
theorem
  for S, T being TopStruct for F being Subset-Family of S, G being
  Subset-Family of T st the TopStruct of S = the TopStruct of T & F = G & F is
  closed holds G is closed
proof
  let S, T be TopStruct, F be Subset-Family of S, G be Subset-Family of T such
  that
A1: the TopStruct of S = the TopStruct of T and
A2: F = G & F is closed;
  let P be Subset of T such that
A3: P in G;
  reconsider R = P as Subset of S by A1;
  R is closed by A2,A3;
  then [#]S \ R is open;
  hence [#]T \ P in the topology of T by A1;
end;
