
theorem
  for I being non empty set
  for J being TopSpace-yielding non-Empty ManySortedSet of I
  for p being Permutation of I
  holds product J, product(J*p) are_homeomorphic
proof
  let I be non empty set;
  let J be TopSpace-yielding non-Empty ManySortedSet of I;
  let p be Permutation of I;
  reconsider q = p" as Permutation of I;
  now
    let i be Element of I;
    A1: J = J*id dom J by RELAT_1:52
      .= J*id I by PARTFUN1:def 2
      .= J*(p*q) by FUNCT_2:61
      .= (J*p)*q by RELAT_1:36;
    thus J.i, ((J*p)*q).i are_homeomorphic by A1;
  end;
  hence product J, product(J*p) are_homeomorphic by Th79;
end;
