let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X st ( ( X1 is nowhere_dense & X2 is boundary ) or ( X1 is boundary & X2 is nowhere_dense ) ) holds
X1 union X2 is boundary
let X1, X2 be non empty SubSpace of X; :: thesis: ( ( ( X1 is nowhere_dense & X2 is boundary ) or ( X1 is boundary & X2 is nowhere_dense ) ) implies X1 union X2 is boundary )
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;
assume ( ( X1 is nowhere_dense & X2 is boundary ) or ( X1 is boundary & X2 is nowhere_dense ) ) ; :: thesis: X1 union X2 is boundary
then ( ( A1 is nowhere_dense & A2 is boundary ) or ( A1 is boundary & A2 is nowhere_dense ) ) ;
then A1 \/ A2 is boundary by TOPS_3:30;
then for A being Subset of X st A = the carrier of (X1 union X2) holds
A is boundary by TSEP_1:def 2;
hence X1 union X2 is boundary ; :: thesis: verum