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 ( for v being Element of V holds v in W ) 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 ( for v being Element of V holds v in W ) holds

W = V

let W be strict Subspace of V; :: thesis: ( ( for v being Element of V holds v in W ) implies W = V )

assume for v being Element of V holds v in W ; :: thesis: W = V

then A1: for v being Element of V holds

( v in W iff v in V ) ;

V is Subspace of V by Th24;

hence W = V by A1, Th30; :: thesis: verum

for W being strict Subspace of V st ( for v being Element of V holds v in W ) 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 ( for v being Element of V holds v in W ) holds

W = V

let W be strict Subspace of V; :: thesis: ( ( for v being Element of V holds v in W ) implies W = V )

assume for v being Element of V holds v in W ; :: thesis: W = V

then A1: for v being Element of V holds

( v in W iff v in V ) ;

V is Subspace of V by Th24;

hence W = V by A1, Th30; :: thesis: verum