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 Th39:
  5 divides 2|^(2*n+1) - 2|^(n+1) + 1 iff n mod 4 = 1 or n mod 4 = 2
  proof
    consider k such that
A1: n = 4*k or n = 4*k+1 or n = 4*k+2 or n = 4*k+3 by NUMBER02:24;
    thus 5 divides an(n) implies n mod 4 = 1 or n mod 4 = 2
    proof
      assume 5 divides an(n);
      then an(n) mod 5 = 0 by INT_1:62;
      hence thesis by A1,Lm6,Lm7,Lm34,Lm37,NAT_D:21;
    end;
A2: an(4*k+1) mod 5 = 0 by Lm35;
A3: an(4*k+2) mod 5 = 0 by Lm36;
    4*k+0 = 4*k;
    hence thesis by A1,A2,A3,Lm8,NAT_D:21,INT_1:62;
  end;
