reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;

theorem Th15:
  divisors(2|^n,4,3) = {} & divisors(2|^n,4,1) = {1}
  proof
    thus divisors(2|^n,4,3)={}
    proof
      assume divisors(2|^n,4,3) <> {};
      then consider k be object such that
A1:   k in divisors(2|^n,4,3) by XBOOLE_0:def 1;
      reconsider k as Nat by A1;
A2:   k mod 4 = 3 & k divides 2|^n by A1,Th12;
      k <> 1 by A2,PEPIN:5;
      hence thesis by A2,Lm4;
    end;
    1 mod 4 = 1 & 1 divides 2|^n by NAT_D:6,PEPIN:5;
    then
A3: 1 in divisors(2|^n,4,1);
    divisors(2|^n,4,1) c= {1}
    proof
      let k be object;
      assume
A4:   k in divisors(2|^n,4,1) & not k in {1};
      then reconsider k as Nat;
A5:   k mod 4 = 1 & k divides 2|^n by A4,Th12;
      k<>1 by A4,TARSKI:def 1;
      hence thesis by A5,Lm4;
    end;
    hence thesis by A3,ZFMISC_1:33;
  end;
