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

theorem Th43:
  (m is odd or m = 2*n+1) implies 2|^m mod 3 = 2
  proof
    defpred P[Nat] means 2|^(2*$1+1) mod 3 = 2;
A1: P[0] by NAT_D:24;
A2: for k st P[k] holds P[k+1]
    proof
      let k such that
A3:   P[k];
A4:   2*(k+1)+1 = 2*k+1+2;
      (2|^(2*k+1) * 4) mod 3 = (2*1) mod 3 by A3,Lm8,Lm10,NAT_D:67
      .= 2 by NAT_D:24;
      hence thesis by A4,Lm2,NEWTON:8;
    end;
A5: for k holds P[k] from NAT_1:sch 2(A1,A2);
    m is odd implies ex k st m = 2*k+1 by ABIAN:9;
    hence thesis by A5;
  end;
