rng (SMoebius X) c= REAL

proof

let y be object ; :: according to TARSKI:def 3 :: thesis:
assume y in rng (SMoebius X) ; :: thesis:
then consider i being object such that
A1: i in dom (SMoebius X) and
A2: (SMoebius X) . i = y by FUNCT_1:def 3;
dom (SMoebius X) = NAT by PARTFUN1:def 2;
then reconsider i = i as Nat by A1;

per cases ;

suppose A3: i in support (SMoebius X) ; :: thesis:

then i in X /\ SCNAT by Def5;
then y = Moebius i by A2, A3, Def5;
hence y in REAL by XREAL_0:def 1; :: thesis:

end;

suppose not i in support (SMoebius X) ; :: thesis:

then (SMoebius X) . i = In (0,REAL) by PRE_POLY:def 7;
hence y in REAL by A2; :: thesis:

end;

end;

end;

hence SMoebius X is real-valued by VALUED_0:def 3; :: thesis: