reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
          right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;

theorem Th28:
  R is well-unital implies p `^ 0 = 1_(O,R) & p `^ 1 = p
proof
  set PR=Polynom-Ring (O, R);
  reconsider P=p as Element of PR by POLYNOM1:def 11;
  reconsider E=1_(O,R) as Element of PR by POLYNOM1:def 11;
  assume
A1: R is well-unital;
A2: for H being Element of PR holds H * E = H & E * H = H
  proof
    let H be Element of PR;
    reconsider h=H as Polynomial of O, R by POLYNOM1:def 11;
    h *' 1_(O,R) = h = 1_(O,R) *' h by A1,POLYNOM1:29,30;
    hence thesis by POLYNOM1:def 11;
  end;
A3: PR is unital
  proof
    take E;
    thus thesis by A2;
  end;
  then 1_PR = E by A2,GROUP_1:def 4;
  then (power PR).(P,0) = E by GROUP_1:def 7;
  hence thesis by A3,GROUP_1:50;
end;
