let C be non empty set ; :: thesis: for f being PartFunc of C,COMPLEX
for r being Complex st r <> 0 holds
(r (#) f) ^ = (r ") (#) (f ^)
let f be PartFunc of C,COMPLEX; :: 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) " {0c}) by Def2
.= (dom (r (#) f)) \ (f " {0c}) by A1, Th12
.= (dom f) \ (f " {0c}) by Th4
.= dom (f ^) by Def2
.= dom ((r ") (#) (f ^)) by Th4 ;

now :: thesis: for c being Element of C st c in dom ((r (#) f) ^) holds
((r (#) f) ^) /. c = ((r ") (#) (f ^)) /. c

let c be Element of C; :: 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 (f ^) by A2, Th4;
c in (dom (r (#) f)) \ ((r (#) f) " {0c}) by A3, Def2;
then A5: c in dom (r (#) f) by XBOOLE_0:def 5;
thus ((r (#) f) ^) /. c = ((r (#) f) /. c) " by A3, Def2
.= (r * (f /. c)) " by A5, Th4
.= (r ") * ((f /. c) ") by XCMPLX_1:204
.= (r ") * ((f ^) /. c) by A4, Def2
.= ((r ") (#) (f ^)) /. c by A2, A3, Th4 ; :: thesis: verum

end;

hence (r (#) f) ^ = (r ") (#) (f ^) by A2, PARTFUN2:1; :: thesis: verum