let x be Vector of V; :: according to HAHNBAN1:def 13 :: thesis: for r being Real holds (RtoC l) . ([**r,0**] * x) = r * ((RtoC l) . x)
let r be Real; :: thesis: (RtoC l) . ([**r,0**] * x) = r * ((RtoC l) . x)
addLoopStr(# the carrier of V, the addF of V, the ZeroF of V #) = addLoopStr(# the carrier of (RealVS V), the addF of (RealVS V), the ZeroF of (RealVS V) #) by Def17;
then reconsider x1 = x as VECTOR of (RealVS V) ;
[**r,0**] * x = r * x1 by Def17;
hence (RtoC l) . ([**r,0**] * x) = r * ((RtoC l) . x) by HAHNBAN:def 3; :: thesis: verum