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 Th40:
  5 divides 2|^(2*n+1) + 2|^(n+1) + 1 iff n mod 4 = 0 or n mod 4 = 3
  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;
A2: 4*k+0 = 4*k;
    thus 5 divides bn(n) implies n mod 4 = 0 or n mod 4 = 3
    proof
      assume 5 divides bn(n);
      then bn(n) mod 5 = 0 by INT_1:62;
      hence thesis by A1,A2,Lm8,Lm35,Lm36,NAT_D:21;
    end;
A3: bn(4*k) mod 5 = 0 by Lm34;
    bn(4*k+3) mod 5 = 0 by Lm37;
    hence thesis by A1,A3,Lm6,Lm7,NAT_D:21,INT_1:62;
  end;
