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 u, v being Element of V

for W being Subspace of V st u in W & v in W holds

u + v in 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 u, v being Element of V

for W being Subspace of V st u in W & v in W holds

u + v in W

let u, v be Element of V; :: thesis: for W being Subspace of V st u in W & v in W holds

u + v in W

let W be Subspace of V; :: thesis: ( u in W & v in W implies u + v in W )

reconsider VW = the carrier of W as Subset of V by Def2;

assume ( u in W & v in W ) ; :: thesis: u + v in W

then A1: ( u in the carrier of W & v in the carrier of W ) ;

VW is linearly-closed by Lm2;

then u + v in the carrier of W by A1;

hence u + v in W ; :: thesis: verum

for u, v being Element of V

for W being Subspace of V st u in W & v in W holds

u + v in 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 u, v being Element of V

for W being Subspace of V st u in W & v in W holds

u + v in W

let u, v be Element of V; :: thesis: for W being Subspace of V st u in W & v in W holds

u + v in W

let W be Subspace of V; :: thesis: ( u in W & v in W implies u + v in W )

reconsider VW = the carrier of W as Subset of V by Def2;

assume ( u in W & v in W ) ; :: thesis: u + v in W

then A1: ( u in the carrier of W & v in the carrier of W ) ;

VW is linearly-closed by Lm2;

then u + v in the carrier of W by A1;

hence u + v in W ; :: thesis: verum