
theorem Th3:
  for K be Field, V be VectSp of K for x be object, v be Vector of V
  holds x in Lin{v} iff ex a be Element of K st x =a*v
proof
  let K be Field, V be VectSp of K, x be object, v be Vector of V;
  thus x in Lin{v} implies ex a be Element of K st x = a * v
  proof
    assume x in Lin{v};
    then consider l be Linear_Combination of {v} such that
A1: x = Sum(l) by VECTSP_7:7;
    Sum(l) = l.v * v by VECTSP_6:17;
    hence thesis by A1;
  end;
  given a be Element of K such that
A2: x = a * v;
  deffunc F(set) = 0.K;
  consider f be Function of the carrier of V, the carrier of K such that
A3: f.v = a and
A4: for z be Vector of V st z <> v holds f.z = F(z) from FUNCT_2:sch 6;
  reconsider f as Element of Funcs(the carrier of V,the carrier of K) by
FUNCT_2:8;
  now
    let z be Vector of V;
    assume not z in {v};
    then z <> v by TARSKI:def 1;
    hence f.z = 0.K by A4;
  end;
  then reconsider f as Linear_Combination of V by VECTSP_6:def 1;
  Carrier f c= {v}
  proof
    let x be object;
    assume
A5: x in Carrier f;
    then f.x <> 0.K by VECTSP_6:2;
    then x = v by A4,A5;
    hence thesis by TARSKI:def 1;
  end;
  then reconsider f as Linear_Combination of {v} by VECTSP_6:def 4;
  Sum(f) = x by A2,A3,VECTSP_6:17;
  hence thesis by VECTSP_7:7;
end;
