reserve k,m,n for Nat;
reserve R for commutative Ring,
        p,q for Polynomial of R,
        z0,z1 for Element of R;

theorem Th18:
  for L being non empty ZeroStr
    for p being Polynomial of L st len even_part p <> 0
      holds len even_part p is odd
  proof
    let L be non empty ZeroStr;
    let p be Polynomial of L such that A1:len even_part p <>0;
    set E=even_part p;
    assume len E is even;
    then consider n such that
    A2: 2*n =len E by ABIAN:def 2;
    A3: len E is_at_least_length_of E by ALGSEQ_1:def 3;
    n<>0 by A1,A2;
   then reconsider n1=n-1 as Nat;
   n+n1 is_at_least_length_of E
   proof
     let k such that A4:k >=n+n1;
     assume A5:E.k <>0.L;
     then k is even & n+n1 =2*n-1 by HURWITZ2:def 1;
     then k> n+n1 by A4,XXREAL_0:1;
     then k>= n+n1+1 by NAT_1:13;
     hence thesis by A5,A2,A3,ALGSEQ_1:def 2;
   end;
   then n+n <= n+n1 by ALGSEQ_1:def 3,A2;
   then n1+1 <= n1 by XREAL_1:8;
   hence thesis by NAT_1:13;
 end;
