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 being Subset of V st V1 is Coset of (0). V holds

ex v being Element of V st V1 = {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 V1 being Subset of V st V1 is Coset of (0). V holds

ex v being Element of V st V1 = {v}

let V1 be Subset of V; :: thesis: ( V1 is Coset of (0). V implies ex v being Element of V st V1 = {v} )

assume V1 is Coset of (0). V ; :: thesis: ex v being Element of V st V1 = {v}

then consider v being Element of V such that

A1: V1 = v + ((0). V) by Def6;

take v ; :: thesis: V1 = {v}

thus V1 = {v} by A1, Th46; :: thesis: verum

for V1 being Subset of V st V1 is Coset of (0). V holds

ex v being Element of V st V1 = {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 V1 being Subset of V st V1 is Coset of (0). V holds

ex v being Element of V st V1 = {v}

let V1 be Subset of V; :: thesis: ( V1 is Coset of (0). V implies ex v being Element of V st V1 = {v} )

assume V1 is Coset of (0). V ; :: thesis: ex v being Element of V st V1 = {v}

then consider v being Element of V such that

A1: V1 = v + ((0). V) by Def6;

take v ; :: thesis: V1 = {v}

thus V1 = {v} by A1, Th46; :: thesis: verum