let V be RealUnitarySpace; :: 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 VECTOR of V such that
A2: C = v + W by Def5;
C <> {} by A2, Th37;
then 0. V in v + W by A1, A2, RLSUB_1:1;
hence C = the carrier of W by A2, Th41; :: thesis: verum

end;

thus ( C = the carrier of W implies C is linearly-closed ) by Lm1; :: thesis: verum