set Y = { x where x is Element of X : x is atom } ;

z = 0. X by Th2;

then 0. X is atom ;

then 0. X in { x where x is Element of X : x is atom } ;

hence { x where x is Element of X : x is atom } is non empty Subset of X by A1, TARSKI:def 3; :: thesis: verum

A1: now :: thesis: for y being object st y in { x where x is Element of X : x is atom } holds

y in the carrier of X

for z being Element of X st z \ (0. X) = 0. X holds y in the carrier of X

let y be object ; :: thesis: ( y in { x where x is Element of X : x is atom } implies y in the carrier of X )

assume y in { x where x is Element of X : x is atom } ; :: thesis: y in the carrier of X

then ex x being Element of X st

( y = x & x is atom ) ;

hence y in the carrier of X ; :: thesis: verum

end;assume y in { x where x is Element of X : x is atom } ; :: thesis: y in the carrier of X

then ex x being Element of X st

( y = x & x is atom ) ;

hence y in the carrier of X ; :: thesis: verum

z = 0. X by Th2;

then 0. X is atom ;

then 0. X in { x where x is Element of X : x is atom } ;

hence { x where x is Element of X : x is atom } is non empty Subset of X by A1, TARSKI:def 3; :: thesis: verum