theorem
  for Y0 being SubSpace of X modified_with_respect_to X0 st the carrier
  of Y0 = the carrier of X0 holds Y0 is open SubSpace of X
  modified_with_respect_to X0
proof
  let Y0 be SubSpace of X modified_with_respect_to X0;
  assume
A1: the carrier of Y0 = the carrier of X0;
  reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
  set Y = X modified_with_respect_to X0;
  reconsider B = the carrier of Y0 as Subset of Y by TSEP_1:1;
  Y = X modified_with_respect_to A by Def10;
  then B is open by A1,Th94;
  hence thesis by TSEP_1:16;
end;
