reserve V,X,Y for RealLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a for Real;
reserve V1,V2,V3 for Subset of V;
reserve x for object;
reserve W,W1,W2 for Subspace of V;
reserve w,w1,w2 for VECTOR of W;
reserve D for non empty set;
reserve d1 for Element of D;
reserve A for BinOp of D;
reserve M for Function of [:REAL,D:],D;
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,Th43;
    then 0.V in v + W by A1,A2,Th1;
    hence thesis by A2,Th47;
  end;
  thus thesis by Lm1;
end;
