let V be ComplexLinearSpace; :: thesis: for V1 being Subset of V st V1 is Coset of (0). V holds
ex v being VECTOR of V st V1 = {v}
let V1 be Subset of V; :: thesis: ( V1 is Coset of (0). V implies ex v being VECTOR of V st V1 = {v} )
assume V1 is Coset of (0). V ; :: thesis: ex v being VECTOR of V st V1 = {v}
then consider v being VECTOR of V such that
A1: V1 = v + ((0). V) by Def12;
take v ; :: thesis: V1 = {v}
thus V1 = {v} by A1, Th64; :: thesis: verum