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 Th37:
  for L being add-associative right_zeroed right_complementable
      non empty addLoopStr
  for p being Polynomial of L holds
  p||(len p-'1) + Leading-Monomial p = p
  proof
    let L be add-associative right_zeroed right_complementable
        non empty addLoopStr;
    let p be Polynomial of L;
    set l = Leading-Monomial p;
    set m = len p-'1;
    let n be Element of NAT;
A1: (p||m + l).n = (p||m).n + l.n by NORMSP_1:def 2;
    per cases;
    suppose
A2:   n = m;
      (p||m).m = 0.L by Th32;
      hence thesis by A1,A2,POLYNOM4:def 1;
    end;
    suppose
A3:   n <> m;
      then l.n = 0.L by POLYNOM4:def 1;
      hence thesis by A1,A3,FUNCT_7:32;
    end;
  end;
