reserve T for TopStruct;
reserve GX for TopSpace;

theorem
  for T1, T2, S1, S2 being TopStruct st the TopStruct of T1 = the
  TopStruct of T2 & the TopStruct of S1 = the TopStruct of S2 holds S1 is
  SubSpace of T1 implies S2 is SubSpace of T2
proof
  let T1, T2, S1, S2 be TopStruct such that
A1: the TopStruct of T1 = the TopStruct of T2 & the TopStruct of S1 =
  the TopStruct of S2;
  assume that
A2: [#]S1 c= [#]T1 and
A3: for P being Subset of S1 holds P in the topology of S1 iff ex Q
  being Subset of T1 st Q in the topology of T1 & P = Q /\ [#]S1;
  thus [#]S2 c= [#]T2 by A1,A2;
  thus thesis by A1,A3;
end;
