reserve X for non empty TopSpace;

theorem Th4:
  for Y0, Y1 being TopStruct, D0 being Subset of Y0, D1 being
Subset of Y1 st the TopStruct of Y0 = the TopStruct of Y1 & D0 = D1 holds D0 is
  anti-discrete implies D1 is anti-discrete
proof
  let Y0, Y1 be TopStruct, D0 be Subset of Y0, D1 be Subset of Y1;
  assume
A1: the TopStruct of Y0 = the TopStruct of Y1;
  assume
A2: D0 = D1;
  assume
A3: D0 is anti-discrete;
  now
    let D be Subset of Y1;
    reconsider E = D as Subset of Y0 by A1;
    assume D is open;
    then E in the topology of Y0 by A1,PRE_TOPC:def 2;
    then E is open by PRE_TOPC:def 2;
    hence D1 misses D or D1 c= D by A2,A3;
  end;
  hence thesis;
end;
