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 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;
