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 strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for W being strict Subspace of V st the carrier of W = the carrier of V holds

W = V

let V be non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for W being strict Subspace of V st the carrier of W = the carrier of V holds

W = V

let W be strict Subspace of V; :: thesis: ( the carrier of W = the carrier of V implies W = V )

assume A1: the carrier of W = the carrier of V ; :: thesis: W = V

V is Subspace of V by Th24;

hence W = V by A1, Th29; :: thesis: verum

for W being strict Subspace of V st the carrier of W = the carrier of V holds

W = V

let V be non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for W being strict Subspace of V st the carrier of W = the carrier of V holds

W = V

let W be strict Subspace of V; :: thesis: ( the carrier of W = the carrier of V implies W = V )

assume A1: the carrier of W = the carrier of V ; :: thesis: W = V

V is Subspace of V by Th24;

hence W = V by A1, Th29; :: thesis: verum