set h = f <#> g;

A1: dom (f <#> g) = (dom f) /\ (dom g) by Def43;

rng (f <#> g) c= R_PFuncs (DOMS Y)

A1: dom (f <#> g) = (dom f) /\ (dom g) by Def43;

rng (f <#> g) c= R_PFuncs (DOMS Y)

proof

hence
f <#> g is PartFunc of (X /\ (dom g)),(R_PFuncs (DOMS Y))
by A1, RELSET_1:4; :: thesis: verum
let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng (f <#> g) or y in R_PFuncs (DOMS Y) )

assume y in rng (f <#> g) ; :: thesis: y in R_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= REAL by A3, A4, XREAL_0:def 1;

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),REAL by A6, A5, RELSET_1:4;

hence y in R_PFuncs (DOMS Y) by Def12; :: thesis: verum

end;assume y in rng (f <#> g) ; :: thesis: y in R_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= REAL by A3, A4, XREAL_0:def 1;

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),REAL by A6, A5, RELSET_1:4;

hence y in R_PFuncs (DOMS Y) by Def12; :: thesis: verum