let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for v being Element of V
for l being Linear_Combination of {v} holds Sum l = (l . v) * v
let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for v being Element of V
for l being Linear_Combination of {v} holds Sum l = (l . v) * v
let v be Element of V; :: thesis: for l being Linear_Combination of {v} holds Sum l = (l . v) * v
let l be Linear_Combination of {v}; :: thesis: Sum l = (l . v) * v
A1: Carrier l c= {v} by Def4;

now :: thesis: Sum l = (l . v) * v

per cases ( Carrier l = {} or Carrier l = {v} ) by A1, ZFMISC_1:33;

suppose Carrier l = {} ; :: thesis: Sum l = (l . v) * v

then A2: l = ZeroLC V by Def3;
hence Sum l = 0. V by Lm1
.= (0. GF) * v by VECTSP_1:14
.= (l . v) * v by A2, Th3 ;
:: thesis: verum

end;

suppose Carrier l = {v} ; :: thesis: Sum l = (l . v) * v

then consider F being FinSequence of V such that
A3: ( F is one-to-one & rng F = {v} ) and
A4: Sum l = Sum (l (#) F) by Def6;
F = <*v*> by A3, FINSEQ_3:97;
then l (#) F = <*((l . v) * v)*> by Th10;
hence Sum l = (l . v) * v by A4, RLVECT_1:44; :: thesis: verum

end;

end;

end;

hence Sum l = (l . v) * v ; :: thesis: verum