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 Th61:
  for A being Subset of X, B being Subset of X0 st A c= B holds A
  is everywhere_dense implies B is everywhere_dense
proof
  let A be Subset of X, B be Subset of X0;
  assume
A1: A c= B;
  then reconsider C = A as Subset of X0 by XBOOLE_1:1;
  assume A is everywhere_dense;
  then Int A is dense;
  then Int C is dense by Th56,Th59;
  then Int B is dense by A1,TOPS_1:19,44;
  hence thesis;
end;
