reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i for Integer;
reserve r for Real;
reserve p for Prime;

theorem Th21:
  r|^(2*n+1) + 1 = (r+1) * Sum OddEvenPowers(r,2*n+1)
  proof
    defpred P[Nat] means r|^(2*$1+1) + 1 = (r+1) * Sum OddEvenPowers(r,2*$1+1);
A1: P[0]
    proof
      Sum OddEvenPowers(r,1) = 1 by Th18;
      hence r|^(2*0+1) + 1 = (r+1) * Sum OddEvenPowers(r,2*0+1);
    end;
A2: P[k] implies P[k+1]
    proof
      assume
A3:   P[k];
A4:   Sum OddEvenPowers(r,2*k+3) =
      r|^(2*k+2) - r|^(2*k+1) + Sum OddEvenPowers(r,2*k+1) by Th20;
A5:   r*r|^(2*k+2) = r|^(2*k+2+1) by NEWTON:6;
      r*r|^(2*k+1) = r|^(2*k+1+1) by NEWTON:6;
      then r|^(2*(k+1)+1) = (r+1)*r|^(2*k+2) - (r+1)*r|^(2*k+1) + r|^(2*k+1)
      by A5;
      hence r|^(2*(k+1)+1) + 1
       = (r+1)*(r|^(2*k+2) - r|^(2*k+1) + Sum OddEvenPowers(r,2*k+1)) by A3
      .= (r+1)*Sum OddEvenPowers(r,2*(k+1)+1) by A4;
    end;
    P[k] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
