reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve c for Complex;

theorem Th20:
  11 divides 2|^341-2
  proof
A1: (2|^10)|^34 = 2|^(10*34) by NEWTON:9;
A2: 1 mod 11 = 1 by NAT_D:24;
    2|^10 = 2*2*2*2*2*2*2*2*2*2 by Th6
    .= 11*93+1;
    then 2|^10 mod 11 = 1 by NAT_D:def 2;
    then (2|^10)|^34 mod 11 = 1 by NEWTON05:15;
    then
A3: 2|^340,1 are_congruent_mod 11 by A1,A2,NAT_D:64;
    2|^(340+1)-2 = (2|^340*2|^1)-(2*1) by NEWTON:8
    .= 2*(2|^340-1);
    hence thesis by A3,INT_2:2;
  end;
