reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve p for Prime;

theorem Th42:
  (m is even or m = 2*n) implies 2|^m mod 3 = 1
  proof
    defpred P[Nat] means 2|^(2*$1) mod 3 = 1;
A1: P[0] by Lm8,NEWTON:4;
A2: for k st P[k] holds P[k+1]
    proof
      let k such that
A3:   P[k];
A4:   2*(k+1) = 2*k+2;
      (2|^(2*k) * 4) mod 3 = (1 * 1) mod 3 by A3,Lm10,NAT_D:67
      .= 1 by NAT_D:24;
      hence thesis by A4,Lm2,NEWTON:8;
    end;
    for k holds P[k] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
