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 W being Subspace of V

for C being Coset of W holds

( 0. V in C iff C = the carrier of W )

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 W being Subspace of V

for C being Coset of W holds

( 0. V in C iff C = the carrier of W )

let W be Subspace of V; :: thesis: for C being Coset of W holds

( 0. V in C iff C = the carrier of W )

let C be Coset of W; :: thesis: ( 0. V in C iff C = the carrier of W )

ex v being Element of V st C = v + W by Def6;

hence ( 0. V in C iff C = the carrier of W ) by Th48; :: thesis: verum

for W being Subspace of V

for C being Coset of W holds

( 0. V in C iff C = the carrier of W )

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 W being Subspace of V

for C being Coset of W holds

( 0. V in C iff C = the carrier of W )

let W be Subspace of V; :: thesis: for C being Coset of W holds

( 0. V in C iff C = the carrier of W )

let C be Coset of W; :: thesis: ( 0. V in C iff C = the carrier of W )

ex v being Element of V st C = v + W by Def6;

hence ( 0. V in C iff C = the carrier of W ) by Th48; :: thesis: verum