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

theorem Th85:
  for x,a be Element of F_Complex st x is algebraic & a <> 0.F_Complex &
  a in the carrier of FQ_Ring(x) holds
  ex b be Element of F_Complex st b in the carrier of FQ_Ring(x)
  & a*b = 1.F_Complex
  proof
    let x,a be Element of F_Complex;
    set COPolynomFRat = the carrier of Polynom-Ring F_Rat;
    set M = {h where h is Polynomial of F_Rat:Ext_eval(h,x)=0.F_Complex};
    assume that
A1:  x is algebraic and
A2:  a <> 0.F_Complex and
A3:  a in the carrier of FQ_Ring(x);
     consider f,g be Element of Polynom-Ring F_Rat such that
A4:  {f}-Ideal = Ann_Poly(x,F_Rat) and
     not(g in Ann_Poly(x,F_Rat)) and
A6:  a = hom_Ext_eval(x,F_Rat).g and
A7:  {f}-Ideal,{g}-Ideal are_co-prime by A1,A2,A3,Th84;
     1.Polynom-Ring F_Rat in {f}-Ideal+{g}-Ideal by A7; then
     1.Polynom-Ring F_Rat in {p+q where p,q is Element of Polynom-Ring F_Rat:
     p in {f}-Ideal & q in {g}-Ideal} by IDEAL_1:def 19; then
     consider p,q be Element of Polynom-Ring F_Rat such that
A10:  1.Polynom-Ring F_Rat = p+q and
A11:  p in {f}-Ideal and
A12:  q in {g}-Ideal;
A14: {g}-Ideal = the set of all g*s where s is Element of Polynom-Ring F_Rat
      by IDEAL_1:64;
      consider s be Element of Polynom-Ring F_Rat such that
A15:  q = g * s by A12,A14;
      reconsider p1=p,q1=q, g1=g,s1=s as Polynomial of F_Rat
        by POLYNOM3:def 10;
A16:  p+q = p1+q1 by POLYNOM3:def 10;
      consider p2 be Polynomial of F_Rat such that
A17:  p2 = p and
A18:  Ext_eval(p2,x)=0.F_Complex by A4,A11;
      set b = Ext_eval(s1,x);
A20:   b = hom_Ext_eval(x,F_Rat).s1 by Def9;
A21: dom hom_Ext_eval(x,F_Rat) = the carrier of Polynom-Ring F_Rat
     by FUNCT_2:def 1;
A22: b in the carrier of FQ_Ring(x) by A20,A21,FUNCT_1:def 3;
     1.F_Complex = Ext_eval(1_.(F_Rat),x) by Th3,Th18
     .= Ext_eval(p1+q1,x) by A10,POLYNOM3:def 10,A16
     .= 0.F_Complex + Ext_eval(q1,x) by A17,A18,Th3,Th19
     .= Ext_eval(g1 *'s1,x) by A15,POLYNOM3:def 10
     .= Ext_eval(g1,x) * Ext_eval(s1,x) by Th3,Th24
     .= a*b by A6,Def9;
     hence thesis by A22;
end;
