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

hereby :: thesis: verum

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

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

X1 meet X2 is SubSpace of X1 by A2, TSEP_1:27;
hence X1 meet X2 is boundary by A3, Th34; :: thesis: verum

end;

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

X1 meet X2 is SubSpace of X2 by A2, TSEP_1:27;
hence X1 meet X2 is boundary by A4, Th34; :: thesis: verum

end;

end;

end;