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