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 L being Linear_Combination of V holds

( L + (ZeroLC V) = L & (ZeroLC V) + L = L )

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 L being Linear_Combination of V holds

( L + (ZeroLC V) = L & (ZeroLC V) + L = L )

let L be Linear_Combination of V; :: thesis: ( L + (ZeroLC V) = L & (ZeroLC V) + L = L )

thus L + (ZeroLC V) = L :: thesis: (ZeroLC V) + L = L

for L being Linear_Combination of V holds

( L + (ZeroLC V) = L & (ZeroLC V) + L = L )

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 L being Linear_Combination of V holds

( L + (ZeroLC V) = L & (ZeroLC V) + L = L )

let L be Linear_Combination of V; :: thesis: ( L + (ZeroLC V) = L & (ZeroLC V) + L = L )

thus L + (ZeroLC V) = L :: thesis: (ZeroLC V) + L = L

proof

hence
(ZeroLC V) + L = L
; :: thesis: verum
let v be Element of V; :: according to VECTSP_6:def 7 :: thesis: (L + (ZeroLC V)) . v = L . v

thus (L + (ZeroLC V)) . v = (L . v) + ((ZeroLC V) . v) by Th22

.= (L . v) + (0. GF) by Th3

.= L . v by RLVECT_1:4 ; :: thesis: verum

end;thus (L + (ZeroLC V)) . v = (L . v) + ((ZeroLC V) . v) by Th22

.= (L . v) + (0. GF) by Th3

.= L . v by RLVECT_1:4 ; :: thesis: verum