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, f1 being continuous
Function of X1,Y, f2 being continuous Function of X2,Y st X = X1 union X2 & (f1
  union f2)|X1 = f1 & (f1 union f2)|X2 = f2 holds f1 union f2 is continuous
  Function of X,Y
proof
  let X1, X2 be closed non empty SubSpace of X, f1 be continuous Function of
  X1,Y, f2 be continuous Function of X2,Y such that
A1: X = X1 union X2 and
A2: (f1 union f2)|X1 = f1 & (f1 union f2)|X2 = f2;
  reconsider g = f1 union f2 as Function of X,Y by A1;
  g|X1 = f1 & g|X2 = f2 by A1,A2,Def5;
  hence thesis by A1,Th121;
end;
