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;

theorem
  for X0 being non empty SubSpace of X, f0 being Function of X0,X st for
x being Point of X st x in the carrier of X0 holds x = f0.x holds incl X0 = f0
proof
  let X0 be non empty SubSpace of X, f0 be Function of X0,X;
  assume for x being Point of X st x in the carrier of X0 holds x = f0.x;
  then for x be Point of X st x in the carrier of X0 holds (id X).x = f0.x;
  hence thesis by Th53;
end;
