reserve x,y,y1,y2 for object;
reserve GF for add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr,
  V,X,Y for Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital non
  empty ModuleStr over GF;
reserve a for Element of GF;
reserve u,u1,u2,v,v1,v2 for Element of V;
reserve W,W1,W2 for Subspace of V;
reserve V1 for Subset of V;
reserve w,w1,w2 for Element of W;
reserve B,C for Coset of W;

theorem
  C is linearly-closed iff C = the carrier of W
proof
  thus C is linearly-closed implies C = the carrier of W
  proof
    assume
A1: C is linearly-closed;
    consider v such that
A2: C = v + W by Def6;
    C <> {} by A2,Th44;
    then 0.V in v + W by A1,A2,Th1;
    hence thesis by A2,Th48;
  end;
  thus thesis by Lm2;
end;
