SET007 Axioms: SET007+149.ax
%------------------------------------------------------------------------------
% File : SET007+149 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Real Function One-Side Differentiability
% 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 : fdiff_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 30 ( 0 unt; 0 def)
% Number of atoms : 451 ( 55 equ)
% Maximal formula atoms : 42 ( 15 avg)
% Number of connectives : 463 ( 42 ~; 6 |; 247 &)
% ( 6 <=>; 162 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 13 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 18 ( 17 usr; 0 prp; 1-3 aty)
% Number of functors : 31 ( 31 usr; 3 con; 0-4 aty)
% Number of variables : 136 ( 128 !; 8 ?)
% 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_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,np__0)
& r1_tarski(k1_rcomp_1(k5_real_1(B,C),B),k4_relset_1(k1_numbers,k1_numbers,A)) )
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v1_fdiff_1(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v5_seqm_3(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ~ ( k1_rfunct_2(D) = k1_tarski(B)
& r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,A))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,C,E)) ) ) ) ) ) ) ) ).
fof(t2_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,np__0)
& r1_tarski(k1_rcomp_1(B,k3_real_1(B,C)),k4_relset_1(k1_numbers,k1_numbers,A)) )
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v1_fdiff_1(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v5_seqm_3(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ~ ( k1_rfunct_2(D) = k1_tarski(B)
& r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,A))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,E),np__0) ) ) ) ) ) ) ) ).
fof(t3_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ( ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v1_fdiff_1(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v5_seqm_3(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(D) = k1_tarski(B)
& r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,A))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,C,E)) ) )
=> v4_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(A,k9_seq_1(C,D)),k2_rfunct_2(A,D)))) ) ) )
& r1_tarski(k1_tarski(B),k4_relset_1(k1_numbers,k1_numbers,A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v1_fdiff_1(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v1_fdiff_1(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_numbers)
& v5_seqm_3(E)
& m2_relset_1(E,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(E) = k1_tarski(B)
& r1_tarski(k1_rfunct_2(k9_seq_1(C,E)),k4_relset_1(k1_numbers,k1_numbers,A))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,C,F)) )
& r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,A))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,D,F)) ) )
=> k2_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(A,k9_seq_1(C,E)),k2_rfunct_2(A,E)))) = k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E)))) ) ) ) ) ) ) ) ).
fof(t4_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ( ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v1_fdiff_1(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v5_seqm_3(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(D) = k1_tarski(B)
& r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,A))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,E),np__0) ) )
=> v4_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(A,k9_seq_1(C,D)),k2_rfunct_2(A,D)))) ) ) )
& r1_tarski(k1_tarski(B),k4_relset_1(k1_numbers,k1_numbers,A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v1_fdiff_1(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v1_fdiff_1(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_numbers)
& v5_seqm_3(E)
& m2_relset_1(E,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(E) = k1_tarski(B)
& r1_tarski(k1_rfunct_2(k9_seq_1(C,E)),k4_relset_1(k1_numbers,k1_numbers,A))
& r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,A))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,F),np__0) )
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,D,F),np__0) ) )
=> k2_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(A,k9_seq_1(C,E)),k2_rfunct_2(A,E)))) = k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E)))) ) ) ) ) ) ) ) ).
fof(d1_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r1_fdiff_3(A,B)
<=> ( r2_hidden(B,k4_relset_1(k1_numbers,k1_numbers,A))
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( r1_tarski(k1_rfunct_2(C),k3_xboole_0(k12_prob_1(B),k4_relset_1(k1_numbers,k1_numbers,A)))
& v4_seq_2(C)
& k2_seq_2(C) = B )
=> ( v4_seq_2(k2_rfunct_2(A,C))
& k2_seq_1(k1_numbers,k1_numbers,A,B) = k2_seq_2(k2_rfunct_2(A,C)) ) ) ) ) ) ) ) ).
fof(d2_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_fdiff_3(A,B)
<=> ( r2_hidden(B,k4_relset_1(k1_numbers,k1_numbers,A))
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( r1_tarski(k1_rfunct_2(C),k3_xboole_0(k4_limfunc1(B),k4_relset_1(k1_numbers,k1_numbers,A)))
& v4_seq_2(C)
& k2_seq_2(C) = B )
=> ( v4_seq_2(k2_rfunct_2(A,C))
& k2_seq_1(k1_numbers,k1_numbers,A,B) = k2_seq_2(k2_rfunct_2(A,C)) ) ) ) ) ) ) ) ).
fof(d3_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r3_fdiff_3(A,B)
<=> ( ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,np__0)
& r1_tarski(k1_rcomp_1(B,k3_real_1(B,C)),k4_relset_1(k1_numbers,k1_numbers,A)) )
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v1_fdiff_1(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v5_seqm_3(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(D) = k1_tarski(B)
& r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,A))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,E),np__0) ) )
=> v4_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(A,k9_seq_1(C,D)),k2_rfunct_2(A,D)))) ) ) ) ) ) ) ) ).
fof(d4_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r4_fdiff_3(A,B)
<=> ( ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,np__0)
& r1_tarski(k1_rcomp_1(k5_real_1(B,C),B),k4_relset_1(k1_numbers,k1_numbers,A)) )
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v1_fdiff_1(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v5_seqm_3(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(D) = k1_tarski(B)
& r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,A))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,C,E)) ) )
=> v4_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(A,k9_seq_1(C,D)),k2_rfunct_2(A,D)))) ) ) ) ) ) ) ) ).
fof(t5_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r4_fdiff_3(A,B)
=> r1_fdiff_3(A,B) ) ) ) ).
fof(t6_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r1_fdiff_3(A,B)
& k2_seq_1(k1_numbers,k1_numbers,A,B) != C
& ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,np__0)
& r1_tarski(k1_rcomp_1(k5_real_1(B,D),B),k4_relset_1(k1_numbers,k1_numbers,A)) )
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k1_rcomp_1(k5_real_1(B,D),B),k4_relset_1(k1_numbers,k1_numbers,A))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( r2_hidden(E,k1_rcomp_1(k5_real_1(B,D),B))
& k2_seq_1(k1_numbers,k1_numbers,A,E) = C ) ) ) ) ) ) ) ) ).
fof(t7_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r3_fdiff_3(A,B)
=> r2_fdiff_3(A,B) ) ) ) ).
fof(t8_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r2_fdiff_3(A,B)
& k2_seq_1(k1_numbers,k1_numbers,A,B) != C
& ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,np__0)
& r1_tarski(k1_rcomp_1(B,k3_real_1(B,D)),k4_relset_1(k1_numbers,k1_numbers,A)) )
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k1_rcomp_1(B,k3_real_1(B,D)),k4_relset_1(k1_numbers,k1_numbers,A))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( r2_hidden(E,k1_rcomp_1(B,k3_real_1(B,D)))
& k2_seq_1(k1_numbers,k1_numbers,A,E) = C ) ) ) ) ) ) ) ) ).
fof(d5_fdiff_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r4_fdiff_3(B,A)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( C = k1_fdiff_3(A,B)
<=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v1_fdiff_1(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_numbers)
& v5_seqm_3(E)
& m2_relset_1(E,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(E) = k1_tarski(A)
& r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,B))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,D,F)) ) )
=> C = k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(B,k9_seq_1(D,E)),k2_rfunct_2(B,E)))) ) ) ) ) ) ) ) ) ).
fof(d6_fdiff_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r3_fdiff_3(B,A)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( C = k2_fdiff_3(A,B)
<=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v1_fdiff_1(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_numbers)
& v5_seqm_3(E)
& m2_relset_1(E,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(E) = k1_tarski(A)
& r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,B))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,D,F),np__0) ) )
=> C = k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(B,k9_seq_1(D,E)),k2_rfunct_2(B,E)))) ) ) ) ) ) ) ) ) ).
fof(t9_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( ( r4_fdiff_3(A,B)
& k1_fdiff_3(B,A) = C )
=> ( ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,np__0)
& r1_tarski(k1_rcomp_1(k5_real_1(B,D),B),k4_relset_1(k1_numbers,k1_numbers,A)) )
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v1_fdiff_1(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_numbers)
& v5_seqm_3(E)
& m2_relset_1(E,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(E) = k1_tarski(B)
& r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,A))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,D,F)) ) )
=> ( v4_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E))))
& k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E)))) = C ) ) ) ) ) )
& ( ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v1_fdiff_1(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_numbers)
& v5_seqm_3(E)
& m2_relset_1(E,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(E) = k1_tarski(B)
& r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,A))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,D,F)) ) )
=> ( v4_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E))))
& k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E)))) = C ) ) ) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k1_rcomp_1(k5_real_1(B,D),B),k4_relset_1(k1_numbers,k1_numbers,A)) ) )
| ( r4_fdiff_3(A,B)
& k1_fdiff_3(B,A) = C ) ) ) ) ) ) ) ).
fof(t10_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r4_fdiff_3(A,C)
& r4_fdiff_3(B,C) )
=> ( r4_fdiff_3(k6_seq_1(k1_numbers,k1_numbers,A,B),C)
& k1_fdiff_3(C,k6_seq_1(k1_numbers,k1_numbers,A,B)) = k3_real_1(k1_fdiff_3(C,A),k1_fdiff_3(C,B)) ) ) ) ) ) ).
fof(t11_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r4_fdiff_3(A,C)
& r4_fdiff_3(B,C) )
=> ( r4_fdiff_3(k7_seq_1(k1_numbers,k1_numbers,A,B),C)
& k1_fdiff_3(C,k7_seq_1(k1_numbers,k1_numbers,A,B)) = k5_real_1(k1_fdiff_3(C,A),k1_fdiff_3(C,B)) ) ) ) ) ) ).
fof(t12_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r4_fdiff_3(A,C)
& r4_fdiff_3(B,C) )
=> ( r4_fdiff_3(k8_seq_1(k1_numbers,k1_numbers,A,B),C)
& k1_fdiff_3(C,k8_seq_1(k1_numbers,k1_numbers,A,B)) = k3_real_1(k4_real_1(k1_fdiff_3(C,A),k2_seq_1(k1_numbers,k1_numbers,B,C)),k4_real_1(k1_fdiff_3(C,B),k2_seq_1(k1_numbers,k1_numbers,A,C))) ) ) ) ) ) ).
fof(t13_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r4_fdiff_3(A,C)
& r4_fdiff_3(B,C) )
=> ( k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0
| ( r4_fdiff_3(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C)
& k1_fdiff_3(C,k2_rfunct_1(k1_numbers,k1_numbers,A,B)) = k6_real_1(k5_real_1(k4_real_1(k1_fdiff_3(C,A),k2_seq_1(k1_numbers,k1_numbers,B,C)),k4_real_1(k1_fdiff_3(C,B),k2_seq_1(k1_numbers,k1_numbers,A,C))),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) ) ) ).
fof(t14_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r4_fdiff_3(A,B)
=> ( k2_seq_1(k1_numbers,k1_numbers,A,B) = np__0
| ( r4_fdiff_3(k4_rfunct_1(k1_numbers,k1_numbers,A),B)
& k1_fdiff_3(B,k4_rfunct_1(k1_numbers,k1_numbers,A)) = k1_real_1(k6_real_1(k1_fdiff_3(B,A),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,A,B)))) ) ) ) ) ) ).
fof(t15_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( ( r3_fdiff_3(A,B)
& k2_fdiff_3(B,A) = C )
=> ( ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,np__0)
& r1_tarski(k1_rcomp_1(B,k3_real_1(B,D)),k4_relset_1(k1_numbers,k1_numbers,A)) )
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v1_fdiff_1(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_numbers)
& v5_seqm_3(E)
& m2_relset_1(E,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(E) = k1_tarski(B)
& r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,A))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,D,F),np__0) ) )
=> ( v4_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E))))
& k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E)))) = C ) ) ) ) ) )
& ( ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v1_fdiff_1(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_numbers)
& v5_seqm_3(E)
& m2_relset_1(E,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(E) = k1_tarski(B)
& r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,A))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,D,F),np__0) ) )
=> ( v4_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E))))
& k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E)))) = C ) ) ) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k1_rcomp_1(B,k3_real_1(B,D)),k4_relset_1(k1_numbers,k1_numbers,A)) ) )
| ( r3_fdiff_3(A,B)
& k2_fdiff_3(B,A) = C ) ) ) ) ) ) ) ).
fof(t16_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r3_fdiff_3(A,C)
& r3_fdiff_3(B,C) )
=> ( r3_fdiff_3(k6_seq_1(k1_numbers,k1_numbers,A,B),C)
& k2_fdiff_3(C,k6_seq_1(k1_numbers,k1_numbers,A,B)) = k3_real_1(k2_fdiff_3(C,A),k2_fdiff_3(C,B)) ) ) ) ) ) ).
fof(t17_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r3_fdiff_3(A,C)
& r3_fdiff_3(B,C) )
=> ( r3_fdiff_3(k7_seq_1(k1_numbers,k1_numbers,A,B),C)
& k2_fdiff_3(C,k7_seq_1(k1_numbers,k1_numbers,A,B)) = k5_real_1(k2_fdiff_3(C,A),k2_fdiff_3(C,B)) ) ) ) ) ) ).
fof(t18_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r3_fdiff_3(A,C)
& r3_fdiff_3(B,C) )
=> ( r3_fdiff_3(k8_seq_1(k1_numbers,k1_numbers,A,B),C)
& k2_fdiff_3(C,k8_seq_1(k1_numbers,k1_numbers,A,B)) = k3_real_1(k4_real_1(k2_fdiff_3(C,A),k2_seq_1(k1_numbers,k1_numbers,B,C)),k4_real_1(k2_fdiff_3(C,B),k2_seq_1(k1_numbers,k1_numbers,A,C))) ) ) ) ) ) ).
fof(t19_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r3_fdiff_3(A,C)
& r3_fdiff_3(B,C) )
=> ( k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0
| ( r3_fdiff_3(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C)
& k2_fdiff_3(C,k2_rfunct_1(k1_numbers,k1_numbers,A,B)) = k6_real_1(k5_real_1(k4_real_1(k2_fdiff_3(C,A),k2_seq_1(k1_numbers,k1_numbers,B,C)),k4_real_1(k2_fdiff_3(C,B),k2_seq_1(k1_numbers,k1_numbers,A,C))),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) ) ) ).
fof(t20_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r3_fdiff_3(A,B)
=> ( k2_seq_1(k1_numbers,k1_numbers,A,B) = np__0
| ( r3_fdiff_3(k4_rfunct_1(k1_numbers,k1_numbers,A),B)
& k2_fdiff_3(B,k4_rfunct_1(k1_numbers,k1_numbers,A)) = k1_real_1(k6_real_1(k2_fdiff_3(B,A),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,A,B)))) ) ) ) ) ) ).
fof(t21_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ( r3_fdiff_3(A,B)
& r4_fdiff_3(A,B)
& k2_fdiff_3(B,A) = k1_fdiff_3(B,A) )
=> ( r1_fdiff_1(A,B)
& k1_fdiff_1(A,B) = k2_fdiff_3(B,A)
& k1_fdiff_1(A,B) = k1_fdiff_3(B,A) ) ) ) ) ).
fof(t22_fdiff_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r1_fdiff_1(A,B)
=> ( r3_fdiff_3(A,B)
& r4_fdiff_3(A,B)
& k1_fdiff_1(A,B) = k2_fdiff_3(B,A)
& k1_fdiff_1(A,B) = k1_fdiff_3(B,A) ) ) ) ) ).
fof(dt_k1_fdiff_3,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& v1_funct_1(B)
& m1_relset_1(B,k1_numbers,k1_numbers) )
=> m1_subset_1(k1_fdiff_3(A,B),k1_numbers) ) ).
fof(dt_k2_fdiff_3,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& v1_funct_1(B)
& m1_relset_1(B,k1_numbers,k1_numbers) )
=> m1_subset_1(k2_fdiff_3(A,B),k1_numbers) ) ).
%------------------------------------------------------------------------------