defpred S_{1}[ set ] means $1 is a_partition of Y;

consider X being set such that

A1: for x being set holds

( x in X iff ( x in bool (bool Y) & S_{1}[x] ) )
from XFAMILY:sch 1();

take X ; :: thesis: for x being set holds

( x in X iff x is a_partition of Y )

let x be set ; :: thesis: ( x in X iff x is a_partition of Y )

thus ( x in X implies x is a_partition of Y ) by A1; :: thesis: ( x is a_partition of Y implies x in X )

assume x is a_partition of Y ; :: thesis: x in X

hence x in X by A1; :: thesis: verum

consider X being set such that

A1: for x being set holds

( x in X iff ( x in bool (bool Y) & S

take X ; :: thesis: for x being set holds

( x in X iff x is a_partition of Y )

let x be set ; :: thesis: ( x in X iff x is a_partition of Y )

thus ( x in X implies x is a_partition of Y ) by A1; :: thesis: ( x is a_partition of Y implies x in X )

assume x is a_partition of Y ; :: thesis: x in X

hence x in X by A1; :: thesis: verum