
theorem
  for M being non empty set, T being non empty TopSpace, x, y being
Point of product (M --> T) holds x in Cl {y} iff for i being Element of M holds
  x.i in Cl {y.i}
proof
  let M be non empty set, T be non empty TopSpace, x, y be Point of product (M
  --> T);
  reconsider J = M --> T as TopStruct-yielding non-Empty ManySortedSet of M;
  reconsider x9 = x, y9 = y as Point of product J;
  thus x in Cl {y} implies for i being Element of M holds x.i in Cl {y.i}
  proof
    assume
A1: x in Cl {y};
    let i be Element of M;
    x9.i in Cl {y9.i} by A1,Th29;
    hence thesis;
  end;
  assume for i being Element of M holds x.i in Cl {y.i};
  then for i being Element of M holds x9.i in Cl {y9.i};
  hence thesis by Th29;
end;
