let V be RealLinearSpace; :: thesis: for v being VECTOR of V

for L being Linear_Combination of V st Carrier L = {v} holds

Sum L = (L . v) * v

let v be VECTOR of V; :: thesis: for L being Linear_Combination of V st Carrier L = {v} holds

Sum L = (L . v) * v

let L be Linear_Combination of V; :: thesis: ( Carrier L = {v} implies Sum L = (L . v) * v )

assume Carrier L = {v} ; :: thesis: Sum L = (L . v) * v

then L is Linear_Combination of {v} by Def6;

hence Sum L = (L . v) * v by Th32; :: thesis: verum

for L being Linear_Combination of V st Carrier L = {v} holds

Sum L = (L . v) * v

let v be VECTOR of V; :: thesis: for L being Linear_Combination of V st Carrier L = {v} holds

Sum L = (L . v) * v

let L be Linear_Combination of V; :: thesis: ( Carrier L = {v} implies Sum L = (L . v) * v )

assume Carrier L = {v} ; :: thesis: Sum L = (L . v) * v

then L is Linear_Combination of {v} by Def6;

hence Sum L = (L . v) * v by Th32; :: thesis: verum