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;
reserve X1,X2 for TopStruct;
reserve X1,X2 for TopSpace;
reserve D1 for Subset of X1,
  D2 for Subset of X2;

theorem
  D2 c= D1 & the TopStruct of X1 = the TopStruct of X2 implies (D1 is
  nowhere_dense implies D2 is nowhere_dense)
proof
  assume
A1: D2 c= D1;
  assume
A2: the TopStruct of X1 = the TopStruct of X2;
  assume D1 is nowhere_dense;
  then Cl D1 is boundary;
  then Cl D2 is boundary by A1,A2,Th81,Th82;
  hence thesis;
end;
