
theorem
  4127 is prime
proof
  now
    4127 = 2*2063 + 1; hence not 2 divides 4127 by NAT_4:9;
    4127 = 3*1375 + 2; hence not 3 divides 4127 by NAT_4:9;
    4127 = 5*825 + 2; hence not 5 divides 4127 by NAT_4:9;
    4127 = 7*589 + 4; hence not 7 divides 4127 by NAT_4:9;
    4127 = 11*375 + 2; hence not 11 divides 4127 by NAT_4:9;
    4127 = 13*317 + 6; hence not 13 divides 4127 by NAT_4:9;
    4127 = 17*242 + 13; hence not 17 divides 4127 by NAT_4:9;
    4127 = 19*217 + 4; hence not 19 divides 4127 by NAT_4:9;
    4127 = 23*179 + 10; hence not 23 divides 4127 by NAT_4:9;
    4127 = 29*142 + 9; hence not 29 divides 4127 by NAT_4:9;
    4127 = 31*133 + 4; hence not 31 divides 4127 by NAT_4:9;
    4127 = 37*111 + 20; hence not 37 divides 4127 by NAT_4:9;
    4127 = 41*100 + 27; hence not 41 divides 4127 by NAT_4:9;
    4127 = 43*95 + 42; hence not 43 divides 4127 by NAT_4:9;
    4127 = 47*87 + 38; hence not 47 divides 4127 by NAT_4:9;
    4127 = 53*77 + 46; hence not 53 divides 4127 by NAT_4:9;
    4127 = 59*69 + 56; hence not 59 divides 4127 by NAT_4:9;
    4127 = 61*67 + 40; hence not 61 divides 4127 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 4127 & n is prime
  holds not n divides 4127 by XPRIMET1:36;
  hence thesis by NAT_4:14;
end;
