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 )

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

thus
( C = the carrier of W implies C is linearly-closed )
by Lm2; :: thesis: verum
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;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