
theorem Th37:
  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_0 TopSpace holds
  product J is T_0
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_0 TopSpace;
  for x, y being Point of product J st x <> y holds Cl {x} <> Cl {y}
  proof
    let x, y be Point of product J such that
A2: x <> y and
A3: Cl {x} = Cl {y};
    y in the carrier of product J;
    then y in product Carrier J by WAYBEL18:def 3;
    then dom y = dom Carrier J by CARD_3:9;
    then
A4: dom y = M by PARTFUN1:def 2;
    x in the carrier of product J;
    then x in product Carrier J by WAYBEL18:def 3;
    then dom x = dom Carrier J by CARD_3:9;
    then dom x = M by PARTFUN1:def 2;
    then consider i being object such that
A5: i in M and
A6: x.i <> y.i by A2,A4,FUNCT_1:2;
    reconsider i as Element of M by A5;
A7: pi(Cl {y}, i) = Cl {y.i} by Th31;
    J.i is T_0 TopSpace & pi(Cl {x}, i) = Cl {x.i} by A1,Th31;
    hence contradiction by A3,A6,A7,TSP_1:def 5;
  end;
  hence thesis by TSP_1:def 5;
end;
