set Y = { x where x is Element of X : 0. X <= x } ;

A1: now :: thesis: for y being object st y in { x where x is Element of X : 0. X <= x } holds
y in the carrier of X

let y be object ; :: thesis: ( y in { x where x is Element of X : 0. X <= x } implies y in the carrier of X )
assume y in { x where x is Element of X : 0. X <= x } ; :: thesis: y in the carrier of X
then ex x being Element of X st
( y = x & 0. X <= x ) ;
hence y in the carrier of X ; :: thesis: verum

end;

(0. X) ` = 0. X by Def5;
then 0. X <= 0. X ;
then 0. X in { x where x is Element of X : 0. X <= x } ;
hence { x where x is Element of X : 0. X <= x } is non empty Subset of X by A1, TARSKI:def 3; :: thesis: verum