set h = f <#> g;
A1: dom (f <#> g) = (dom f) /\ (dom g) by Def43;
rng (f <#> g) c= N_PFuncs (DOMS Y)

proof

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng (f <#> g) or y in N_PFuncs (DOMS Y) )
assume y in rng (f <#> g) ; :: thesis: y in N_PFuncs (DOMS Y)
then consider x being object such that
A2: x in dom (f <#> g) and
A3: (f <#> g) . x = y by FUNCT_1:def 3;
reconsider y = y as Function by A3;
A4: (f <#> g) . x = (f . x) (#) (g . x) by A2, Def43;
A5: rng y c= NAT by A3, A4, ORDINAL1:def 12;
x in dom f by A1, A2, XBOOLE_0:def 4;
then f . x in Y by PARTFUN1:4;
then dom (f . x) in { (dom m) where m is Element of Y : verum } ;
then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;
dom y = dom (f . x) by A3, A4, VALUED_1:def 5;
then y is PartFunc of (DOMS Y),NAT by A6, A5, RELSET_1:4;
hence y in N_PFuncs (DOMS Y) by Def18; :: thesis: verum

end;

hence f <#> g is PartFunc of (X /\ (dom g)),(N_PFuncs (DOMS Y)) by A1, RELSET_1:4; :: thesis: verum