let x be VECTOR of (RealVS V); :: according to HAHNBAN:def 5 :: thesis: for b₁ being Element of REAL holds (CtoR l) . (b₁ * x) = b₁ * ((CtoR l) . x)
let r be Real; :: thesis: (CtoR l) . (r * x) = r * ((CtoR l) . x)
addLoopStr(# the carrier of V,the addF of V,the U2 of V #) = addLoopStr(# the carrier of (RealVS V),the addF of (RealVS V),the U2 of (RealVS V) #) by Def22;
then reconsider x1 = x as Vector of V ;
[**r,0 **] * x1 = r * x by Def22;
hence (CtoR l) . (r * x) = r * ((CtoR l) . x) by Def18; :: thesis: verum