let X be set ; :: thesis: for c being Complex
for Y being complex-functions-membered set
for f being PartFunc of X,Y holds f [+] c = f <+> ((dom f) --> c)
let c be Complex; :: thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y holds f [+] c = f <+> ((dom f) --> c)
let Y be complex-functions-membered set ; :: thesis: for f being PartFunc of X,Y holds f [+] c = f <+> ((dom f) --> c)
let f be PartFunc of X,Y; :: thesis: f [+] c = f <+> ((dom f) --> c)
set g = (dom f) --> c;
A1: dom (f [+] c) = dom f by VALUED_2:def 37;
A2: dom (f <+> ((dom f) --> c)) = (dom f) /\ (dom ((dom f) --> c)) by VALUED_2:def 41;

now :: thesis: for x being object st x in dom (f [+] c) holds
(f [+] c) . x = (f <+> ((dom f) --> c)) . x

let x be object ; :: thesis: ( x in dom (f [+] c) implies (f [+] c) . x = (f <+> ((dom f) --> c)) . x )
assume A4: x in dom (f [+] c) ; :: thesis: (f [+] c) . x = (f <+> ((dom f) --> c)) . x
hence (f [+] c) . x = (f . x) + c by VALUED_2:def 37
.= (f . x) + (((dom f) --> c) . x) by A1, A4, FUNCOP_1:7
.= (f <+> ((dom f) --> c)) . x by A1, A2, A4, VALUED_2:def 41 ;
:: thesis: verum

end;

hence f [+] c = f <+> ((dom f) --> c) by A1, A2; :: thesis: verum