
theorem Th2:
  for Y being TopStruct, Y0 being SubSpace of Y for F being Subset
of Y st F is closed holds ex F0 being Subset of Y0 st F0 is closed & F0 = F /\
  the carrier of Y0
proof
  let Y be TopStruct, Y0 be SubSpace of Y;
  let F be Subset of Y;
  assume
A1: F is closed;
  [#]Y0 = the carrier of Y0 & [#]Y = the carrier of Y;
  then reconsider A = the carrier of Y0 as Subset of Y by PRE_TOPC:def 4;
  reconsider F0 = F /\ A as Subset of Y0 by XBOOLE_1:17;
  take F0;
  thus F0 is closed by A1,Def2;
  thus thesis;
end;
