reserve X for TopStruct,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A, B for Subset of X;
reserve D for Subset of X;
reserve Y0 for SubSpace of X;
reserve X0 for SubSpace of X;
reserve X0 for non empty SubSpace of X;

theorem Th69:
  for C, A being Subset of X, B being Subset of X0 st C is open &
  C c= the carrier of X0 & A c= C & A = B holds A is nowhere_dense implies B is
  nowhere_dense
proof
  let C, A be Subset of X, B be Subset of X0;
  assume
A1: C is open;
  assume
A2: C c= the carrier of X0;
  assume that
A3: A c= C and
A4: A = B;
  assume
A5: A is nowhere_dense;
A6: now
    assume C <> {};
    then consider X1 being strict non empty SubSpace of X such that
A7: C = the carrier of X1 by TSEP_1:10;
    reconsider E = B as Subset of X1 by A3,A4,A7;
    E is nowhere_dense & X1 is SubSpace of X0 by A1,A2,A4,A5,A7,Lm1,TSEP_1:4;
    hence thesis by Th68;
  end;
  now
    assume C = {};
    then B = {}X0 by A3,A4,XBOOLE_1:3;
    hence thesis;
  end;
  hence thesis by A6;
end;
