reserve i,j,n,n1,n2,m,k,l,u for Nat,
        i1,i2,i3,i4,i5,i6 for Element of n,
        p,q for n-element XFinSequence of NAT,
        a,b,c,d,e,f for Integer;

theorem
  for n being Ordinal,p being Polynomial of n, F_Real holds
    degree p = degree |. p .|
proof
  let n be Ordinal,p be Polynomial of n, F_Real;
  per cases;
  suppose p = 0_(n,F_Real);
    hence thesis;
  end;
  suppose
A1:   p <> 0_(n,F_Real);
    then consider s be bag of n such that
A2:   s in Support p &
    degree p = degree s by Def3;
A3:   |.p.| <> 0_(n,F_Real) by A1,Th7;
    then consider sM be bag of n such that
A4:   sM in Support |.p.| &
    degree |.p.| = degree sM by Def3;
    Support |.p.| = Support p by Th3;
    then degree p <= degree |.p.|<= degree p by A2,A1,Def3,A4,A3;
    hence thesis by XXREAL_0:1;
  end;
end;
