SET007 Axioms: SET007+887.ax
%------------------------------------------------------------------------------
% File : SET007+887 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Preliminaries to Mathematical Morphology and Its Properties
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : mathmorp [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 105 ( 0 unt; 0 def)
% Number of atoms : 534 ( 78 equ)
% Maximal formula atoms : 11 ( 5 avg)
% Number of connectives : 433 ( 4 ~; 2 |; 63 &)
% ( 8 <=>; 356 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 9 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 9 ( 8 usr; 0 prp; 1-3 aty)
% Number of functors : 38 ( 38 usr; 6 con; 0-4 aty)
% Number of variables : 370 ( 364 !; 6 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k2_mathmorp(A,k2_mathmorp(A,B)) = B ) ) ).
fof(t2_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> k1_mathmorp(A,B,k1_struct_0(k15_euclid(A),k16_euclid(A))) = k1_struct_0(k15_euclid(A),B) ) ) ).
fof(t3_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ( r1_tarski(B,C)
=> r1_tarski(k1_mathmorp(A,D,B),k1_mathmorp(A,D,C)) ) ) ) ) ) ).
fof(t4_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( C = k1_xboole_0
=> k1_mathmorp(A,B,C) = k1_xboole_0 ) ) ) ) ).
fof(t5_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k3_mathmorp(A,B,k1_struct_0(k15_euclid(A),k16_euclid(A))) = B ) ) ).
fof(t6_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k4_mathmorp(A,B,k1_struct_0(k15_euclid(A),k16_euclid(A))) = B ) ) ).
fof(t7_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> k4_mathmorp(A,B,k1_struct_0(k15_euclid(A),C)) = k1_mathmorp(A,C,B) ) ) ) ).
fof(t8_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( C = k1_xboole_0
=> k3_mathmorp(A,B,C) = k1_euclid(A) ) ) ) ) ).
fof(t9_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r1_tarski(B,C)
=> ( r1_tarski(k3_mathmorp(A,B,D),k3_mathmorp(A,C,D))
& r1_tarski(k4_mathmorp(A,B,D),k4_mathmorp(A,C,D)) ) ) ) ) ) ) ).
fof(t10_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r1_tarski(B,C)
=> ( r1_tarski(k3_mathmorp(A,D,C),k3_mathmorp(A,D,B))
& r1_tarski(k4_mathmorp(A,D,B),k4_mathmorp(A,D,C)) ) ) ) ) ) ) ).
fof(t11_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r2_hidden(k16_euclid(A),B)
=> ( r1_tarski(k3_mathmorp(A,C,B),C)
& r1_tarski(C,k4_mathmorp(A,C,B)) ) ) ) ) ) ).
fof(t12_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k4_mathmorp(A,B,C) = k4_mathmorp(A,C,B) ) ) ) ).
fof(t13_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ( r1_tarski(k1_mathmorp(A,D,B),k1_mathmorp(A,E,C))
<=> r1_tarski(k1_mathmorp(A,k20_euclid(A,D,E),B),C) ) ) ) ) ) ) ).
fof(t14_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> k3_mathmorp(A,k1_mathmorp(A,D,B),C) = k1_mathmorp(A,D,k3_mathmorp(A,B,C)) ) ) ) ) ).
fof(t15_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> k4_mathmorp(A,k1_mathmorp(A,D,B),C) = k1_mathmorp(A,D,k4_mathmorp(A,B,C)) ) ) ) ) ).
fof(t16_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> k1_mathmorp(A,D,k1_mathmorp(A,C,B)) = k1_mathmorp(A,k17_euclid(A,C,D),B) ) ) ) ) ).
fof(t17_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> k3_mathmorp(A,B,k1_mathmorp(A,D,C)) = k1_mathmorp(A,k19_euclid(A,D),k3_mathmorp(A,B,C)) ) ) ) ) ).
fof(t18_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> k4_mathmorp(A,B,k1_mathmorp(A,D,C)) = k1_mathmorp(A,D,k4_mathmorp(A,B,C)) ) ) ) ) ).
fof(t19_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ( r2_hidden(D,B)
=> r1_tarski(k1_mathmorp(A,D,C),k4_mathmorp(A,C,B)) ) ) ) ) ) ).
fof(t20_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> r1_tarski(B,k3_mathmorp(A,k4_mathmorp(A,B,C),C)) ) ) ) ).
fof(t21_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k1_mathmorp(A,k16_euclid(A),B) = B ) ) ).
fof(t22_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> k3_mathmorp(A,B,k1_struct_0(k15_euclid(A),C)) = k1_mathmorp(A,k19_euclid(A,C),B) ) ) ) ).
fof(t23_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k3_mathmorp(A,B,k4_mathmorp(A,C,D)) = k3_mathmorp(A,k3_mathmorp(A,B,C),D) ) ) ) ) ).
fof(t24_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k3_mathmorp(A,B,k4_mathmorp(A,C,D)) = k3_mathmorp(A,k3_mathmorp(A,B,D),C) ) ) ) ) ).
fof(t25_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> r1_tarski(k4_mathmorp(A,B,k3_mathmorp(A,C,D)),k3_mathmorp(A,k4_mathmorp(A,B,C),D)) ) ) ) ) ).
fof(t26_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k4_mathmorp(A,B,k4_mathmorp(A,C,D)) = k4_mathmorp(A,k4_mathmorp(A,B,C),D) ) ) ) ) ).
fof(t27_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> k1_mathmorp(A,D,k4_subset_1(u1_struct_0(k15_euclid(A)),B,C)) = k4_subset_1(u1_struct_0(k15_euclid(A)),k1_mathmorp(A,D,B),k1_mathmorp(A,D,C)) ) ) ) ) ).
fof(t28_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> k1_mathmorp(A,D,k5_subset_1(u1_struct_0(k15_euclid(A)),B,C)) = k5_subset_1(u1_struct_0(k15_euclid(A)),k1_mathmorp(A,D,B),k1_mathmorp(A,D,C)) ) ) ) ) ).
fof(t29_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k3_mathmorp(A,B,k4_subset_1(u1_struct_0(k15_euclid(A)),C,D)) = k5_subset_1(u1_struct_0(k15_euclid(A)),k3_mathmorp(A,B,C),k3_mathmorp(A,B,D)) ) ) ) ) ).
fof(t30_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k4_mathmorp(A,B,k4_subset_1(u1_struct_0(k15_euclid(A)),C,D)) = k4_subset_1(u1_struct_0(k15_euclid(A)),k4_mathmorp(A,B,C),k4_mathmorp(A,B,D)) ) ) ) ) ).
fof(t31_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> r1_tarski(k4_subset_1(u1_struct_0(k15_euclid(A)),k3_mathmorp(A,B,C),k3_mathmorp(A,D,C)),k3_mathmorp(A,k4_subset_1(u1_struct_0(k15_euclid(A)),B,D),C)) ) ) ) ) ).
fof(t32_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k4_mathmorp(A,k4_subset_1(u1_struct_0(k15_euclid(A)),B,C),D) = k4_subset_1(u1_struct_0(k15_euclid(A)),k4_mathmorp(A,B,D),k4_mathmorp(A,C,D)) ) ) ) ) ).
fof(t33_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k3_mathmorp(A,k5_subset_1(u1_struct_0(k15_euclid(A)),B,C),D) = k5_subset_1(u1_struct_0(k15_euclid(A)),k3_mathmorp(A,B,D),k3_mathmorp(A,C,D)) ) ) ) ) ).
fof(t34_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> r1_tarski(k4_mathmorp(A,k5_subset_1(u1_struct_0(k15_euclid(A)),B,C),D),k5_subset_1(u1_struct_0(k15_euclid(A)),k4_mathmorp(A,B,D),k4_mathmorp(A,C,D))) ) ) ) ) ).
fof(t35_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> r1_tarski(k4_mathmorp(A,B,k5_subset_1(u1_struct_0(k15_euclid(A)),C,D)),k5_subset_1(u1_struct_0(k15_euclid(A)),k4_mathmorp(A,B,C),k4_mathmorp(A,B,D))) ) ) ) ) ).
fof(t36_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> r1_tarski(k4_subset_1(u1_struct_0(k15_euclid(A)),k3_mathmorp(A,B,C),k3_mathmorp(A,B,D)),k3_mathmorp(A,B,k5_subset_1(u1_struct_0(k15_euclid(A)),C,D))) ) ) ) ) ).
fof(t37_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k3_subset_1(u1_struct_0(k15_euclid(A)),k3_mathmorp(A,k3_subset_1(u1_struct_0(k15_euclid(A)),B),C)) = k4_mathmorp(A,B,k2_mathmorp(A,C)) ) ) ) ).
fof(t38_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k3_subset_1(u1_struct_0(k15_euclid(A)),k3_mathmorp(A,B,C)) = k4_mathmorp(A,k3_subset_1(u1_struct_0(k15_euclid(A)),B),k2_mathmorp(A,C)) ) ) ) ).
fof(d5_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k5_mathmorp(A,B,C) = k4_mathmorp(A,k3_mathmorp(A,B,C),C) ) ) ) ).
fof(d6_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k6_mathmorp(A,B,C) = k3_mathmorp(A,k4_mathmorp(A,B,C),C) ) ) ) ).
fof(t39_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k3_subset_1(u1_struct_0(k15_euclid(A)),k5_mathmorp(A,k3_subset_1(u1_struct_0(k15_euclid(A)),B),k2_mathmorp(A,C))) = k6_mathmorp(A,B,C) ) ) ) ).
fof(t40_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k3_subset_1(u1_struct_0(k15_euclid(A)),k6_mathmorp(A,k3_subset_1(u1_struct_0(k15_euclid(A)),B),k2_mathmorp(A,C))) = k5_mathmorp(A,B,C) ) ) ) ).
fof(t41_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r1_tarski(k5_mathmorp(A,B,C),B)
& r1_tarski(B,k6_mathmorp(A,B,C)) ) ) ) ) ).
fof(t42_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k5_mathmorp(A,B,B) = B ) ) ).
fof(t43_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r1_tarski(k3_mathmorp(A,k5_mathmorp(A,B,C),C),k3_mathmorp(A,B,C))
& r1_tarski(k4_mathmorp(A,k5_mathmorp(A,B,C),C),k4_mathmorp(A,B,C)) ) ) ) ) ).
fof(t44_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r1_tarski(k3_mathmorp(A,B,C),k3_mathmorp(A,k6_mathmorp(A,B,C),C))
& r1_tarski(k4_mathmorp(A,B,C),k4_mathmorp(A,k6_mathmorp(A,B,C),C)) ) ) ) ) ).
fof(t45_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r1_tarski(B,C)
=> ( r1_tarski(k5_mathmorp(A,B,D),k5_mathmorp(A,C,D))
& r1_tarski(k6_mathmorp(A,B,D),k6_mathmorp(A,C,D)) ) ) ) ) ) ) ).
fof(t46_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> k5_mathmorp(A,k1_mathmorp(A,D,B),C) = k1_mathmorp(A,D,k5_mathmorp(A,B,C)) ) ) ) ) ).
fof(t47_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> k6_mathmorp(A,k1_mathmorp(A,D,B),C) = k1_mathmorp(A,D,k6_mathmorp(A,B,C)) ) ) ) ) ).
fof(t48_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r1_tarski(B,C)
=> r1_tarski(k5_mathmorp(A,D,C),k4_mathmorp(A,k3_mathmorp(A,D,B),C)) ) ) ) ) ) ).
fof(t49_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r1_tarski(B,C)
=> r1_tarski(k6_mathmorp(A,D,B),k3_mathmorp(A,k4_mathmorp(A,D,C),B)) ) ) ) ) ) ).
fof(t50_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( k4_mathmorp(A,B,C) = k4_mathmorp(A,k6_mathmorp(A,B,C),C)
& k3_mathmorp(A,B,C) = k3_mathmorp(A,k5_mathmorp(A,B,C),C) ) ) ) ) ).
fof(t51_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( k4_mathmorp(A,B,C) = k5_mathmorp(A,k4_mathmorp(A,B,C),C)
& k3_mathmorp(A,B,C) = k6_mathmorp(A,k3_mathmorp(A,B,C),C) ) ) ) ) ).
fof(t52_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k5_mathmorp(A,k5_mathmorp(A,B,C),C) = k5_mathmorp(A,B,C) ) ) ) ).
fof(t53_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k6_mathmorp(A,k6_mathmorp(A,B,C),C) = k6_mathmorp(A,B,C) ) ) ) ).
fof(t54_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> r1_tarski(k5_mathmorp(A,B,C),k5_mathmorp(A,k4_subset_1(u1_struct_0(k15_euclid(A)),B,D),C)) ) ) ) ) ).
fof(t55_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( B = k5_mathmorp(A,B,C)
=> r1_tarski(k5_mathmorp(A,D,B),k5_mathmorp(A,D,C)) ) ) ) ) ) ).
fof(t56_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( B = k1_xboole_0
=> k7_mathmorp(np__0,A,B) = k1_xboole_0 ) ) ) ).
fof(t57_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> k7_mathmorp(np__0,A,B) = k1_struct_0(k15_euclid(A),k16_euclid(A)) ) ) ).
fof(t58_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k7_mathmorp(np__1,A,B) = B ) ) ).
fof(t59_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> r1_tarski(k7_mathmorp(np__2,A,B),k4_mathmorp(A,B,B)) ) ) ).
fof(t60_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> k7_mathmorp(k3_xcmplx_0(C,D),A,B) = k7_mathmorp(C,A,k7_mathmorp(D,A,B)) ) ) ) ) ).
fof(t61_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( v1_xreal_0(D)
=> ( r1_tarski(B,C)
=> r1_tarski(k7_mathmorp(D,A,B),k7_mathmorp(D,A,C)) ) ) ) ) ) ).
fof(t62_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( v1_xreal_0(D)
=> k7_mathmorp(D,A,k1_mathmorp(A,C,B)) = k1_mathmorp(A,k18_euclid(D,A,C),k7_mathmorp(D,A,B)) ) ) ) ) ).
fof(t63_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( v1_xreal_0(D)
=> k7_mathmorp(D,A,k4_mathmorp(A,B,C)) = k4_mathmorp(A,k7_mathmorp(D,A,B),k7_mathmorp(D,A,C)) ) ) ) ) ).
fof(t64_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( v1_xreal_0(D)
=> ( D != np__0
=> k7_mathmorp(D,A,k3_mathmorp(A,B,C)) = k3_mathmorp(A,k7_mathmorp(D,A,B),k7_mathmorp(D,A,C)) ) ) ) ) ) ).
fof(t65_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( v1_xreal_0(D)
=> ( D != np__0
=> k7_mathmorp(D,A,k5_mathmorp(A,B,C)) = k5_mathmorp(A,k7_mathmorp(D,A,B),k7_mathmorp(D,A,C)) ) ) ) ) ) ).
fof(t66_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( v1_xreal_0(D)
=> ( D != np__0
=> k7_mathmorp(D,A,k6_mathmorp(A,B,C)) = k6_mathmorp(A,k7_mathmorp(D,A,B),k7_mathmorp(D,A,C)) ) ) ) ) ) ).
fof(d8_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k8_mathmorp(A,B,C,D) = k5_subset_1(u1_struct_0(k15_euclid(A)),k3_mathmorp(A,B,C),k3_mathmorp(A,k3_subset_1(u1_struct_0(k15_euclid(A)),B),D)) ) ) ) ) ).
fof(d9_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k9_mathmorp(A,B,C,D) = k4_subset_1(u1_struct_0(k15_euclid(A)),B,k8_mathmorp(A,B,C,D)) ) ) ) ) ).
fof(d10_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k10_mathmorp(A,B,C,D) = k6_subset_1(u1_struct_0(k15_euclid(A)),B,k8_mathmorp(A,B,C,D)) ) ) ) ) ).
fof(t67_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( B = k1_xboole_0
=> k8_mathmorp(A,C,B,D) = k3_mathmorp(A,k3_subset_1(u1_struct_0(k15_euclid(A)),C),D) ) ) ) ) ) ).
fof(t68_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( B = k1_xboole_0
=> k8_mathmorp(A,C,D,B) = k3_mathmorp(A,C,D) ) ) ) ) ) ).
fof(t69_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r2_hidden(k16_euclid(A),B)
=> r1_tarski(k8_mathmorp(A,C,B,D),C) ) ) ) ) ) ).
fof(t70_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r2_hidden(k16_euclid(A),B)
=> k5_subset_1(u1_struct_0(k15_euclid(A)),k8_mathmorp(A,C,D,B),C) = k1_xboole_0 ) ) ) ) ) ).
fof(t71_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r2_hidden(k16_euclid(A),B)
=> k9_mathmorp(A,C,B,D) = C ) ) ) ) ) ).
fof(t72_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r2_hidden(k16_euclid(A),B)
=> k10_mathmorp(A,C,D,B) = C ) ) ) ) ) ).
fof(t73_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k9_mathmorp(A,B,C,D) = k3_subset_1(u1_struct_0(k15_euclid(A)),k10_mathmorp(A,k3_subset_1(u1_struct_0(k15_euclid(A)),B),D,C)) ) ) ) ) ).
fof(t74_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k10_mathmorp(A,B,C,D) = k3_subset_1(u1_struct_0(k15_euclid(A)),k9_mathmorp(A,k3_subset_1(u1_struct_0(k15_euclid(A)),B),D,C)) ) ) ) ) ).
fof(t75_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( v1_jordan1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( v1_xreal_0(E)
=> ( ( r1_xreal_0(np__0,E)
& r1_xreal_0(E,np__1)
& r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k17_euclid(A,k18_euclid(E,A,C),k18_euclid(k6_xcmplx_0(np__1,E),A,D)),B) ) ) ) ) ) ) ) ).
fof(d11_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( v1_jordan1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( v1_xreal_0(E)
=> ( ( r1_xreal_0(np__0,E)
& r1_xreal_0(E,np__1)
& r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k17_euclid(A,k18_euclid(E,A,C),k18_euclid(k6_xcmplx_0(np__1,E),A,D)),B) ) ) ) ) ) ) ) ).
fof(t76_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( v1_jordan1(B,A)
=> v1_jordan1(k2_mathmorp(A,B),A) ) ) ) ).
fof(t77_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( ( v1_jordan1(B,A)
& v1_jordan1(C,A) )
=> ( v1_jordan1(k4_mathmorp(A,B,C),A)
& v1_jordan1(k3_mathmorp(A,B,C),A) ) ) ) ) ) ).
fof(t78_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( ( v1_jordan1(B,A)
& v1_jordan1(C,A) )
=> ( v1_jordan1(k5_mathmorp(A,B,C),A)
& v1_jordan1(k6_mathmorp(A,B,C),A) ) ) ) ) ) ).
fof(t79_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( v1_jordan1(B,A)
=> ( r1_xreal_0(C,np__0)
| r1_xreal_0(D,np__0)
| k7_mathmorp(k2_xcmplx_0(D,C),A,B) = k4_mathmorp(A,k7_mathmorp(D,A,B),k7_mathmorp(C,A,B)) ) ) ) ) ) ) ).
fof(dt_k1_mathmorp,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> m1_subset_1(k1_mathmorp(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k2_mathmorp,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> m1_subset_1(k2_mathmorp(A,B),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k3_mathmorp,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> m1_subset_1(k3_mathmorp(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k4_mathmorp,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> m1_subset_1(k4_mathmorp(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k5_mathmorp,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> m1_subset_1(k5_mathmorp(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k6_mathmorp,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> m1_subset_1(k6_mathmorp(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k7_mathmorp,axiom,
! [A,B,C] :
( ( v1_xreal_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(B)))) )
=> m1_subset_1(k7_mathmorp(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(B)))) ) ).
fof(dt_k8_mathmorp,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> m1_subset_1(k8_mathmorp(A,B,C,D),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k9_mathmorp,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> m1_subset_1(k9_mathmorp(A,B,C,D),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(dt_k10_mathmorp,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> m1_subset_1(k10_mathmorp(A,B,C,D),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ).
fof(d1_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k1_mathmorp(A,B,C) = a_3_0_mathmorp(A,B,C) ) ) ) ).
fof(d2_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k2_mathmorp(A,B) = a_2_0_mathmorp(A,B) ) ) ).
fof(d3_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k3_mathmorp(A,B,C) = a_3_1_mathmorp(A,B,C) ) ) ) ).
fof(d4_mathmorp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> k4_mathmorp(A,B,C) = a_3_2_mathmorp(A,B,C) ) ) ) ).
fof(d7_mathmorp,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(B))))
=> k7_mathmorp(A,B,C) = a_3_3_mathmorp(A,B,C) ) ) ) ).
fof(fraenkel_a_3_0_mathmorp,axiom,
! [A,B,C,D] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m1_subset_1(C,u1_struct_0(k15_euclid(B)))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(B)))) )
=> ( r2_hidden(A,a_3_0_mathmorp(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(B)))
& A = k17_euclid(B,E,C)
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_2_0_mathmorp,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(B)))) )
=> ( r2_hidden(A,a_2_0_mathmorp(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(B)))
& A = k19_euclid(B,D)
& r2_hidden(D,C) ) ) ) ).
fof(fraenkel_a_3_1_mathmorp,axiom,
! [A,B,C,D] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(B))))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(B)))) )
=> ( r2_hidden(A,a_3_1_mathmorp(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(B)))
& A = E
& r1_tarski(k1_mathmorp(B,E,D),C) ) ) ) ).
fof(fraenkel_a_3_2_mathmorp,axiom,
! [A,B,C,D] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(B))))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(B)))) )
=> ( r2_hidden(A,a_3_2_mathmorp(B,C,D))
<=> ? [E,F] :
( m1_subset_1(E,u1_struct_0(k15_euclid(B)))
& m1_subset_1(F,u1_struct_0(k15_euclid(B)))
& A = k17_euclid(B,E,F)
& r2_hidden(E,C)
& r2_hidden(F,D) ) ) ) ).
fof(fraenkel_a_3_3_mathmorp,axiom,
! [A,B,C,D] :
( ( v1_xreal_0(B)
& m2_subset_1(C,k1_numbers,k5_numbers)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(C)))) )
=> ( r2_hidden(A,a_3_3_mathmorp(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(C)))
& A = k18_euclid(B,C,E)
& r2_hidden(E,D) ) ) ) ).
%------------------------------------------------------------------------------