SET007 Axioms: SET007+346.ax
%------------------------------------------------------------------------------
% File : SET007+346 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Brouwer Fixed Point Theorem for Intervals
% 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 : treal_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 35 ( 3 unt; 0 def)
% Number of atoms : 257 ( 47 equ)
% Maximal formula atoms : 17 ( 7 avg)
% Number of connectives : 238 ( 16 ~; 0 |; 84 &)
% ( 2 <=>; 136 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 10 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 15 ( 14 usr; 0 prp; 1-3 aty)
% Number of functors : 29 ( 29 usr; 7 con; 0-5 aty)
% Number of variables : 122 ( 116 !; 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_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(C,D) )
=> r1_tarski(k1_rcomp_1(B,C),k1_rcomp_1(A,D)) ) ) ) ) ) ).
fof(t2_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(C,D)
& r1_xreal_0(B,C) )
=> k4_subset_1(k1_numbers,k1_rcomp_1(A,C),k1_rcomp_1(B,D)) = k1_rcomp_1(A,D) ) ) ) ) ) ).
fof(t3_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(C,D)
& r1_xreal_0(B,C) )
=> k5_subset_1(k1_numbers,k1_rcomp_1(A,C),k1_rcomp_1(B,D)) = k1_rcomp_1(B,C) ) ) ) ) ) ).
fof(t4_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( C = k1_rcomp_1(A,B)
=> v4_pre_topc(C,k3_topmetr) ) ) ) ) ).
fof(t5_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> ( v1_borsuk_1(k4_topmetr(A,B),k3_topmetr)
& m1_pre_topc(k4_topmetr(A,B),k3_topmetr) ) ) ) ) ).
fof(t6_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(C,D)
& r1_xreal_0(B,C) )
=> ( v1_borsuk_1(k4_topmetr(B,C),k4_topmetr(A,D))
& m1_pre_topc(k4_topmetr(B,C),k4_topmetr(A,D)) ) ) ) ) ) ) ).
fof(t7_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(C,D)
& r1_xreal_0(B,C) )
=> ( k4_topmetr(A,D) = k1_tsep_1(k3_topmetr,k4_topmetr(A,C),k4_topmetr(B,D))
& k4_topmetr(B,C) = k2_tsep_1(k3_topmetr,k4_topmetr(A,C),k4_topmetr(B,D)) ) ) ) ) ) ) ).
fof(d1_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> k1_treal_1(A,B) = A ) ) ) ).
fof(d2_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> k2_treal_1(A,B) = B ) ) ) ).
fof(t8_treal_1,axiom,
( k23_borsuk_1 = k1_treal_1(np__0,np__1)
& k24_borsuk_1 = k2_treal_1(np__0,np__1) ) ).
fof(t9_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(B,C) )
=> ( k1_treal_1(A,B) = k1_treal_1(A,C)
& k2_treal_1(B,C) = k2_treal_1(A,C) ) ) ) ) ) ).
fof(d3_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_topmetr(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k4_topmetr(A,B)))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)))
& m2_relset_1(E,u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B))) )
=> ( E = k3_treal_1(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,u1_struct_0(k4_topmetr(np__0,np__1)))
=> ! [G] :
( v1_xreal_0(G)
=> ! [H] :
( v1_xreal_0(H)
=> ! [I] :
( v1_xreal_0(I)
=> ( ( F = G
& H = C
& I = D )
=> k8_funct_2(u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)),E,F) = k2_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(np__1,G),H),k3_xcmplx_0(G,I)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t10_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_topmetr(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k4_topmetr(A,B)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k4_topmetr(np__0,np__1)))
=> ! [F] :
( v1_xreal_0(F)
=> ! [G] :
( v1_xreal_0(G)
=> ! [H] :
( v1_xreal_0(H)
=> ( ( E = F
& G = C
& H = D )
=> k8_funct_2(u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)),k3_treal_1(A,B,C,D),E) = k2_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(H,G),F),G) ) ) ) ) ) ) ) ) ) ) ).
fof(t11_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_topmetr(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k4_topmetr(A,B)))
=> ( v1_funct_1(k3_treal_1(A,B,C,D))
& v1_funct_2(k3_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)))
& v5_pre_topc(k3_treal_1(A,B,C,D),k4_topmetr(np__0,np__1),k4_topmetr(A,B))
& m2_relset_1(k3_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B))) ) ) ) ) ) ) ).
fof(t12_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_topmetr(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k4_topmetr(A,B)))
=> ( k8_funct_2(u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)),k3_treal_1(A,B,C,D),k1_treal_1(np__0,np__1)) = C
& k8_funct_2(u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)),k3_treal_1(A,B,C,D),k2_treal_1(np__0,np__1)) = D ) ) ) ) ) ) ).
fof(t13_treal_1,axiom,
k3_treal_1(np__0,np__1,k1_treal_1(np__0,np__1),k2_treal_1(np__0,np__1)) = k7_grcat_1(k4_topmetr(np__0,np__1)) ).
fof(d4_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_topmetr(np__0,np__1)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k4_topmetr(np__0,np__1)))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)))
& m2_relset_1(E,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1))) )
=> ( E = k4_treal_1(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,u1_struct_0(k4_topmetr(A,B)))
=> ! [G] :
( v1_xreal_0(G)
=> ! [H] :
( v1_xreal_0(H)
=> ! [I] :
( v1_xreal_0(I)
=> ( ( F = G
& H = C
& I = D )
=> k8_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)),E,F) = k7_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(B,G),H),k3_xcmplx_0(k6_xcmplx_0(G,A),I)),k6_xcmplx_0(B,A)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t14_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_topmetr(np__0,np__1)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k4_topmetr(np__0,np__1)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k4_topmetr(A,B)))
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ! [H] :
( m1_subset_1(H,k1_numbers)
=> ( ( E = F
& G = C
& H = D )
=> k8_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)),k4_treal_1(A,B,C,D),E) = k2_xcmplx_0(k3_xcmplx_0(k7_xcmplx_0(k5_real_1(H,G),k6_xcmplx_0(B,A)),F),k7_xcmplx_0(k6_xcmplx_0(k3_xcmplx_0(B,G),k3_xcmplx_0(A,H)),k6_xcmplx_0(B,A))) ) ) ) ) ) ) ) ) ) ) ).
fof(t15_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_topmetr(np__0,np__1)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k4_topmetr(np__0,np__1)))
=> ( v1_funct_1(k4_treal_1(A,B,C,D))
& v1_funct_2(k4_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)))
& v5_pre_topc(k4_treal_1(A,B,C,D),k4_topmetr(A,B),k4_topmetr(np__0,np__1))
& m2_relset_1(k4_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1))) ) ) ) ) ) ) ).
fof(t16_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_topmetr(np__0,np__1)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k4_topmetr(np__0,np__1)))
=> ( k8_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)),k4_treal_1(A,B,C,D),k1_treal_1(A,B)) = C
& k8_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)),k4_treal_1(A,B,C,D),k2_treal_1(A,B)) = D ) ) ) ) ) ) ).
fof(t17_treal_1,axiom,
k4_treal_1(np__0,np__1,k1_treal_1(np__0,np__1),k2_treal_1(np__0,np__1)) = k7_grcat_1(k4_topmetr(np__0,np__1)) ).
fof(t18_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> ( k7_grcat_1(k4_topmetr(A,B)) = k7_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)),k4_treal_1(A,B,k1_treal_1(np__0,np__1),k2_treal_1(np__0,np__1)),k3_treal_1(A,B,k1_treal_1(A,B),k2_treal_1(A,B)))
& k7_grcat_1(k4_topmetr(np__0,np__1)) = k7_funct_2(u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)),k3_treal_1(A,B,k1_treal_1(A,B),k2_treal_1(A,B)),k4_treal_1(A,B,k1_treal_1(np__0,np__1),k2_treal_1(np__0,np__1))) ) ) ) ) ).
fof(t19_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> ( k7_grcat_1(k4_topmetr(A,B)) = k7_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)),k4_treal_1(A,B,k2_treal_1(np__0,np__1),k1_treal_1(np__0,np__1)),k3_treal_1(A,B,k2_treal_1(A,B),k1_treal_1(A,B)))
& k7_grcat_1(k4_topmetr(np__0,np__1)) = k7_funct_2(u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)),k3_treal_1(A,B,k2_treal_1(A,B),k1_treal_1(A,B)),k4_treal_1(A,B,k2_treal_1(np__0,np__1),k1_treal_1(np__0,np__1))) ) ) ) ) ).
fof(t20_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> ( v3_tops_2(k3_treal_1(A,B,k1_treal_1(A,B),k2_treal_1(A,B)),k4_topmetr(np__0,np__1),k4_topmetr(A,B))
& k2_tops_2(k4_topmetr(np__0,np__1),k4_topmetr(A,B),k3_treal_1(A,B,k1_treal_1(A,B),k2_treal_1(A,B))) = k4_treal_1(A,B,k1_treal_1(np__0,np__1),k2_treal_1(np__0,np__1))
& v3_tops_2(k4_treal_1(A,B,k1_treal_1(np__0,np__1),k2_treal_1(np__0,np__1)),k4_topmetr(A,B),k4_topmetr(np__0,np__1))
& k2_tops_2(k4_topmetr(A,B),k4_topmetr(np__0,np__1),k4_treal_1(A,B,k1_treal_1(np__0,np__1),k2_treal_1(np__0,np__1))) = k3_treal_1(A,B,k1_treal_1(A,B),k2_treal_1(A,B)) ) ) ) ) ).
fof(t21_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> ( v3_tops_2(k3_treal_1(A,B,k2_treal_1(A,B),k1_treal_1(A,B)),k4_topmetr(np__0,np__1),k4_topmetr(A,B))
& k2_tops_2(k4_topmetr(np__0,np__1),k4_topmetr(A,B),k3_treal_1(A,B,k2_treal_1(A,B),k1_treal_1(A,B))) = k4_treal_1(A,B,k2_treal_1(np__0,np__1),k1_treal_1(np__0,np__1))
& v3_tops_2(k4_treal_1(A,B,k2_treal_1(np__0,np__1),k1_treal_1(np__0,np__1)),k4_topmetr(A,B),k4_topmetr(np__0,np__1))
& k2_tops_2(k4_topmetr(A,B),k4_topmetr(np__0,np__1),k4_treal_1(A,B,k2_treal_1(np__0,np__1),k1_treal_1(np__0,np__1))) = k3_treal_1(A,B,k2_treal_1(A,B),k1_treal_1(A,B)) ) ) ) ) ).
fof(t22_treal_1,axiom,
v1_connsp_1(k22_borsuk_1) ).
fof(t23_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> v1_connsp_1(k4_topmetr(A,B)) ) ) ) ).
fof(t24_treal_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,u1_struct_0(k22_borsuk_1),u1_struct_0(k22_borsuk_1))
& v5_pre_topc(A,k22_borsuk_1,k22_borsuk_1)
& m2_relset_1(A,u1_struct_0(k22_borsuk_1),u1_struct_0(k22_borsuk_1)) )
=> ? [B] :
( m1_subset_1(B,u1_struct_0(k22_borsuk_1))
& k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k22_borsuk_1),A,B) = B ) ) ).
fof(t25_treal_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(A,B)))
& v5_pre_topc(C,k4_topmetr(A,B),k4_topmetr(A,B))
& m2_relset_1(C,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(A,B))) )
=> ? [D] :
( m1_subset_1(D,u1_struct_0(k4_topmetr(A,B)))
& k8_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(A,B)),C,D) = D ) ) ) ) ) ).
fof(t26_treal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& m1_pre_topc(A,k3_topmetr) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,k3_topmetr) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ~ ( ? [D] :
( m1_subset_1(D,k1_numbers)
& ? [E] :
( m1_subset_1(E,k1_numbers)
& r1_xreal_0(D,E)
& r1_tarski(k1_rcomp_1(D,E),u1_struct_0(A))
& r1_tarski(k1_rcomp_1(D,E),u1_struct_0(B))
& r1_tarski(k4_pre_topc(A,B,C,k1_rcomp_1(D,E)),k1_rcomp_1(D,E)) ) )
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D) != D ) ) ) ) ) ).
fof(t27_treal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& m1_pre_topc(A,k3_topmetr) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,k3_topmetr) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ~ ( ? [D] :
( m1_subset_1(D,k1_numbers)
& ? [E] :
( m1_subset_1(E,k1_numbers)
& r1_xreal_0(D,E)
& r1_tarski(k1_rcomp_1(D,E),u1_struct_0(A))
& r1_tarski(k4_pre_topc(A,B,C,k1_rcomp_1(D,E)),k1_rcomp_1(D,E)) ) )
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D) != D ) ) ) ) ) ).
fof(dt_k1_treal_1,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> m1_subset_1(k1_treal_1(A,B),u1_struct_0(k4_topmetr(A,B))) ) ).
fof(dt_k2_treal_1,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> m1_subset_1(k2_treal_1(A,B),u1_struct_0(k4_topmetr(A,B))) ) ).
fof(dt_k3_treal_1,axiom,
! [A,B,C,D] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& m1_subset_1(C,u1_struct_0(k4_topmetr(A,B)))
& m1_subset_1(D,u1_struct_0(k4_topmetr(A,B))) )
=> ( v1_funct_1(k3_treal_1(A,B,C,D))
& v1_funct_2(k3_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)))
& m2_relset_1(k3_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B))) ) ) ).
fof(dt_k4_treal_1,axiom,
! [A,B,C,D] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& m1_subset_1(C,u1_struct_0(k4_topmetr(np__0,np__1)))
& m1_subset_1(D,u1_struct_0(k4_topmetr(np__0,np__1))) )
=> ( v1_funct_1(k4_treal_1(A,B,C,D))
& v1_funct_2(k4_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)))
& m2_relset_1(k4_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1))) ) ) ).
%------------------------------------------------------------------------------