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

theorem Th4:
  A^d = ((A`)^f)`
proof
  for x being object holds x in A^d iff x in ((A`)^f)`
  proof
    let x be object;
A1: ((A`)^f)={x2:ex y st y in A` & x2 in U_FT y} by FIN_TOPO:def 12;
    thus x in A^d implies x in ((A`)^f)`
    proof
A2:   (A`)^f={x2:ex y st y in A` & x2 in U_FT y} by FIN_TOPO:def 12;
      assume
A3:   x in A^d;
      then not(ex x2 st x2=x & ex y st y in A` & x2 in U_FT y ) by Th2;
      then not x in (A`)^f by A2;
      hence thesis by A3,SUBSET_1:29;
    end;
    assume
A4: x in ((A`)^f)`;
    then not x in (A`)^f by XBOOLE_0:def 5;
    then for y st y in A` holds not x in U_FT y by A1;
    hence thesis by A4;
  end;
  hence thesis by TARSKI:2;
end;
