
theorem Th29:
  for V being RealUnitarySpace, v being VECTOR of V, x being set
  holds x in Lin{v} iff ex a being Real st x = a * v
proof
  deffunc F(set) = In(0,REAL);
  let V be RealUnitarySpace;
  let v be VECTOR of V;
  let x be set;
  thus x in Lin{v} implies ex a being Real st x = a * v
  proof
    assume x in Lin{v};
    then consider l being Linear_Combination of {v} such that
A1: x = Sum(l) by Th1;
    Sum(l) = l.v * v by RLVECT_2:32;
    hence thesis by A1;
  end;
  given a be Real such that
A2: x = a * v;
  reconsider a as Element of REAL by XREAL_0:def 1;
  consider f being Function of the carrier of V, REAL such that
A3: f.v = a and
A4: for z being 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,REAL) 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 by A4;
  end;
  then reconsider f as Linear_Combination of V by RLVECT_2:def 3;
  Carrier f c= {v}
  proof
    let x be object;
    assume
A5: x in Carrier f;
    then f.x <> 0 by RLVECT_2:19;
    then x = v by A4,A5;
    hence thesis by TARSKI:def 1;
  end;
  then reconsider f as Linear_Combination of {v} by RLVECT_2:def 6;
  Sum(f) = x by A2,A3,RLVECT_2:32;
  hence thesis by Th1;
end;
