reserve c for Complex;
reserve r for Real;
reserve m,n for Nat;
reserve f for complex-valued Function;
reserve f,g for differentiable Function of REAL,REAL;
reserve L for non empty ZeroStr;
reserve x for Element of L;

theorem
  for L being non empty ZeroStr, p being Polynomial of L holds
  n > len p-'1 implies p||n = p
  proof
    let L be non empty ZeroStr, p be Polynomial of L such that
A1: n > len p-'1;
    let m be Element of NAT;
    per cases;
    suppose p = 0_.L;
      hence thesis;
    end;
    suppose that
A2:   m = n and
A3:   p <> 0_.L;
      0 <> len p by A3,POLYNOM4:5;
      then 0-1 < len p-1 by XREAL_1:14;
      then -1+1 <= len p-1 by INT_1:7;
      then len p-'1+1 = len p by NAT_D:72;
      then n >= len p by A1,NAT_1:13;
      hence p.m = 0.L by A2,ALGSEQ_1:8
      .= (p||n).m by A2,Th32;
    end;
    suppose m <> n;
      hence (p||n).m = p.m by FUNCT_7:32;
    end;
  end;
