consider X being set such that

A1: for x being object holds

( x in X iff ( x in PFuncs (D,NAT) & S_{1}[x] ) )
from XBOOLE_0:sch 1();

take X ; :: thesis: for f being object holds

( f in X iff f is PartFunc of D,NAT )

let f be object ; :: thesis: ( f in X iff f is PartFunc of D,NAT )

thus ( f in X implies f is PartFunc of D,NAT ) by A1; :: thesis: ( f is PartFunc of D,NAT implies f in X )

assume A2: f is PartFunc of D,NAT ; :: thesis: f in X

then f in PFuncs (D,NAT) by PARTFUN1:45;

hence f in X by A1, A2; :: thesis: verum

A1: for x being object holds

( x in X iff ( x in PFuncs (D,NAT) & S

take X ; :: thesis: for f being object holds

( f in X iff f is PartFunc of D,NAT )

let f be object ; :: thesis: ( f in X iff f is PartFunc of D,NAT )

thus ( f in X implies f is PartFunc of D,NAT ) by A1; :: thesis: ( f is PartFunc of D,NAT implies f in X )

assume A2: f is PartFunc of D,NAT ; :: thesis: f in X

then f in PFuncs (D,NAT) by PARTFUN1:45;

hence f in X by A1, A2; :: thesis: verum