reserve X for non empty TopSpace,
  A,B for Subset of X;
reserve Y1,Y2 for non empty SubSpace of X;
reserve X1, X2 for non empty SubSpace of X;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X for non discrete non empty TopSpace;

theorem
  for X being non trivial TopSpace holds (for X0 being
  proper SubSpace of X holds X0 is non everywhere_dense) implies X is
  almost_discrete
proof
  let X be non trivial TopSpace;
  assume
A1: for X0 being proper SubSpace of X holds X0 is non everywhere_dense;
  now
    let A be Subset of X;
    assume
A2: A <> the carrier of X;
    now
      per cases;
      suppose
        A is empty;
        hence A is not everywhere_dense by TOPS_3:34;
      end;
      suppose
        A is non empty;
        then consider X0 being strict non empty SubSpace of X such that
A3:     A = the carrier of X0 by TSEP_1:10;
        A is proper by A2,SUBSET_1:def 6;
        then reconsider X0 as proper strict SubSpace of X by A3,TEX_2:8;
        X0 is non everywhere_dense by A1;
        hence A is not everywhere_dense by A3;
      end;
    end;
    hence A is not everywhere_dense;
  end;
  hence thesis by TEX_1:32;
end;
