
theorem Th25:
  for S, T being TopStruct, A being Subset of S, B being Subset of
T st A = B & the TopStruct of S = the TopStruct of T & A is compact holds B is
  compact
proof
  let S, T be TopStruct, A be Subset of S, B be Subset of T such that
A1: A = B and
A2: the TopStruct of S = the TopStruct of T and
A3: for F being Subset-Family of S st F is Cover of A & F is open ex G
  being Subset-Family of S st G c= F & G is Cover of A & G is finite;
  let F be Subset-Family of T such that
A4: F is Cover of B & F is open;
  reconsider K = F as Subset-Family of S by A2;
  consider L being Subset-Family of S such that
A5: L c= K & L is Cover of A & L is finite by A1,A2,A3,A4,WAYBEL_9:19;
  reconsider G = L as Subset-Family of T by A2;
  take G;
  thus thesis by A1,A5;
end;
