reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;

theorem Th6:
  for X being TopStruct, X0 being SubSpace of X holds
  the TopStruct of X0 is strict SubSpace of X
proof
  let X be TopStruct, X0 be SubSpace of X;
  reconsider S = the TopStruct of X0 as TopStruct;
  S is SubSpace of X
  proof
A1: [#] (X0) = the carrier of X0;
    hence [#](S) c= [#](X) by PRE_TOPC:def 4;
    let P be Subset of S;
    thus P in the topology of S implies ex Q being Subset of X st Q in the
    topology of X & P = Q /\ [#](S) by A1,PRE_TOPC:def 4;
    given Q being Subset of X such that
A2: Q in the topology of X & P = Q /\ [#](S);
    thus thesis by A1,A2,PRE_TOPC:def 4;
  end;
  hence thesis;
end;
