ).
% minf(1)
thf(fact_791_pinf_I7_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z3 @ X8 )
=> ( ord_less_nat @ T2 @ X8 ) ) ).
% pinf(7)
thf(fact_792_pinf_I7_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ Z3 @ X8 )
=> ( ord_less_real @ T2 @ X8 ) ) ).
% pinf(7)
thf(fact_793_pinf_I5_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z3 @ X8 )
=> ~ ( ord_less_nat @ X8 @ T2 ) ) ).
% pinf(5)
thf(fact_794_pinf_I5_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ Z3 @ X8 )
=> ~ ( ord_less_real @ X8 @ T2 ) ) ).
% pinf(5)
thf(fact_795_pinf_I4_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z3 @ X8 )
=> ( X8 != T2 ) ) ).
% pinf(4)
thf(fact_796_pinf_I4_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ Z3 @ X8 )
=> ( X8 != T2 ) ) ).
% pinf(4)
thf(fact_797_pinf_I3_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z3 @ X8 )
=> ( X8 != T2 ) ) ).
% pinf(3)
thf(fact_798_pinf_I3_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ Z3 @ X8 )
=> ( X8 != T2 ) ) ).
% pinf(3)
thf(fact_799_pinf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q2: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z3 @ X8 )
=> ( ( ( P @ X8 )
| ( Q2 @ X8 ) )
= ( ( P6 @ X8 )
| ( Q4 @ X8 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_800_pinf_I2_J,axiom,
! [P: real > $o,P6: real > $o,Q2: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ Z3 @ X8 )
=> ( ( ( P @ X8 )
| ( Q2 @ X8 ) )
= ( ( P6 @ X8 )
| ( Q4 @ X8 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_801_pinf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q2: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z3 @ X8 )
=> ( ( ( P @ X8 )
& ( Q2 @ X8 ) )
= ( ( P6 @ X8 )
& ( Q4 @ X8 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_802_pinf_I1_J,axiom,
! [P: real > $o,P6: real > $o,Q2: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ Z3 @ X8 )
=> ( ( ( P @ X8 )
& ( Q2 @ X8 ) )
= ( ( P6 @ X8 )
& ( Q4 @ X8 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_803_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A2: real,X: real] :
( ( ( times_times_real @ A2 @ X )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_804_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: real] :
( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_805_vector__space__over__itself_Oscale__zero__right,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_806_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A2: real,X: real,Y: real] :
( ( ( times_times_real @ A2 @ X )
= ( times_times_real @ A2 @ Y ) )
= ( ( X = Y )
| ( A2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_807_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A2: real,X: real,Y: real] :
( ( A2 != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ X )
= ( times_times_real @ A2 @ Y ) )
=> ( X = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_808_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A2: real,X: real,B2: real] :
( ( ( times_times_real @ A2 @ X )
= ( times_times_real @ B2 @ X ) )
= ( ( A2 = B2 )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_809_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: real,A2: real,B2: real] :
( ( X != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ X )
= ( times_times_real @ B2 @ X ) )
=> ( A2 = B2 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_810_minf_I8_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z3 )
=> ~ ( ord_less_eq_nat @ T2 @ X8 ) ) ).
% minf(8)
thf(fact_811_minf_I8_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z3 )
=> ~ ( ord_less_eq_real @ T2 @ X8 ) ) ).
% minf(8)
thf(fact_812_minf_I6_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z3 )
=> ( ord_less_eq_nat @ X8 @ T2 ) ) ).
% minf(6)
thf(fact_813_minf_I6_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z3 )
=> ( ord_less_eq_real @ X8 @ T2 ) ) ).
% minf(6)
thf(fact_814_pinf_I8_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z3 @ X8 )
=> ( ord_less_eq_nat @ T2 @ X8 ) ) ).
% pinf(8)
thf(fact_815_pinf_I8_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ Z3 @ X8 )
=> ( ord_less_eq_real @ T2 @ X8 ) ) ).
% pinf(8)
thf(fact_816_pinf_I6_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z3 @ X8 )
=> ~ ( ord_less_eq_nat @ X8 @ T2 ) ) ).
% pinf(6)
thf(fact_817_pinf_I6_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X8: real] :
( ( ord_less_real @ Z3 @ X8 )
=> ~ ( ord_less_eq_real @ X8 @ T2 ) ) ).
% pinf(6)
thf(fact_818_inf__period_I2_J,axiom,
! [P: real > $o,D2: real,Q2: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D2 ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D2 ) ) ) )
=> ! [X8: real,K3: real] :
( ( ( P @ X8 )
| ( Q2 @ X8 ) )
= ( ( P @ ( minus_minus_real @ X8 @ ( times_times_real @ K3 @ D2 ) ) )
| ( Q2 @ ( minus_minus_real @ X8 @ ( times_times_real @ K3 @ D2 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_819_assms_I6_J,axiom,
ord_less_eq_nat @ ( groups4561878855575611511st_nat @ l ) @ n ).
% assms(6)
thf(fact_820_assms_I3_J,axiom,
l != nil_nat ).
% assms(3)
thf(fact_821_Suc_I7_J,axiom,
ord_less_eq_nat @ ( groups4561878855575611511st_nat @ la ) @ na ).
% Suc(7)
thf(fact_822_assms_I4_J,axiom,
ord_less_nat @ i @ ( size_size_list_nat @ l ) ).
% assms(4)
thf(fact_823_Suc_Oprems_I3_J,axiom,
la != nil_nat ).
% Suc.prems(3)
thf(fact_824_False,axiom,
( ( size_size_list_nat @ la )
!= one_one_nat ) ).
% False
thf(fact_825_Suc_Oprems_I7_J,axiom,
! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ la ) )
=> ( ord_less_nat @ zero_zero_nat @ ( nth_nat @ la @ J3 ) ) ) ).
% Suc.prems(7)
thf(fact_826_assms_I7_J,axiom,
! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ l ) )
=> ( ord_less_nat @ zero_zero_nat @ ( nth_nat @ l @ J3 ) ) ) ).
% assms(7)
thf(fact_827_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_828_subset__iff,axiom,
( ord_le3632134057777142183omplex
= ( ^ [A6: set_mat_complex,B6: set_mat_complex] :
! [T: mat_complex] :
( ( member_mat_complex @ T @ A6 )
=> ( member_mat_complex @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_829_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [T: real] :
( ( member_real @ T @ A6 )
=> ( member_real @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_830_subset__eq,axiom,
( ord_le3632134057777142183omplex
= ( ^ [A6: set_mat_complex,B6: set_mat_complex] :
! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ A6 )
=> ( member_mat_complex @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_831_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [X4: real] :
( ( member_real @ X4 @ A6 )
=> ( member_real @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_832_subsetI,axiom,
! [A4: set_mat_complex,B4: set_mat_complex] :
( ! [X3: mat_complex] :
( ( member_mat_complex @ X3 @ A4 )
=> ( member_mat_complex @ X3 @ B4 ) )
=> ( ord_le3632134057777142183omplex @ A4 @ B4 ) ) ).
% subsetI
thf(fact_833_subsetI,axiom,
! [A4: set_real,B4: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( member_real @ X3 @ B4 ) )
=> ( ord_less_eq_set_real @ A4 @ B4 ) ) ).
% subsetI
thf(fact_834_subsetD,axiom,
! [A4: set_mat_complex,B4: set_mat_complex,C2: mat_complex] :
( ( ord_le3632134057777142183omplex @ A4 @ B4 )
=> ( ( member_mat_complex @ C2 @ A4 )
=> ( member_mat_complex @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_835_subsetD,axiom,
! [A4: set_real,B4: set_real,C2: real] :
( ( ord_less_eq_set_real @ A4 @ B4 )
=> ( ( member_real @ C2 @ A4 )
=> ( member_real @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_836_in__mono,axiom,
! [A4: set_mat_complex,B4: set_mat_complex,X: mat_complex] :
( ( ord_le3632134057777142183omplex @ A4 @ B4 )
=> ( ( member_mat_complex @ X @ A4 )
=> ( member_mat_complex @ X @ B4 ) ) ) ).
% in_mono
thf(fact_837_in__mono,axiom,
! [A4: set_real,B4: set_real,X: real] :
( ( ord_less_eq_set_real @ A4 @ B4 )
=> ( ( member_real @ X @ A4 )
=> ( member_real @ X @ B4 ) ) ) ).
% in_mono
thf(fact_838__092_060open_0621_A_060_Alength_Al_092_060close_062,axiom,
ord_less_nat @ one_one_nat @ ( size_size_list_nat @ la ) ).
% \<open>1 < length l\<close>
thf(fact_839_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_840_sum__list_ONil,axiom,
( ( groups6723090944982001619t_real @ nil_real )
= zero_zero_real ) ).
% sum_list.Nil
thf(fact_841_sum__list_ONil,axiom,
( ( groups4561878855575611511st_nat @ nil_nat )
= zero_zero_nat ) ).
% sum_list.Nil
thf(fact_842_assms_I5_J,axiom,
ord_less_nat @ j @ ( nth_nat @ l @ i ) ).
% assms(5)
thf(fact_843_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_nat,Z2: list_nat] : ( Y3 = Z2 ) )
= ( ^ [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I4 )
= ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_844_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X9: nat] : ( P @ I4 @ X9 ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_845_nth__equalityI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys2 @ I3 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_846_psubsetD,axiom,
! [A4: set_mat_complex,B4: set_mat_complex,C2: mat_complex] :
( ( ord_le5598786136212072115omplex @ A4 @ B4 )
=> ( ( member_mat_complex @ C2 @ A4 )
=> ( member_mat_complex @ C2 @ B4 ) ) ) ).
% psubsetD
thf(fact_847_psubsetD,axiom,
! [A4: set_real,B4: set_real,C2: real] :
( ( ord_less_set_real @ A4 @ B4 )
=> ( ( member_real @ C2 @ A4 )
=> ( member_real @ C2 @ B4 ) ) ) ).
% psubsetD
thf(fact_848_psubset__imp__ex__mem,axiom,
! [A4: set_mat_complex,B4: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ A4 @ B4 )
=> ? [B: mat_complex] : ( member_mat_complex @ B @ ( minus_8760755521168068590omplex @ B4 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_849_psubset__imp__ex__mem,axiom,
! [A4: set_real,B4: set_real] :
( ( ord_less_set_real @ A4 @ B4 )
=> ? [B: real] : ( member_real @ B @ ( minus_minus_set_real @ B4 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_850_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_851_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_852_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_853_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_854_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_855_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_856_semiring__norm_I138_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% semiring_norm(138)
thf(fact_857_semiring__norm_I138_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% semiring_norm(138)
thf(fact_858_arithmetic__simps_I78_J,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% arithmetic_simps(78)
thf(fact_859_arithmetic__simps_I78_J,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% arithmetic_simps(78)
thf(fact_860_arithmetic__simps_I79_J,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% arithmetic_simps(79)
thf(fact_861_arithmetic__simps_I79_J,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% arithmetic_simps(79)
thf(fact_862_comm__monoid__mult__class_Omult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_863_comm__monoid__mult__class_Omult__1,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_864_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_865_mult_Ocomm__neutral,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% mult.comm_neutral
thf(fact_866_semiring__norm_I114_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% semiring_norm(114)
thf(fact_867_semiring__norm_I114_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% semiring_norm(114)
thf(fact_868_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
% one_neq_zero
thf(fact_869_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
% one_neq_zero
thf(fact_870_vector__space__over__itself_Ovector__space__assms_I4_J,axiom,
! [X: real] :
( ( times_times_real @ one_one_real @ X )
= X ) ).
% vector_space_over_itself.vector_space_assms(4)
thf(fact_871_le__numeral__extra_I1_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% le_numeral_extra(1)
thf(fact_872_le__numeral__extra_I1_J,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% le_numeral_extra(1)
thf(fact_873_le__numeral__extra_I2_J,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% le_numeral_extra(2)
thf(fact_874_le__numeral__extra_I2_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% le_numeral_extra(2)
thf(fact_875_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_876_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_877_mult__cancel__left1,axiom,
! [C2: real,B2: real] :
( ( C2
= ( times_times_real @ C2 @ B2 ) )
= ( ( C2 = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_878_mult__cancel__left2,axiom,
! [C2: real,A2: real] :
( ( ( times_times_real @ C2 @ A2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A2 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_879_mult__cancel__right1,axiom,
! [C2: real,B2: real] :
( ( C2
= ( times_times_real @ B2 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_880_mult__cancel__right2,axiom,
! [A2: real,C2: real] :
( ( ( times_times_real @ A2 @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A2 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_881_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_882_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_883_less__numeral__extra_I2_J,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% less_numeral_extra(2)
thf(fact_884_less__numeral__extra_I2_J,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% less_numeral_extra(2)
thf(fact_885_verit__comp__simplify_I28_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% verit_comp_simplify(28)
thf(fact_886_verit__comp__simplify_I28_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% verit_comp_simplify(28)
thf(fact_887_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_888_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_889_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_890_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_891_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_892_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q2: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I3: nat] :
( ( Q2 @ I3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) )
& ( ord_less_eq_real @ ( X3 @ I3 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X8: nat > real,I: nat] : ( ord_less_eq_nat @ ( L2 @ X8 @ I ) @ one_one_nat )
& ! [X8: nat > real,I: nat] :
( ( ( P @ X8 )
& ( Q2 @ I )
& ( ( X8 @ I )
= zero_zero_real ) )
=> ( ( L2 @ X8 @ I )
= zero_zero_nat ) )
& ! [X8: nat > real,I: nat] :
( ( ( P @ X8 )
& ( Q2 @ I )
& ( ( X8 @ I )
= one_one_real ) )
=> ( ( L2 @ X8 @ I )
= one_one_nat ) )
& ! [X8: nat > real,I: nat] :
( ( ( P @ X8 )
& ( Q2 @ I )
& ( ( L2 @ X8 @ I )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X8 @ I ) @ ( F @ X8 @ I ) ) )
& ! [X8: nat > real,I: nat] :
( ( ( P @ X8 )
& ( Q2 @ I )
& ( ( L2 @ X8 @ I )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X8 @ I ) @ ( X8 @ I ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_893_Suc_I6_J,axiom,
ord_less_nat @ j @ ( nth_nat @ la @ ( suc @ ia ) ) ).
% Suc(6)
thf(fact_894_Suc_I5_J,axiom,
ord_less_nat @ ( suc @ ia ) @ ( size_size_list_nat @ la ) ).
% Suc(5)
thf(fact_895_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ Y2 @ Z3 )
=> ( R2 @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_896_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_897_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_898_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_899_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_900_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_901_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_902_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_903_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_904_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_905_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_906_unit__vecs__last_Ocases,axiom,
! [X: product_prod_nat_nat] :
( ! [N2: nat] :
( X
!= ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) )
=> ~ ! [N2: nat,I3: nat] :
( X
!= ( product_Pair_nat_nat @ N2 @ ( suc @ I3 ) ) ) ) ).
% unit_vecs_last.cases
thf(fact_907_unit__vecs__first_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [N2: nat] : ( P @ N2 @ zero_zero_nat )
=> ( ! [N2: nat,I3: nat] :
( ( P @ N2 @ I3 )
=> ( P @ N2 @ ( suc @ I3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% unit_vecs_first.induct
thf(fact_908_nat_Osimps_I3_J,axiom,
! [X2: nat] :
( ( suc @ X2 )
!= zero_zero_nat ) ).
% nat.simps(3)
thf(fact_909_old_Onat_Osimps_I3_J,axiom,
! [Nat: nat] :
( ( suc @ Nat )
!= zero_zero_nat ) ).
% old.nat.simps(3)
thf(fact_910_old_Onat_Osimps_I2_J,axiom,
! [Nat: nat] :
( zero_zero_nat
!= ( suc @ Nat ) ) ).
% old.nat.simps(2)
thf(fact_911_nat_OdiscI,axiom,
! [Nat2: nat,X2: nat] :
( ( Nat2
= ( suc @ X2 ) )
=> ( Nat2 != zero_zero_nat ) ) ).
% nat.discI
thf(fact_912_nat_Oinduct,axiom,
! [P: nat > $o,Nat2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat2 ) ) ) ).
% nat.induct
thf(fact_913_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [X22: nat] :
( Y
!= ( suc @ X22 ) ) ) ).
% nat.exhaust
thf(fact_914_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_915_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_916_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_917_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_918_Suc__not__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_not_Zero
thf(fact_919_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_920_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_921_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_922_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_923_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_924_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_925_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_926_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_927_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_928_le__simps_I3_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% le_simps(3)
thf(fact_929_le__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% le_simps(2)
thf(fact_930_not__less__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_simps(2)
thf(fact_931_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_932_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_933_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_934_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_935_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_936_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_937_ex__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_938_all__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_939_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_940_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_941_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_942_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_943_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_944_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J4: nat] :
( ( M
= ( suc @ J4 ) )
& ( ord_less_nat @ J4 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_945_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_946_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_947_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_948_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_949_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_950_old_Onat_Oinject,axiom,
! [Nat2: nat,Nat: nat] :
( ( ( suc @ Nat2 )
= ( suc @ Nat ) )
= ( Nat2 = Nat ) ) ).
% old.nat.inject
thf(fact_951_nat_Oinject,axiom,
! [X2: nat,Y22: nat] :
( ( ( suc @ X2 )
= ( suc @ Y22 ) )
= ( X2 = Y22 ) ) ).
% nat.inject
thf(fact_952_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_953_not__less__simps_I1_J,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_simps(1)
thf(fact_954_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_955_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_956_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_957_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_958_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_959_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_960_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_961_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_962_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_963_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_964_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_965_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_966_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_967_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_968_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_969_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_970_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_971_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_972_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_973_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_974_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_975_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_976_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I: nat] :
( ( ord_less_eq_nat @ I @ K2 )
=> ~ ( P @ I ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_977_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_978_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_979_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_980_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_981_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_982_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_983_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_984_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_985_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_986_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_987_prod__decode__aux_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [K2: nat,M5: nat] :
( ( ~ ( ord_less_eq_nat @ M5 @ K2 )
=> ( P @ ( suc @ K2 ) @ ( minus_minus_nat @ M5 @ ( suc @ K2 ) ) ) )
=> ( P @ K2 @ M5 ) )
=> ( P @ A0 @ A1 ) ) ).
% prod_decode_aux.induct
thf(fact_988_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_989_a__def,axiom,
( a
= ( hd_nat @ la ) ) ).
% a_def
thf(fact_990_inf__concat_Ocases,axiom,
! [X: produc8199716216217303280at_nat] :
( ! [N2: nat > nat] :
( X
!= ( produc72220940542539688at_nat @ N2 @ zero_zero_nat ) )
=> ~ ! [N2: nat > nat,K2: nat] :
( X
!= ( produc72220940542539688at_nat @ N2 @ ( suc @ K2 ) ) ) ) ).
% inf_concat.cases
thf(fact_991_inf__concat__simple_Oinduct,axiom,
! [P: ( nat > nat ) > nat > $o,A0: nat > nat,A1: nat] :
( ! [F2: nat > nat] : ( P @ F2 @ zero_zero_nat )
=> ( ! [F2: nat > nat,N2: nat] :
( ( P @ F2 @ N2 )
=> ( P @ F2 @ ( suc @ N2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% inf_concat_simple.induct
thf(fact_992_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq_nat @ Xa @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_993_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K5: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K5 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K5 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K5 ) @ ( minus_minus_nat @ M3 @ ( suc @ K5 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_994_delete__index__def,axiom,
( delete_index
= ( ^ [I4: nat,I5: nat] : ( if_nat @ ( ord_less_nat @ I5 @ I4 ) @ I5 @ ( minus_minus_nat @ I5 @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% delete_index_def
thf(fact_995_permutation__delete__expand,axiom,
( permutation_delete
= ( ^ [P3: nat > nat,I4: nat,J4: nat] : ( if_nat @ ( ord_less_nat @ ( P3 @ ( if_nat @ ( ord_less_nat @ J4 @ I4 ) @ J4 @ ( suc @ J4 ) ) ) @ ( P3 @ I4 ) ) @ ( P3 @ ( if_nat @ ( ord_less_nat @ J4 @ I4 ) @ J4 @ ( suc @ J4 ) ) ) @ ( minus_minus_nat @ ( P3 @ ( if_nat @ ( ord_less_nat @ J4 @ I4 ) @ J4 @ ( suc @ J4 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% permutation_delete_expand
thf(fact_996_seq__mono__lemma,axiom,
! [M: nat,D5: nat > real,E: nat > real] :
( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_real @ ( D5 @ N2 ) @ ( E @ N2 ) ) )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_real @ ( E @ N2 ) @ ( E @ M ) ) )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ord_less_real @ ( D5 @ N4 ) @ ( E @ M ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_997_real__sup__exists,axiom,
! [P: real > $o] :
( ? [X_12: real] : ( P @ X_12 )
=> ( ? [Z4: real] :
! [X3: real] :
( ( P @ X3 )
=> ( ord_less_real @ X3 @ Z4 ) )
=> ? [S4: real] :
! [Y5: real] :
( ( ? [X4: real] :
( ( P @ X4 )
& ( ord_less_real @ Y5 @ X4 ) ) )
= ( ord_less_real @ Y5 @ S4 ) ) ) ) ).
% real_sup_exists
thf(fact_998_Bolzano,axiom,
! [A2: real,B2: real,P: real > real > $o] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ! [A: real,B: real,C: real] :
( ( P @ A @ B )
=> ( ( P @ B @ C )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( P @ A @ C ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A2 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B2 )
=> ? [D7: real] :
( ( ord_less_real @ zero_zero_real @ D7 )
& ! [A: real,B: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B )
& ( ord_less_real @ ( minus_minus_real @ B @ A ) @ D7 ) )
=> ( P @ A @ B ) ) ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Bolzano
thf(fact_999_minus__one__less,axiom,
! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ one_one_real ) @ X ) ).
% minus_one_less
thf(fact_1000_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1001_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_1002_complete__real,axiom,
! [S5: set_real] :
( ? [X8: real] : ( member_real @ X8 @ S5 )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real @ X3 @ S5 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y2: real] :
( ! [X8: real] :
( ( member_real @ X8 @ S5 )
=> ( ord_less_eq_real @ X8 @ Y2 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S5 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y2 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_1003_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1004_mat__assoc__test_I7_J,axiom,
! [A4: mat_complex,N: nat,B4: mat_complex,C4: mat_complex,D2: mat_complex] :
( ( member_mat_complex @ A4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ B4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ C4 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( member_mat_complex @ D2 @ ( carrier_mat_complex @ N @ N ) )
=> ( ( times_8009071140041733218omplex @ ( plus_p8323303612493835998omplex @ A4 @ B4 ) @ ( plus_p8323303612493835998omplex @ B4 @ C4 ) )
= ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ ( plus_p8323303612493835998omplex @ ( times_8009071140041733218omplex @ A4 @ B4 ) @ ( times_8009071140041733218omplex @ B4 @ B4 ) ) @ ( times_8009071140041733218omplex @ A4 @ C4 ) ) @ ( times_8009071140041733218omplex @ B4 @ C4 ) ) ) ) ) ) ) ).
% mat_assoc_test(7)
thf(fact_1005_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1006_nat__distrib_I1_J,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% nat_distrib(1)
thf(fact_1007_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1008_times__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% times_nat.simps(2)
thf(fact_1009_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1010_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1011_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1012_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1013_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1014_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1015_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1016_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1017_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( split_block_complex @ ba @ a @ a )
= ( produc1901862033385395287omplex @ ( produc8911724726559533635omplex @ ( split_block_complex @ ba @ a @ a ) ) @ ( produc2861545499953221015omplex @ ( produc2697000228617323907omplex @ ( produc943930114779824517omplex @ ( split_block_complex @ ba @ a @ a ) ) ) @ ( produc3658446505030690647omplex @ ( produc9163778666669654339omplex @ ( produc7343567217041670085omplex @ ( produc943930114779824517omplex @ ( split_block_complex @ ba @ a @ a ) ) ) ) @ ( produc4897211011226852997omplex @ ( produc7343567217041670085omplex @ ( produc943930114779824517omplex @ ( split_block_complex @ ba @ a @ a ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------