
theorem Th2:
  for V being RealUnitarySpace, A being Subset of V, x being set st
  x in A holds x in Lin(A)
proof
  deffunc F(set) = zz;
  let V be RealUnitarySpace;
  let A be Subset of V;
  let x be set;
  assume
A1: x in A;
  then reconsider v = x as VECTOR of V;
  consider f being Function of the carrier of V, REAL such that
A2: f.v = jj and
A3: for u being VECTOR of V st u <> v holds f.u = F(u) from FUNCT_2:sch 6;
  reconsider f as Element of Funcs(the carrier of V, REAL) by FUNCT_2:8;
  ex T being finite Subset of V st for u being VECTOR of V st not u in T
  holds f.u = 0
  proof
    take T = {v};
    let u be VECTOR of V;
    assume not u in T;
    then u <> v by TARSKI:def 1;
    hence thesis by A3;
  end;
  then reconsider f as Linear_Combination of V by RLVECT_2:def 3;
A4: Carrier(f) c= {v}
  proof
    let x be object;
    assume x in Carrier(f);
    then consider u being VECTOR of V such that
A5: x = u and
A6: f.u <> 0;
    u = v by A3,A6;
    hence thesis by A5,TARSKI:def 1;
  end;
  then reconsider f as Linear_Combination of {v} by RLVECT_2:def 6;
A7: Sum(f) = 1 * v by A2,RLVECT_2:32
    .= v by RLVECT_1:def 8;
  {v} c= A by A1,ZFMISC_1:31;
  then Carrier(f) c= A by A4;
  then reconsider f as Linear_Combination of A by RLVECT_2:def 6;
  Sum(f) = v by A7;
  hence thesis by Th1;
end;
