set h = f <-> g;

A1: dom (f <-> g) = (dom f) /\ (dom g) by Th61;

rng (f <-> g) c= C_PFuncs (DOMS Y)

A1: dom (f <-> g) = (dom f) /\ (dom g) by Th61;

rng (f <-> g) c= C_PFuncs (DOMS Y)

proof

hence
f <-> g is PartFunc of (X /\ (dom g)),(C_PFuncs (DOMS Y))
by A1, RELSET_1:4, XBOOLE_1:27; :: thesis: verum
let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng (f <-> g) or y in C_PFuncs (DOMS Y) )

assume y in rng (f <-> g) ; :: thesis: y in C_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;

A4: (f <-> g) . x = (f . x) - (g . x) by A2, Th62;

then reconsider y = y as Function by A3;

A5: rng y c= COMPLEX by A3, A4, XCMPLX_0:def 2;

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;

(f <-> g) . x = (f . x) - (g . x) by A2, Th62;

then dom y = dom (f . x) by A3, VALUED_1:def 2;

then y is PartFunc of (DOMS Y),COMPLEX by A6, A5, RELSET_1:4;

hence y in C_PFuncs (DOMS Y) by Def8; :: thesis: verum

end;assume y in rng (f <-> g) ; :: thesis: y in C_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;

A4: (f <-> g) . x = (f . x) - (g . x) by A2, Th62;

then reconsider y = y as Function by A3;

A5: rng y c= COMPLEX by A3, A4, XCMPLX_0:def 2;

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;

(f <-> g) . x = (f . x) - (g . x) by A2, Th62;

then dom y = dom (f . x) by A3, VALUED_1:def 2;

then y is PartFunc of (DOMS Y),COMPLEX by A6, A5, RELSET_1:4;

hence y in C_PFuncs (DOMS Y) by Def8; :: thesis: verum