let X be non empty TopSpace; :: thesis: for X0 being non empty everywhere_dense SubSpace of X
for A being Subset of X
for B being Subset of X0 st A = B holds
( B is everywhere_dense iff A is everywhere_dense )
let X0 be non empty everywhere_dense SubSpace of X; :: thesis: for A being Subset of X
for B being Subset of X0 st A = B holds
( B is everywhere_dense iff A is everywhere_dense )
let A be Subset of X; :: thesis: for B being Subset of X0 st A = B holds
( B is everywhere_dense iff A is everywhere_dense )
let B be Subset of X0; :: thesis: ( A = B implies ( B is everywhere_dense iff A is everywhere_dense ) )
assume A1: A = B ; :: thesis: ( B is everywhere_dense iff A is everywhere_dense )
reconsider C = the carrier of X0 as Subset of X by TSEP_1:1;
C is everywhere_dense by Th16;
hence ( B is everywhere_dense iff A is everywhere_dense ) by A1, TOPS_3:64; :: thesis: verum