theorem
  for X1, X2 being non empty SubSpace of X st X = X1 union X2 for f1
being continuous Function of X1,Y, f2 being continuous Function of X2,Y st (f1
  union f2)|X1 = f1 & (f1 union f2)|X2 = f2 holds X1,X2 are_weakly_separated
  implies f1 union f2 is continuous Function of X,Y
proof
  let X1, X2 be non empty SubSpace of X such that
A1: X = X1 union X2;
  let f1 be continuous Function of X1,Y, f2 be continuous Function of X2,Y
  such that
A2: (f1 union f2)|X1 = f1 & (f1 union f2)|X2 = f2;
  reconsider g = f1 union f2 as Function of X,Y by A1;
  assume
A3: X1,X2 are_weakly_separated;
  g|X1 = f1 & g|X2 = f2 by A1,A2,Def5;
  hence thesis by A1,A3,Th120;
end;
