reserve r for Real,
  X for set,
  f, g, h for real-valued Function;

theorem Th13:
  for X, Y being non empty TopSpace, X1, X2 being non empty
  SubSpace of X for f1 being Function of X1,Y, f2 being Function of X2,Y st X1
  misses X2 or f1|(X1 meet X2) = f2|(X1 meet X2) holds rng (f1 union f2) c= rng
  f1 \/ rng f2
proof
  let X, Y be non empty TopSpace, X1, X2 be non empty SubSpace of X;
  let f1 be Function of X1,Y, f2 be Function of X2,Y;
  set F = f1 union f2;
  assume
A1: X1 misses X2 or f1|(X1 meet X2) = f2|(X1 meet X2);
  thus rng F c= rng f1 \/ rng f2
  proof
A2: the carrier of X1 union X2 = (the carrier of X1) \/ the carrier of X2
    by TSEP_1:def 2;
A3: the carrier of X2 = dom f2 by FUNCT_2:def 1;
    let y be object;
A4: the carrier of X1 = dom f1 by FUNCT_2:def 1;
    assume y in rng F;
    then consider x being object such that
A5: x in dom F and
A6: F.x = y by FUNCT_1:def 3;
A7: x is Point of X by A5,PRE_TOPC:25;
    per cases by A5,A2,XBOOLE_0:def 3;
    suppose
      x in the carrier of X1;
      then f1.x in rng f1 & F.x = f1.x by A1,A4,A7,Th12,FUNCT_1:def 3;
      hence thesis by A6,XBOOLE_0:def 3;
    end;
    suppose
      x in the carrier of X2;
      then f2.x in rng f2 & F.x = f2.x by A1,A3,A7,Th12,FUNCT_1:def 3;
      hence thesis by A6,XBOOLE_0:def 3;
    end;
  end;
end;
