reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i for Integer;
reserve r for Real;
reserve p for Prime;

theorem Th30:
  2|^n,2|^(n mod 12) are_congruent_mod 65
  proof
    defpred P[Nat] means 2|^$1,2|^($1 mod 12) are_congruent_mod 65;
A1: P[0] by INT_1:11;
A2: P[k] implies P[k+1]
    proof
      assume P[k];
      then
A3:   2*2|^k,2*2|^(k mod 12) are_congruent_mod 65 by INT_4:11;
A4:   2|^((k mod 12)+1) = 2*2|^(k mod 12) by NEWTON:6;
      per cases by RADIX_1:4;
      suppose
A5:     (k+1) mod 12 = 0;
        set d = (k+1) div 12;
A6:     k+1 = 12*d + ((k+1) mod 12) by INT_1:59;
        2|^12|^d,1|^d are_congruent_mod 65 by Lm1107,Lm1118,GR_CY_3:34;
        then 2|^(12*d),1 are_congruent_mod 65 by NEWTON:9;
        hence thesis by A5,A6,NEWTON:4;
      end;
      suppose (k+1) mod 12 = (k mod 12) + 1;
        hence thesis by A3,A4,NEWTON:6;
      end;
    end;
    P[k] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
