
theorem Th38:
  for M being non empty set, J being TopStruct-yielding non-Empty
  ManySortedSet of M st for i being Element of M holds J.i is T_1 TopSpace-like
  holds product J is T_1
proof
  let M be non empty set, J be TopStruct-yielding non-Empty ManySortedSet of M
  such that
A1: for i be Element of M holds J.i is T_1 TopSpace-like;
  for p being Point of product J holds {p} is closed
  proof
    let p be Point of product J;
    {p} = Cl {p}
    proof
      thus {p} c= Cl {p} by PRE_TOPC:18;
      let a be object;
      assume
A2:   a in Cl {p};
      then reconsider a, p as Element of product J;
A3:   for i being object st i in M holds a.i = p.i
      proof
        let i be object;
        assume i in M;
        then reconsider i as Element of M;
        J.i is TopSpace & J.i is T_1 by A1;
        then
A4:     {p.i} is closed by URYSOHN1:19;
        a.i in Cl {p.i} by A2,Th29;
        then a.i in {p.i} by A4,PRE_TOPC:22;
        hence thesis by TARSKI:def 1;
      end;
      p in the carrier of product J;
      then p in product Carrier J by WAYBEL18:def 3;
      then dom p = dom Carrier J by CARD_3:9;
      then
A5:   dom p = M by PARTFUN1:def 2;
      a in the carrier of product J;
      then a in product Carrier J by WAYBEL18:def 3;
      then dom a = dom Carrier J by CARD_3:9;
      then dom a = M by PARTFUN1:def 2;
      then a = p by A5,A3,FUNCT_1:2;
      hence thesis by TARSKI:def 1;
    end;
    hence thesis;
  end;
  hence thesis by URYSOHN1:19;
end;
