
theorem
  1567 is prime
proof
  now
    1567 = 2*783 + 1; hence not 2 divides 1567 by NAT_4:9;
    1567 = 3*522 + 1; hence not 3 divides 1567 by NAT_4:9;
    1567 = 5*313 + 2; hence not 5 divides 1567 by NAT_4:9;
    1567 = 7*223 + 6; hence not 7 divides 1567 by NAT_4:9;
    1567 = 11*142 + 5; hence not 11 divides 1567 by NAT_4:9;
    1567 = 13*120 + 7; hence not 13 divides 1567 by NAT_4:9;
    1567 = 17*92 + 3; hence not 17 divides 1567 by NAT_4:9;
    1567 = 19*82 + 9; hence not 19 divides 1567 by NAT_4:9;
    1567 = 23*68 + 3; hence not 23 divides 1567 by NAT_4:9;
    1567 = 29*54 + 1; hence not 29 divides 1567 by NAT_4:9;
    1567 = 31*50 + 17; hence not 31 divides 1567 by NAT_4:9;
    1567 = 37*42 + 13; hence not 37 divides 1567 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 1567 & n is prime
  holds not n divides 1567 by XPRIMET1:24;
  hence thesis by NAT_4:14;
end;
