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 Th1:
  V1 <> {} & V1 is linearly-closed implies 0.V in V1
proof
  assume that
A1: V1 <> {} and
A2: V1 is linearly-closed;
  set x = the Element of V1;
  reconsider x as Element of V by A1,TARSKI:def 3;
  x * 0.R in V1 by A1,A2;
  hence thesis by VECTSP_2:32;
end;
