set h = f <++> g;
A1: dom (f <++> g) = (dom f) /\ (dom g) by Def45;
rng (f <++> g) c= N_PFuncs ((DOMS Y1) /\ (DOMS Y2))

proof

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng (f <++> g) or y in N_PFuncs ((DOMS Y1) /\ (DOMS Y2)) )
assume y in rng (f <++> g) ; :: thesis: y in N_PFuncs ((DOMS Y1) /\ (DOMS Y2))
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, Def45;
A5: rng y c= NAT by A3, A4, ORDINAL1:def 12;
x in dom g by A1, A2, XBOOLE_0:def 4;
then g . x in Y2 by PARTFUN1:4;
then dom (g . x) in { (dom m) where m is Element of Y2 : verum } ;
then A6: dom (g . x) c= DOMS Y2 by ZFMISC_1:74;
x in dom f by A1, A2, XBOOLE_0:def 4;
then f . x in Y1 by PARTFUN1:4;
then dom (f . x) in { (dom m) where m is Element of Y1 : verum } ;
then A7: dom (f . x) c= DOMS Y1 by ZFMISC_1:74;
dom y = (dom (f . x)) /\ (dom (g . x)) by A3, A4, VALUED_1:def 1;
then y is PartFunc of ((DOMS Y1) /\ (DOMS Y2)),NAT by A7, A6, A5, RELSET_1:4, XBOOLE_1:27;
hence y in N_PFuncs ((DOMS Y1) /\ (DOMS Y2)) by Def18; :: thesis: verum

end;

hence f <++> g is PartFunc of (X1 /\ X2),(N_PFuncs ((DOMS Y1) /\ (DOMS Y2))) by A1, RELSET_1:4; :: thesis: verum