let X be non empty TopSpace; :: thesis:
let X1, X2 be non empty SubSpace of X; :: thesis:
assume A1: ( X1 is nowhere_dense or X2 is nowhere_dense ) ; :: thesis:
assume A2: X1 meets X2 ; :: thesis:

hereby :: thesis:

per cases by A1;

suppose A3: X1 is nowhere_dense ; :: thesis:

X1 meet X2 is SubSpace of X1 by A2, TSEP_1:27;
hence X1 meet X2 is nowhere_dense by A3, Th40; :: thesis:

end;

suppose A4: X2 is nowhere_dense ; :: thesis:

X1 meet X2 is SubSpace of X2 by A2, TSEP_1:27;
hence X1 meet X2 is nowhere_dense by A4, Th40; :: thesis:

end;

end;

end;