reserve T for non empty RelStr,
  A,B for Subset of T,
  x,x2,y,z for Element of T;

theorem Th3:
  T is filled implies A^d c= A
proof
  assume
A1: T is filled;
  thus A^d c= A
  proof
    let x be object;
    assume
A2: x in A^d;
    then reconsider z=x as Element of T;
    now
      assume not x in A;
      then
A3:   x in A` by A2,SUBSET_1:29;
      x in U_FT z by A1,FIN_TOPO:def 4;
      hence contradiction by A2,A3,Th2;
    end;
    hence thesis;
  end;
end;
