
theorem
  for S, T being TopStruct st the TopStruct of S = the TopStruct of T &
  S is compact holds T is compact
proof
  let S, T be TopStruct such that
A1: the TopStruct of S = the TopStruct of T and
A2: for F being Subset-Family of S st F is Cover of S & F is open ex G
  being Subset-Family of S st G c= F & G is Cover of S & G is finite;
  let F be Subset-Family of T such that
A3: F is Cover of T & F is open;
  reconsider K = F as Subset-Family of S by A1;
  consider L being Subset-Family of S such that
A4: L c= K & L is Cover of S & L is finite by A1,A2,A3,WAYBEL_9:19;
  reconsider G = L as Subset-Family of T by A1;
  take G;
  thus thesis by A1,A4;
end;
