let R be non empty multLoopStr ; :: thesis: for A being non empty Subset of R
for a being Element of R
for F being LeftLinearCombination of A holds F * a is LinearCombination of A
let A be non empty Subset of R; :: thesis: for a being Element of R
for F being LeftLinearCombination of A holds F * a is LinearCombination of A
let a be Element of R; :: thesis: for F being LeftLinearCombination of A holds F * a is LinearCombination of A
let F be LeftLinearCombination of A; :: thesis: F * a is LinearCombination of A
let i be set ; :: according to IDEAL_1:def 8 :: thesis: ( i in dom (F * a) implies ex u, v being Element of R ex a being Element of A st (F * a) /. i = (u * a) * v )
reconsider c = a as Element of R ;
assume i in dom (F * a) ; :: thesis: ex u, v being Element of R ex a being Element of A st (F * a) /. i = (u * a) * v
then A1: i in dom F by POLYNOM1:def 2;
then consider u being Element of R, b being Element of A such that
A2: F /. i = u * b by Def9;
take u ; :: thesis: ex v being Element of R ex a being Element of A st (F * a) /. i = (u * a) * v
take c ; :: thesis: ex a being Element of A st (F * a) /. i = (u * a) * c
take b ; :: thesis: (F * a) /. i = (u * b) * c
thus (F * a) /. i = (u * b) * c by A1, A2, POLYNOM1:def 2; :: thesis: verum