
theorem Th1:
  for Y being TopStruct, Y0 being SubSpace of Y for G being Subset
of Y st G is open holds ex G0 being Subset of Y0 st G0 is open & G0 = G /\ the
  carrier of Y0
proof
  let Y be TopStruct, Y0 be SubSpace of Y;
  let G be Subset of Y;
  assume
A1: G is open;
  [#]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 G0 = G /\ A as Subset of Y0 by XBOOLE_1:17;
  take G0;
  thus G0 is open by A1,Def1;
  thus thesis;
end;
