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;

theorem
  X1 is everywhere_dense & X2 is dense or X1 is dense & X2 is
  everywhere_dense implies X1 meet X2 is dense SubSpace of X
proof
  reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
  reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;
  reconsider A = the carrier of X1 meet X2 as Subset of X by TSEP_1:1;
  assume
  X1 is everywhere_dense & X2 is dense or X1 is dense & X2 is everywhere_dense;
  then
A1: A1 is everywhere_dense & A2 is dense or A1 is dense & A2 is
  everywhere_dense;
  then A1 meets A2 by Lm2;
  then
A2: X1 meets X2 by TSEP_1:def 3;
  A1 /\ A2 is dense by A1,TOPS_3:45;
  then A is dense by A2,TSEP_1:def 4;
  hence thesis by Th9;
end;
