reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;

theorem
  2|^n mod 3 = 1 or 2|^n mod 3 = 2
  proof
    defpred P[Nat] means 2|^$1 mod 3 = 1 or 2|^$1 mod 3 = 2;
A1: P[0] by Lm1,NEWTON:4;
A2: for k st P[k] holds P[k+1]
    proof
      let k;
      (2|^k*2) mod 3 = ((2|^k mod 3) * (2 mod 3)) mod 3 by NAT_D:67;
      hence thesis by Lm1,Lm2,Lm3,NEWTON:6;
    end;
    for k holds P[k] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
