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 W being Subspace of V
for C being Coset of W holds
( C is linearly-closed iff C = the carrier of 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 W being Subspace of V
for C being Coset of W holds
( C is linearly-closed iff C = the carrier of W )
let W be Subspace of V; :: thesis: for C being Coset of W holds
( C is linearly-closed iff C = the carrier of W )
let C be Coset of W; :: thesis: ( C is linearly-closed iff C = the carrier of W )
thus ( C is linearly-closed implies C = the carrier of W ) :: thesis: ( C = the carrier of W implies C is linearly-closed )
proof
assume A1: C is linearly-closed ; :: thesis: C = the carrier of W
consider v being Element of V such that
A2: C = v + W by Def6;
C <> {} by A2, Th44;
then 0. V in v + W by A1, A2, Th1;
hence C = the carrier of W by A2, Th48; :: thesis: verum
end;
thus ( C = the carrier of W implies C is linearly-closed ) by Lm2; :: thesis: verum