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 Th20:
  Sum OddEvenPowers(r,2*k+3) =
   r|^(2*k+2) - r|^(2*k+1) + Sum OddEvenPowers(r,2*k+1)
  proof
A1: OddEvenPowers(r,2*(k+1)+1) =
    <*r|^(2*k+2),-r|^(2*k+1)*>^OddEvenPowers(r,2*k+1) by Th19;
    Sum <*r|^(2*k+2),-r|^(2*k+1)*> = r|^(2*k+2) + -r|^(2*k+1) by RVSUM_1:77;
    hence thesis by A1,RVSUM_1:75;
  end;
