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

theorem Th8:
  (A \/ B)^b = (A^b) \/ (B^b)
proof
  thus (A \/ B)^b c= (A^b) \/ (B^b)
  proof
    let x be object;
    assume
A1: x in (A \/ B)^b;
    then reconsider px=x as Point of T;
    U_FT px meets (A \/ B) by A1,FIN_TOPO:8;
    then U_FT px meets A or U_FT px meets B by XBOOLE_1:70;
    then x in A^b or x in B^b by FIN_TOPO:8;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x be object;
  assume
A2: x in (A^b) \/ (B^b);
  then reconsider px=x as Point of T;
  x in A^b or x in B^b by A2,XBOOLE_0:def 3;
  then U_FT px meets A or U_FT px meets B by FIN_TOPO:8;
  then U_FT px meets (A \/ B) by XBOOLE_1:70;
  hence thesis by FIN_TOPO:8;
end;
