reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T;

theorem Th27:
  for F, G being set holds union UNION (F,G) c= union F \/ union G
proof
  let F, G be set;
  let x be object;
  assume x in union UNION (F,G);
  then consider Y being set such that
A1: x in Y and
A2: Y in UNION (F,G) by TARSKI:def 4;
  consider f,g being set such that
A3: f in F and
A4: g in G and
A5: Y = f \/ g by A2,SETFAM_1:def 4;
  per cases by A1,A5,XBOOLE_0:def 3;
  suppose
    x in f;
    then x in union F by A3,TARSKI:def 4;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose
    x in g;
    then x in union G by A4,TARSKI:def 4;
    hence thesis by XBOOLE_0:def 3;
  end;
end;
