theorem
  z <> 0 & z * v = z * u implies v = u
proof
  assume that
A1: z <> 0 and
A2: z * v = z * u;
  0.V = z * v - z * u by A2,RLVECT_1:15
    .= z * (v - u) by Th9;
  then v - u = 0.V by A1,Th2;
  hence thesis by RLVECT_1:21;
end;
