reserve            x for object,
               X,Y,Z for set,
         i,j,k,l,m,n for Nat,
                 r,s for Real,
                  no for Element of OrderedNAT,
                   A for Subset of [:NAT,NAT:];
reserve X,Y,X1,X2 for non empty set,
          cA1,cB1 for filter_base of X1,
          cA2,cB2 for filter_base of X2,
              cF1 for Filter of X1,
              cF2 for Filter of X2,
             cBa1 for basis of cF1,
             cBa2 for basis of cF2;
reserve T for non empty TopSpace,
        s for Function of [:NAT,NAT:], the carrier of T,
        M for Subset of the carrier of T;
reserve cF3,cF4 for Filter of the carrier of T;

theorem
  x in s.:([:NAT,NAT:] \ [:Segm i,Segm j:]) iff ex n,m being Nat st
  (i <= n or j <= m) & x = s.(n,m)
  proof
    hereby
      assume x in s.:([:NAT,NAT:] \ [:Segm i,Segm j:]);
      then consider y be object such that
A1:   y in dom s and
A2:   y in [:NAT,NAT:] \ [:Segm i,Segm j:] and
A3:   x = s.y by FUNCT_1:def 6;
      reconsider z = y as Element of [:NAT,NAT:] by A1;
A4:   not z`1 in Segm i or not z`2 in Segm j
      proof
        assume not (not z`1 in Segm i or not z`2 in Segm j);
        then
A5:     [z`1,z`2] in [:Segm i,Segm j:] by ZFMISC_1:def 2;
        z is pair by Th4;
        hence thesis by A2,A5,XBOOLE_0:def 5;
      end;
      reconsider n = z`1,m = z`2 as Nat;
      take n,m;
      thus i <= n or j <= m by A4,NAT_1:44;
      z is pair by Th4;
      hence x = s.(n,m) by A3,BINOP_1:def 1;
    end;
    assume ex n,m be Nat st (i <= n or j <= m) & x = s.(n,m);
    then consider n,m be Nat such that
A6: (i <= n or j <= m) and
A7: x = s.(n,m);
    n in NAT & m in NAT by ORDINAL1:def 12; then
A8: [n,m] in [:NAT,NAT:] by ZFMISC_1:def 2;
    not [n,m] in [:Segm i,Segm j:]
    proof
      assume [n,m] in [:Segm i,Segm j:];
      then ex a,b be object st a in Segm i & b in Segm j & [n,m] = [a,b]
        by ZFMISC_1:def 2;
      then n in Segm i & m in Segm j by XTUPLE_0:1;
      hence thesis by A6,NAT_1:44;
    end;
    then
A9: [n,m] in [:NAT,NAT:] \ [:Segm i,Segm j:] by A8,XBOOLE_0:def 5;
A10: x = s.([n,m]) by BINOP_1:def 1,A7;
    [n,m] in dom s by A8,FUNCT_2:def 1;
    hence x in s.:([:NAT,NAT:] \ [:Segm i,Segm j:]) by A9,A10,FUNCT_1:def 6;
  end;
