theorem Th13:
  for f being Function holds union((.:f).:A) c= f.:(union A)
proof
  let f be Function;
  let y be object;
  assume y in union((.:f).:A);
  then consider Z such that
A1: y in Z and
A2: Z in (.:f).:A by TARSKI:def 4;
  consider X being object such that
A3: X in dom(.:f) and
A4: X in A and
A5: Z = (.:f).X by A2,FUNCT_1:def 6;
  reconsider X as set by TARSKI:1;
  y in f.:X by A1,A3,A5,Th7;
  then consider x being object such that
A6: x in dom f and
A7: x in X and
A8: y = f.x by FUNCT_1:def 6;
  x in union A by A4,A7,TARSKI:def 4;
  hence thesis by A6,A8,FUNCT_1:def 6;
end;
