reserve r,t for Real;
reserve i for Integer;
reserve k,n for Nat;
reserve p for Polynomial of F_Real;
reserve e for Element of F_Real;
reserve L for non empty ZeroStr;
reserve z,z0,z1,z2 for Element of L;

theorem Th18:
  z <> 0.L implies
  for p being Polynomial of L st p = 0_.L +* (n,z) holds len p = n+1
  proof
    assume
A1: z <> 0.L;
    let p be Polynomial of L;
    assume
A2: p = 0_.L +* (n,z);
A3: n+1 is_at_least_length_of p
    proof
      let i be Nat such that
A4:   i >= n+1;
      i > n by A4,NAT_1:13;
      hence p.i = (0_.L).i by A2,FUNCT_7:32
      .= 0.L by ORDINAL1:def 12,FUNCOP_1:7;
    end;
    for m being Nat st m is_at_least_length_of p holds n+1 <= m
    proof
      let m be Nat;
      assume
A5:   m is_at_least_length_of p;
      assume
A6:   n+1 > m;
      dom 0_.L = NAT by FUNCOP_1:13;
      then p.n = z by A2,ORDINAL1:def 12,FUNCT_7:31;
      hence contradiction by A1,A5,A6,NAT_1:13;
    end;
    hence thesis by A3,ALGSEQ_1:def 3;
  end;
