 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem
  for x be Element of F_Real st x is irrational holds
  for g be non zero Polynomial of F_Rat st Ext_eval(g,x) = 0 holds deg g >= 2
   proof
     let x be Element of F_Real;
     assume
A1:  x is irrational;
     let g be non zero Polynomial of F_Rat;
     assume
A2:  Ext_eval(g,x) = 0;
     assume not deg g >= 2; then
     consider y,z being Element of F_Rat such that
A4:  g = <%y,z%> by FIELD_9:13;
A5:  0 = In(y,F_Real)+In(z,F_Real)*x by ALGNUM_1:22,LIOUVIL2:4,A4,A2;
     per cases;
     suppose
       z = 0; then
       g = 0_.F_Rat by A5,GAUSSINT:def 14,A4,POLYNOM5:42;
       hence contradiction;
     end;
     suppose
       z <> 0;
       hence contradiction by A1,A5;
     end;
   end;
