let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X st ( X1 is nowhere_dense or X2 is nowhere_dense ) & X1 meets X2 holds
X1 meet X2 is nowhere_dense
let X1, X2 be non empty SubSpace of X; :: thesis: ( ( X1 is nowhere_dense or X2 is nowhere_dense ) & X1 meets X2 implies X1 meet X2 is nowhere_dense )
assume A1: ( X1 is nowhere_dense or X2 is nowhere_dense ) ; :: thesis: ( not X1 meets X2 or X1 meet X2 is nowhere_dense )
assume A2: X1 meets X2 ; :: thesis: X1 meet X2 is nowhere_dense

hereby :: thesis: verum

per cases ( X1 is nowhere_dense or X2 is nowhere_dense ) by A1;

suppose A3: X1 is nowhere_dense ; :: thesis: X1 meet X2 is nowhere_dense

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

end;

suppose A4: X2 is nowhere_dense ; :: thesis: X1 meet X2 is nowhere_dense

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

end;

end;

end;