let f1, f2 be complex-valued Function; :: thesis: for r being Complex holds r (#) (f1 + f2) = (r (#) f1) + (r (#) f2)
let r be Complex; :: thesis: r (#) (f1 + f2) = (r (#) f1) + (r (#) f2)
A1: dom (r (#) (f1 + f2)) = dom (f1 + f2) by VALUED_1:def 5
.= (dom f1) /\ (dom f2) by VALUED_1:def 1
.= (dom f1) /\ (dom (r (#) f2)) by VALUED_1:def 5
.= (dom (r (#) f1)) /\ (dom (r (#) f2)) by VALUED_1:def 5
.= dom ((r (#) f1) + (r (#) f2)) by VALUED_1:def 1 ;
now :: thesis: for c being object st c in dom (r (#) (f1 + f2)) holds
(r (#) (f1 + f2)) . c = ((r (#) f1) + (r (#) f2)) . c
let c be object ; :: thesis: ( c in dom (r (#) (f1 + f2)) implies (r (#) (f1 + f2)) . c = ((r (#) f1) + (r (#) f2)) . c )
assume A2: c in dom (r (#) (f1 + f2)) ; :: thesis: (r (#) (f1 + f2)) . c = ((r (#) f1) + (r (#) f2)) . c
then A3: c in dom (f1 + f2) by VALUED_1:def 5;
A4: c in (dom (r (#) f1)) /\ (dom (r (#) f2)) by ;
then A5: c in dom (r (#) f1) by XBOOLE_0:def 4;
A6: c in dom (r (#) f2) by ;
thus (r (#) (f1 + f2)) . c = r * ((f1 + f2) . c) by
.= r * ((f1 . c) + (f2 . c)) by
.= (r * (f1 . c)) + (r * (f2 . c))
.= ((r (#) f1) . c) + (r * (f2 . c)) by
.= ((r (#) f1) . c) + ((r (#) f2) . c) by
.= ((r (#) f1) + (r (#) f2)) . c by ; :: thesis: verum
end;
hence r (#) (f1 + f2) = (r (#) f1) + (r (#) f2) by ; :: thesis: verum