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

theorem Th7:
  for X being TopStruct for X1,X2 being TopSpace st
  X1 = the TopStruct of X2 holds X1 is SubSpace of X iff X2 is SubSpace of X
proof
  let X be TopStruct;
  let X1,X2 be TopSpace such that
A1: X1 = the TopStruct of X2;
  thus X1 is SubSpace of X implies X2 is SubSpace of X
  proof
A2: [#] (X1) = the carrier of X1;
    assume
A3: X1 is SubSpace of X;
    hence [#](X2) c= [#](X) by A1,A2,PRE_TOPC:def 4;
    let P be Subset of X2;
    thus P in the topology of X2 implies ex Q being Subset of X st Q in the
    topology of X & P = Q /\ [#](X2) by A1,A3,A2,PRE_TOPC:def 4;
    given Q being Subset of X such that
A4: Q in the topology of X & P = Q /\ [#](X2);
    thus thesis by A1,A3,A2,A4,PRE_TOPC:def 4;
  end;
  thus thesis by A1,Th6;
end;
