set h = abs f;
A1:
dom (abs f) = dom f
by Def36;
rng (abs f) c= C_PFuncs (DOMS Y)
proof
let y be
object ;
TARSKI:def 3 ( not y in rng (abs f) or y in C_PFuncs (DOMS Y) )
assume
y in rng (abs f)
;
y in C_PFuncs (DOMS Y)
then consider x being
object such that A2:
x in dom (abs f)
and A3:
(abs f) . x = y
by FUNCT_1:def 3;
A4:
(abs f) . x = abs (f . x)
by A2, Def36;
then reconsider y =
y as
Function by A3;
A5:
rng y c= COMPLEX
by A3, A4, XCMPLX_0:def 2;
f . x in Y
by A1, A2, 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 11;
then
y is
PartFunc of
(DOMS Y),
COMPLEX
by A6, A5, RELSET_1:4;
hence
y in C_PFuncs (DOMS Y)
by Def8;
verum
end;
hence
abs f is PartFunc of X,(C_PFuncs (DOMS Y))
by A1, RELSET_1:4; verum