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

theorem Th11:
  (A^f) \/ (B^f) = (A \/ B)^f
proof
  A^f c= (A \/ B)^f & B^f c= (A \/ B)^f by Th5,XBOOLE_1:7;
  hence (A^f) \/ (B^f) c= (A \/ B)^f by XBOOLE_1:8;
  let z be object;
  assume z in (A \/ B)^f;
  then consider y such that
A1: y in A \/ B and
A2: z in U_FT y by FIN_TOPO:11;
  per cases by A1,XBOOLE_0:def 3;
  suppose
    y in A;
    then z in (A^f) by A2,FIN_TOPO:11;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose
    y in B;
    then z in (B^f) by A2,FIN_TOPO:11;
    hence thesis by XBOOLE_0:def 3;
  end;
end;
