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;

theorem
  (X1 is nowhere_dense or X2 is nowhere_dense) & X1 meets X2 implies X1
  meet X2 is nowhere_dense
proof
  assume
A1: X1 is nowhere_dense or X2 is nowhere_dense;
  assume
A2: X1 meets X2;
  hereby
    per cases by A1;
    suppose
A3:   X1 is nowhere_dense;
      X1 meet X2 is SubSpace of X1 by A2,TSEP_1:27;
      hence thesis by A3,Th40;
    end;
    suppose
A4:   X2 is nowhere_dense;
      X1 meet X2 is SubSpace of X2 by A2,TSEP_1:27;
      hence thesis by A4,Th40;
    end;
  end;
end;
