
theorem Th11:
  for a,b,c being Nat holds a|^(2*b) mod c = (a|^b mod c)*(a|^b mod c) mod c
proof
  let a,b,c be Nat;
  reconsider a,b,c as Element of NAT by ORDINAL1:def 12;
  a|^(2*b) mod c = a|^(b+b) mod c .=(a|^b*a|^b) mod c by NEWTON:8
    .=(a|^b mod c) * (a|^b mod c) mod c by NAT_D:67;
  hence thesis;
end;
