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 39;
A2: dom (f </> ((dom f) --> c)) = (dom f) /\ (dom ((dom f) --> c)) by VALUED_2:71;
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_2:56
.= (f . x) (/) (((dom f) --> c) . x) by
.= (f </> ((dom f) --> c)) . x by ;
:: thesis: verum
end;
hence f [/] c = f </> ((dom f) --> c) by A1, A2; :: thesis: verum