let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X st ( ( X1 is everywhere_dense & X2 is dense ) or ( X1 is dense & X2 is everywhere_dense ) ) holds
X1 meet X2 is dense SubSpace of X
let X1, X2 be non empty SubSpace of X; :: thesis: ( ( ( X1 is everywhere_dense & X2 is dense ) or ( X1 is dense & X2 is everywhere_dense ) ) implies X1 meet X2 is 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 meet X2) as Subset of X by TSEP_1:1;
assume ( ( X1 is everywhere_dense & X2 is dense ) or ( X1 is dense & X2 is everywhere_dense ) ) ; :: thesis: X1 meet X2 is dense SubSpace of X
then A1: ( ( A1 is everywhere_dense & A2 is dense ) or ( A1 is dense & A2 is everywhere_dense ) ) ;
then A1 meets A2 by Lm2;
then A2: X1 meets X2 by TSEP_1:def 3;
A1 /\ A2 is dense by A1, TOPS_3:45;
then A is dense by A2, TSEP_1:def 4;
hence X1 meet X2 is dense SubSpace of X by Th9; :: thesis: verum