reserve i,j for Nat;
reserve A,B for Ring;
reserve K, L for Field;

theorem
  for x be Element of F_Complex st x is algebraic holds FQ_Ring(x) is Field
  proof
    let x be Element of F_Complex;
    assume
A1: x is algebraic;
    for a be Element of FQ_Ring(x) st a <> 0.FQ_Ring(x) holds
    a is left_invertible
    proof
      let a be Element of FQ_Ring(x);
      assume a <> 0.FQ_Ring(x); then
A4:   a <> 0.F_Complex by SUBSET_1:def 8;
      a in FQ(x); then
      reconsider y = a as Element of F_Complex;
      consider b be Element of F_Complex such that
A5:   b in the carrier of FQ_Ring(x) and
A6:   y*b = 1.F_Complex by A1,A4,Th85;
      reconsider a1=y,b1 = b as Element of FQ_Ring(x) by A5;
      b1*a1 = 1.F_Complex by A6,Th50
           .= 1.FQ_Ring(x) by Lm52;
      hence thesis;
    end;
    then FQ_Ring(x) is almost_left_invertible;
    hence FQ_Ring(x) is Field;
  end;
