reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve f for Function of X,Y;
reserve X,Y,Z for non empty TopSpace;
reserve f for Function of X,Y,
  g for Function of Y,Z;
reserve X, Y for non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve f for Function of X,Y;
reserve f for Function of X,Y,
  X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace,
  X0, X1 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X0, X1, X2 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X for non empty TopSpace,
  H, G for Subset of X;
reserve A for Subset of X;
reserve X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;

theorem
  for X1, X2 being closed non empty SubSpace of X st X1 misses X2 for f1
  being continuous Function of X1,Y, f2 being continuous Function of X2,Y holds
  f1 union f2 is continuous Function of X1 union X2,Y
proof
  let X1, X2 be closed non empty SubSpace of X such that
A1: X1 misses X2;
  let f1 be continuous Function of X1,Y, f2 be continuous Function of X2,Y;
  (f1 union f2)|X1 = f1 & (f1 union f2)|X2 = f2 by A1,Def12;
  hence thesis by Th115;
end;
