reserve Y for TopStruct;

theorem
  for Y0 being SubSpace of Y holds the carrier of Y0 = the carrier
  of Y implies Y0 is non proper
proof
  let Y0 be SubSpace of Y;
  reconsider A = the carrier of Y0 as Subset of Y by Lm1;
  assume
A1: the carrier of Y0 = the carrier of Y;
  now
    take A;
    thus A = the carrier of Y0 & A is non proper by A1;
  end;
  hence thesis;
end;
