
theorem
  4111 is prime
proof
  now
    4111 = 2*2055 + 1; hence not 2 divides 4111 by NAT_4:9;
    4111 = 3*1370 + 1; hence not 3 divides 4111 by NAT_4:9;
    4111 = 5*822 + 1; hence not 5 divides 4111 by NAT_4:9;
    4111 = 7*587 + 2; hence not 7 divides 4111 by NAT_4:9;
    4111 = 11*373 + 8; hence not 11 divides 4111 by NAT_4:9;
    4111 = 13*316 + 3; hence not 13 divides 4111 by NAT_4:9;
    4111 = 17*241 + 14; hence not 17 divides 4111 by NAT_4:9;
    4111 = 19*216 + 7; hence not 19 divides 4111 by NAT_4:9;
    4111 = 23*178 + 17; hence not 23 divides 4111 by NAT_4:9;
    4111 = 29*141 + 22; hence not 29 divides 4111 by NAT_4:9;
    4111 = 31*132 + 19; hence not 31 divides 4111 by NAT_4:9;
    4111 = 37*111 + 4; hence not 37 divides 4111 by NAT_4:9;
    4111 = 41*100 + 11; hence not 41 divides 4111 by NAT_4:9;
    4111 = 43*95 + 26; hence not 43 divides 4111 by NAT_4:9;
    4111 = 47*87 + 22; hence not 47 divides 4111 by NAT_4:9;
    4111 = 53*77 + 30; hence not 53 divides 4111 by NAT_4:9;
    4111 = 59*69 + 40; hence not 59 divides 4111 by NAT_4:9;
    4111 = 61*67 + 24; hence not 61 divides 4111 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 4111 & n is prime
  holds not n divides 4111 by XPRIMET1:36;
  hence thesis by NAT_4:14;
end;
