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 a being Element of GF

for v being Element of V

for W being Subspace of V st v in W holds

a * 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 a being Element of GF

for v being Element of V

for W being Subspace of V st v in W holds

a * v in W

let a be Element of GF; :: thesis: for v being Element of V

for W being Subspace of V st v in W holds

a * v in W

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

a * v in W

let W be Subspace of V; :: thesis: ( v in W implies a * v in W )

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

assume v in W ; :: thesis: a * v in W

then A1: v in the carrier of W ;

VW is linearly-closed by Lm2;

then a * v in the carrier of W by A1;

hence a * v in W ; :: thesis: verum

for a being Element of GF

for v being Element of V

for W being Subspace of V st v in W holds

a * 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 a being Element of GF

for v being Element of V

for W being Subspace of V st v in W holds

a * v in W

let a be Element of GF; :: thesis: for v being Element of V

for W being Subspace of V st v in W holds

a * v in W

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

a * v in W

let W be Subspace of V; :: thesis: ( v in W implies a * v in W )

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

assume v in W ; :: thesis: a * v in W

then A1: v in the carrier of W ;

VW is linearly-closed by Lm2;

then a * v in the carrier of W by A1;

hence a * v in W ; :: thesis: verum