
theorem Th13:
  for K be Field, V be VectSp of K, W be Subspace of V for v be
  Vector of V, w be Vector of W st v = w holds Lin{w} = Lin{v}
proof
  let K be Field, V be VectSp of K, W be Subspace of V, v be Vector of V, w be
  Vector of W such that
A1: v = w;
  reconsider W1 = Lin{w} as Subspace of V by VECTSP_4:26;
  now
    let u be Vector of V;
    hereby
      assume u in W1;
      then consider a be Element of K such that
A2:   u = a * w by Th3;
      u = a * v by A1,A2,VECTSP_4:14;
      hence u in Lin{v} by Th3;
    end;
    assume u in Lin{v};
    then consider a be Element of K such that
A3: u = a * v by Th3;
    u = a * w by A1,A3,VECTSP_4:14;
    hence u in W1 by Th3;
  end;
  hence thesis by VECTSP_4:30;
end;
