theorem Th2:
  for GX being TopSpace, A,B being Subset of GX,
  AA being Subset of GX|B st A = AA holds GX|A = GX|B|AA
proof
  let GX be TopSpace, A,B be Subset of GX;
  let AA be Subset of GX|B;
  assume
A1: A = AA;
  the carrier of GX|A = [#](GX|A) .= A by PRE_TOPC:def 5;
  then reconsider GY = GX|A as strict SubSpace of GX|B by A1,TSEP_1:4;
  [#] GY = AA by A1,PRE_TOPC:def 5;
  hence thesis by PRE_TOPC:def 5;
end;
