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 (Omega). V holds
V1 = the carrier of 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 (Omega). V holds
V1 = the carrier of V
let V1 be Subset of V; :: thesis: ( V1 is Coset of (Omega). V implies V1 = the carrier of V )
assume V1 is Coset of (Omega). V ; :: thesis: V1 = the carrier of V
then ex v being Element of V st V1 = v + ((Omega). V) by Def6;
hence V1 = the carrier of V by Th47; :: thesis: verum