let X be set ; :: thesis: for f being complex-valued Function
for r being complex number holds (r (#) f) | X = r (#) (f | X)
let f be complex-valued Function; :: thesis: for r being complex number holds (r (#) f) | X = r (#) (f | X)
let r be complex number ; :: thesis: (r (#) f) | X = r (#) (f | X)
A1: dom ((r (#) f) | X) = (dom (r (#) f)) /\ X by RELAT_1:90
.= (dom f) /\ X by VALUED_1:def 5
.= dom (f | X) by RELAT_1:90
.= dom (r (#) (f | X)) by VALUED_1:def 5 ;

now

let c be set ; :: thesis: ( c in dom ((r (#) f) | X) implies ((r (#) f) | X) . c = (r (#) (f | X)) . c )
assume A2: c in dom ((r (#) f) | X) ; :: thesis: ((r (#) f) | X) . c = (r (#) (f | X)) . c
then c in (dom (r (#) f)) /\ X by RELAT_1:90;
then A3: ( c in dom (r (#) f) & c in X ) by XBOOLE_0:def 4;
then c in dom f by VALUED_1:def 5;
then c in (dom f) /\ X by A3, XBOOLE_0:def 4;
then A4: c in dom (f | X) by RELAT_1:90;
then A5: c in dom (r (#) (f | X)) by VALUED_1:def 5;
thus ((r (#) f) | X) . c = (r (#) f) . c by A2, FUNCT_1:70
.= r * (f . c) by A3, VALUED_1:def 5
.= r * ((f | X) . c) by A4, FUNCT_1:70
.= (r (#) (f | X)) . c by A5, VALUED_1:def 5 ; :: thesis: verum

end;

hence (r (#) f) | X = r (#) (f | X) by A1, FUNCT_1:9; :: thesis: verum