
theorem
  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
  maximal_T_0 implies D1 is maximal_T_0
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 maximal_T_0;
A4: now
    let D be Subset of Y1;
    reconsider E = D as Subset of Y0 by A1;
    assume D is T_0;
    then
A5: E is T_0 by A1,TSP_1:3;
    assume D1 c= D;
    hence D1 = D by A2,A3,A5;
  end;
  D0 is T_0 by A3;
  then D1 is T_0 by A1,A2,TSP_1:3;
  hence thesis by A4;
end;
