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;
  0 * x in V1 by A1,A2;
  hence thesis by RLVECT_1:10;
end;
