theorem Th81:
  for n be positive Nat holds k*(a|^n+1) mod a = k mod a
  proof
    let n be positive Nat;
    consider k such that
    A1: n = 1 + k by NAT_1:10,14;
    a|^(1+k) = a*a|^k by NEWTON:6;
    hence thesis by A1,Th80;
  end;
