let X be non empty TopSpace; :: thesis: for X1 being non empty dense SubSpace of X
for X2 being non empty dense SubSpace of X1 holds X2 is non empty dense SubSpace of X
let X1 be non empty dense SubSpace of X; :: thesis: for X2 being non empty dense SubSpace of X1 holds X2 is non empty dense SubSpace of X
let X2 be non empty dense SubSpace of X1; :: thesis: X2 is non empty dense SubSpace of X
reconsider C = the carrier of X1 as Subset of X by BORSUK_1:1;
now :: thesis: for A being Subset of X st A = the carrier of X2 holds
A is dense
let A be Subset of X; :: thesis: ( A = the carrier of X2 implies A is dense )
assume A1: A = the carrier of X2 ; :: thesis: A is dense
then reconsider B = A as Subset of X1 by BORSUK_1:1;
A2: C is dense by Def1;
B is dense by A1, Def1;
hence A is dense by A2, TOPS_3:60; :: thesis: verum
end;
hence X2 is non empty dense SubSpace of X by Def1, TSEP_1:7; :: thesis: verum