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 Th17:
  for p being Prime, n being Nat st p mod 4 = 1 holds
    card divisors(p|^n,4,1) = n+1 &
    card divisors(p|^n,4,3) = 0
  proof
    let p be Prime,n such that
A1: p mod 4 = 1;
    set X={ k where k is Nat: k mod 4 = 1 & k divides p|^n };
    set Y = { k where k is Nat: k divides p|^n };
A2: divisors(p|^n,4,1) c= Y
    proof
      let x be object;
      assume x in divisors(p|^n,4,1);
      then ex k be Nat st x= k & k mod 4 = 1 & k divides p|^n;
      hence thesis;
    end;
    Y c= divisors(p|^n,4,1)
    proof
      let x be object;
      assume x in Y;
      then consider k be Nat such that
A3:   x= k & k divides p|^n;
      k mod 4 =1 by Th10,A1,A3;
      hence thesis by A3;
    end;
    then Y = divisors(p|^n,4,1) by A2,XBOOLE_0:def 10;
    hence card divisors(p|^n,4,1) = n+1 by Th16;
    assume card divisors(p|^n,4,3) <> 0;
    then divisors(p|^n,4,3)<>{};
    then consider x be object such that
A4: x in divisors(p|^n,4,3) by XBOOLE_0:def 1;
    ex k be Nat st x=k & k mod 4 = 3 & k divides p|^n by A4;
    hence thesis by Th10,A1;
  end;
