reserve T,T1,T2,S for non empty TopSpace;
reserve GY for non empty TopSpace,
  r,s for Real;

theorem Th7:
  for X, Y being non empty TopSpace for A being Subset-Family of Y
  for f being Function of X, Y holds f"(union A) = union (f"A)
proof
  let X, Y be non empty TopSpace, A be Subset-Family of Y, f be Function of X,
  Y;
  thus f"(union A) c= union (f"A)
  proof
    reconsider uA = union A as Subset of Y;
    let x be object;
    assume
A1: x in f"(union A);
    then f.x in uA by FUNCT_2:38;
    then consider YY being set such that
A2: f.x in YY and
A3: YY in A by TARSKI:def 4;
    reconsider fY = f"YY as Subset of X;
A4: fY in f"A by A3,FUNCT_2:def 9;
    x in f"YY by A1,A2,FUNCT_2:38;
    hence thesis by A4,TARSKI:def 4;
  end;
  let x be object;
  assume x in union (f"A);
  then consider YY be set such that
A5: x in YY and
A6: YY in f"A by TARSKI:def 4;
  x in the carrier of X by A5,A6;
  then
A7: x in dom f by FUNCT_2:def 1;
  reconsider B1 = YY as Subset of X by A6;
  consider B being Subset of Y such that
A8: B in A and
A9: B1 = f"B by A6,FUNCT_2:def 9;
  f.x in B by A5,A9,FUNCT_1:def 7;
  then f.x in union A by A8,TARSKI:def 4;
  hence thesis by A7,FUNCT_1:def 7;
end;
