let f be complex-valued Function; :: thesis: for r being Complex st r <> 0 holds
(r (#) f) ^ = (r ") (#) (f ^)
let r be Complex; :: thesis: ( r <> 0 implies (r (#) f) ^ = (r ") (#) (f ^) )
assume A1: r <> 0 ; :: thesis: (r (#) f) ^ = (r ") (#) (f ^)
A2: dom ((r (#) f) ^) = (dom (r (#) f)) \ ((r (#) f) " {0}) by Def2
.= (dom (r (#) f)) \ (f " {0}) by A1, Th7
.= (dom f) \ (f " {0}) by VALUED_1:def 5
.= dom (f ^) by Def2
.= dom ((r ") (#) (f ^)) by VALUED_1:def 5 ;

now :: thesis: for c being object st c in dom ((r (#) f) ^) holds
((r (#) f) ^) . c = ((r ") (#) (f ^)) . c

let c be object ; :: thesis: ( c in dom ((r (#) f) ^) implies ((r (#) f) ^) . c = ((r ") (#) (f ^)) . c )
assume A3: c in dom ((r (#) f) ^) ; :: thesis: ((r (#) f) ^) . c = ((r ") (#) (f ^)) . c
then A4: c in (dom (r (#) f)) \ ((r (#) f) " {0}) by Def2;
A5: c in dom (f ^) by A2, A3, VALUED_1:def 5;
thus ((r (#) f) ^) . c = ((r (#) f) . c) " by A3, Def2
.= (r * (f . c)) " by A4, VALUED_1:def 5
.= (r ") * ((f . c) ") by XCMPLX_1:204
.= (r ") * ((f ^) . c) by A5, Def2
.= ((r ") (#) (f ^)) . c by A2, A3, VALUED_1:def 5 ; :: thesis: verum

end;

hence (r (#) f) ^ = (r ") (#) (f ^) by A2, FUNCT_1:2; :: thesis: verum