
theorem
  7727 is prime
proof
  now
    7727 = 2*3863 + 1; hence not 2 divides 7727 by NAT_4:9;
    7727 = 3*2575 + 2; hence not 3 divides 7727 by NAT_4:9;
    7727 = 5*1545 + 2; hence not 5 divides 7727 by NAT_4:9;
    7727 = 7*1103 + 6; hence not 7 divides 7727 by NAT_4:9;
    7727 = 11*702 + 5; hence not 11 divides 7727 by NAT_4:9;
    7727 = 13*594 + 5; hence not 13 divides 7727 by NAT_4:9;
    7727 = 17*454 + 9; hence not 17 divides 7727 by NAT_4:9;
    7727 = 19*406 + 13; hence not 19 divides 7727 by NAT_4:9;
    7727 = 23*335 + 22; hence not 23 divides 7727 by NAT_4:9;
    7727 = 29*266 + 13; hence not 29 divides 7727 by NAT_4:9;
    7727 = 31*249 + 8; hence not 31 divides 7727 by NAT_4:9;
    7727 = 37*208 + 31; hence not 37 divides 7727 by NAT_4:9;
    7727 = 41*188 + 19; hence not 41 divides 7727 by NAT_4:9;
    7727 = 43*179 + 30; hence not 43 divides 7727 by NAT_4:9;
    7727 = 47*164 + 19; hence not 47 divides 7727 by NAT_4:9;
    7727 = 53*145 + 42; hence not 53 divides 7727 by NAT_4:9;
    7727 = 59*130 + 57; hence not 59 divides 7727 by NAT_4:9;
    7727 = 61*126 + 41; hence not 61 divides 7727 by NAT_4:9;
    7727 = 67*115 + 22; hence not 67 divides 7727 by NAT_4:9;
    7727 = 71*108 + 59; hence not 71 divides 7727 by NAT_4:9;
    7727 = 73*105 + 62; hence not 73 divides 7727 by NAT_4:9;
    7727 = 79*97 + 64; hence not 79 divides 7727 by NAT_4:9;
    7727 = 83*93 + 8; hence not 83 divides 7727 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 7727 & n is prime
  holds not n divides 7727 by XPRIMET1:46;
  hence thesis by NAT_4:14;
end;
