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 v being Element of V
for W being Subspace of V holds
( 0. V in v + W iff 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 v being Element of V
for W being Subspace of V holds
( 0. V in v + W iff v in W )

let v be Element of V; :: thesis: for W being Subspace of V holds
( 0. V in v + W iff v in W )

let W be Subspace of V; :: thesis: ( 0. V in v + W iff v in W )
thus ( 0. V in v + W implies v in W ) :: thesis: ( v in W implies 0. V in v + W )
proof
assume 0. V in v + W ; :: thesis: v in W
then consider u being Element of V such that
A1: 0. V = v + u and
A2: u in W ;
v = - u by ;
hence v in W by ; :: thesis: verum
end;
assume v in W ; :: thesis: 0. V in v + W
then A3: - v in W by Th22;
0. V = v + (- v) by VECTSP_1:19;
hence 0. V in v + W by A3; :: thesis: verum