let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X st ( X1 is everywhere_dense or X2 is everywhere_dense ) holds
X1 union X2 is everywhere_dense SubSpace of X
let X1, X2 be non empty SubSpace of X; :: thesis: ( ( X1 is everywhere_dense or X2 is everywhere_dense ) implies X1 union X2 is everywhere_dense SubSpace of X )
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;
reconsider A = the carrier of (X1 union X2) as Subset of X by TSEP_1:1;
assume ( X1 is everywhere_dense or X2 is everywhere_dense ) ; :: thesis: X1 union X2 is everywhere_dense SubSpace of X
then ( A1 is everywhere_dense or A2 is everywhere_dense ) ;
then A1 \/ A2 is everywhere_dense by TOPS_3:43;
then A is everywhere_dense by TSEP_1:def 2;
hence X1 union X2 is everywhere_dense SubSpace of X by Th16; :: thesis: verum