
theorem
  1583 is prime
proof
  now
    1583 = 2*791 + 1; hence not 2 divides 1583 by NAT_4:9;
    1583 = 3*527 + 2; hence not 3 divides 1583 by NAT_4:9;
    1583 = 5*316 + 3; hence not 5 divides 1583 by NAT_4:9;
    1583 = 7*226 + 1; hence not 7 divides 1583 by NAT_4:9;
    1583 = 11*143 + 10; hence not 11 divides 1583 by NAT_4:9;
    1583 = 13*121 + 10; hence not 13 divides 1583 by NAT_4:9;
    1583 = 17*93 + 2; hence not 17 divides 1583 by NAT_4:9;
    1583 = 19*83 + 6; hence not 19 divides 1583 by NAT_4:9;
    1583 = 23*68 + 19; hence not 23 divides 1583 by NAT_4:9;
    1583 = 29*54 + 17; hence not 29 divides 1583 by NAT_4:9;
    1583 = 31*51 + 2; hence not 31 divides 1583 by NAT_4:9;
    1583 = 37*42 + 29; hence not 37 divides 1583 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 1583 & n is prime
  holds not n divides 1583 by XPRIMET1:24;
  hence thesis by NAT_4:14;
end;
