
theorem Th4:
  for K be Field, V be VectSp of K, v be Vector of V, a,b be Scalar
  of V st v <> 0.V & a * v = b * v holds a = b
proof
  let K be Field, V be VectSp of K, v be Vector of V, a,b be Scalar of V such
  that
A1: v <> 0.V and
A2: a * v = b * v;
  a*v - b*v = 0.V by A2,VECTSP_1:19;
  then (a-b)*v = 0.V by VECTSP_4:82;
  then a-b = 0.K by A1,VECTSP_1:15;
  hence thesis by VECTSP_1:19;
end;
