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 v1, v2 being Element of V
for L being Linear_Combination of V st Carrier L = {v1,v2} & v1 <> v2 holds
Sum L = ((L . v1) * v1) + ((L . v2) * v2)
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 v1, v2 being Element of V
for L being Linear_Combination of V st Carrier L = {v1,v2} & v1 <> v2 holds
Sum L = ((L . v1) * v1) + ((L . v2) * v2)
let v1, v2 be Element of V; :: thesis: for L being Linear_Combination of V st Carrier L = {v1,v2} & v1 <> v2 holds
Sum L = ((L . v1) * v1) + ((L . v2) * v2)
let L be Linear_Combination of V; :: thesis: ( Carrier L = {v1,v2} & v1 <> v2 implies Sum L = ((L . v1) * v1) + ((L . v2) * v2) )
assume that
A1: Carrier L = {v1,v2} and
A2: v1 <> v2 ; :: thesis: Sum L = ((L . v1) * v1) + ((L . v2) * v2)
L is Linear_Combination of {v1,v2} by A1, Def4;
hence Sum L = ((L . v1) * v1) + ((L . v2) * v2) by A2, Th18; :: thesis: verum