let R be non empty associative multLoopStr ; :: thesis: for A being non empty Subset of R
for a being Element of R
for F being RightLinearCombination of A holds F * a is RightLinearCombination of A
let A be non empty Subset of R; :: thesis: for a being Element of R
for F being RightLinearCombination of A holds F * a is RightLinearCombination of A
let a be Element of R; :: thesis: for F being RightLinearCombination of A holds F * a is RightLinearCombination of A
let F be RightLinearCombination of A; :: thesis: F * a is RightLinearCombination of A
let i be set ; :: according to IDEAL_1:def 10 :: thesis: ( i in dom (F * a) implies ex u being Element of R ex a being Element of A st (F * a) /. i = a * u )
assume i in dom (F * a) ; :: thesis: ex u being Element of R ex a being Element of A st (F * a) /. i = a * u
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 = b * u by Def10;
take x = u * a; :: thesis: ex a being Element of A st (F * a) /. i = a * x
take b ; :: thesis: (F * a) /. i = b * x
thus (F * a) /. i = (F /. i) * a by A1, POLYNOM1:def 2
.= b * x by A2, GROUP_1:def 3 ; :: thesis: verum