theorem
  for X1, X2 being open non empty SubSpace of X st X1 meets X2 for f1
being continuous Function of X1,Y, f2 being continuous Function of X2,Y st f1|(
  X1 meet X2) = f2|(X1 meet X2) holds f1 union f2 is continuous Function of X1
  union X2,Y
proof
  let X1, X2 be open non empty SubSpace of X such that
A1: X1 meets X2;
  let f1 be continuous Function of X1,Y, f2 be continuous Function of X2,Y;
  assume f1|(X1 meet X2) = f2|(X1 meet X2);
  then (f1 union f2)|X1 = f1 & (f1 union f2)|X2 = f2 by A1,Th128;
  hence thesis by Th116;
end;
