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;

theorem Th65:
  X1 is SubSpace of X0 implies for x0 being Point of X0 st x0 in
  the carrier of X1 holds g.x0 = (g|X1).x0
proof
  assume
A1: X1 is SubSpace of X0;
  let x0 be Point of X0;
  assume x0 in the carrier of X1;
  hence g.x0 = (g|(the carrier of X1)).x0 by FUNCT_1:49
    .= (g|X1).x0 by A1,Def5;
end;
