reserve x,y,y1,y2 for object;
reserve R for Ring;
reserve a for Scalar of R;
reserve V,X,Y for RightMod of R;
reserve u,u1,u2,v,v1,v2 for Vector of V;
reserve V1,V2,V3 for Subset of V;

theorem
  V1 is linearly-closed implies for v,u st v in V1 & u in V1 holds v - u in V1
proof
  assume
A1: V1 is linearly-closed;
  let v,u;
  assume that
A2: v in V1 and
A3: u in V1;
  - u in V1 by A1,A3,Th2;
  hence thesis by A1,A2;
end;
