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;

theorem Th53:
  for f0 being Function of X0,Y st for x being Point of X st x in
  the carrier of X0 holds f.x = f0.x holds f|X0 = f0
proof
  let f0 be Function of X0,Y;
  the carrier of X0 is Subset of X by TSEP_1:1;
  hence thesis by FUNCT_2:96;
end;
