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

theorem Th12:
  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) for x being Point of X holds (x
in the carrier of X1 implies (f1 union f2).x = f1.x) & (x in the carrier of X2
  implies (f1 union f2).x = f2.x)
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;
  assume
A1: X1 misses X2 or f1|(X1 meet X2) = f2|(X1 meet X2);
  let x be Point of X;
  set F = f1 union f2;
A2: X1 is SubSpace of X1 union X2 by TSEP_1:22;
  hereby
    assume x in the carrier of X1;
    hence F.x = (F|the carrier of X1).x by FUNCT_1:49
      .= (F|X1).x by A2,TMAP_1:def 5
      .= f1.x by A1,TMAP_1:def 12;
  end;
A3: X2 is SubSpace of X1 union X2 by TSEP_1:22;
  assume x in the carrier of X2;
  hence F.x = (F|the carrier of X2).x by FUNCT_1:49
    .= (F|X2).x by A3,TMAP_1:def 5
    .= f2.x by A1,TMAP_1:def 12;
end;
