reserve Y for TopStruct;

theorem Th19:
  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
  discrete implies D1 is 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 discrete;
  now
    let D be Subset of Y1;
    reconsider E = D as Subset of Y0 by A1;
    assume D c= D1;
    then consider G0 being Subset of Y0 such that
A4: G0 is open and
A5: D0 /\ G0 = E by A2,A3;
    reconsider G = G0 as Subset of Y1 by A1;
    now
      take G;
      G in the topology of Y1 by A1,A4;
      hence G is open;
      thus D1 /\ G = D by A2,A5;
    end;
    hence ex G being Subset of Y1 st G is open & D1 /\ G = D;
  end;
  hence thesis;
end;
