let R be Ring; :: thesis: for V being RightMod of R
for W being Submodule of V
for C being Coset of W holds
( C is linearly-closed iff C = the carrier of W )
let V be RightMod of R; :: thesis: for W being Submodule of V
for C being Coset of W holds
( C is linearly-closed iff C = the carrier of W )
let W be Submodule 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 Vector 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 Lm1; :: thesis: verum