
theorem
  7867 is prime
proof
  now
    7867 = 2*3933 + 1; hence not 2 divides 7867 by NAT_4:9;
    7867 = 3*2622 + 1; hence not 3 divides 7867 by NAT_4:9;
    7867 = 5*1573 + 2; hence not 5 divides 7867 by NAT_4:9;
    7867 = 7*1123 + 6; hence not 7 divides 7867 by NAT_4:9;
    7867 = 11*715 + 2; hence not 11 divides 7867 by NAT_4:9;
    7867 = 13*605 + 2; hence not 13 divides 7867 by NAT_4:9;
    7867 = 17*462 + 13; hence not 17 divides 7867 by NAT_4:9;
    7867 = 19*414 + 1; hence not 19 divides 7867 by NAT_4:9;
    7867 = 23*342 + 1; hence not 23 divides 7867 by NAT_4:9;
    7867 = 29*271 + 8; hence not 29 divides 7867 by NAT_4:9;
    7867 = 31*253 + 24; hence not 31 divides 7867 by NAT_4:9;
    7867 = 37*212 + 23; hence not 37 divides 7867 by NAT_4:9;
    7867 = 41*191 + 36; hence not 41 divides 7867 by NAT_4:9;
    7867 = 43*182 + 41; hence not 43 divides 7867 by NAT_4:9;
    7867 = 47*167 + 18; hence not 47 divides 7867 by NAT_4:9;
    7867 = 53*148 + 23; hence not 53 divides 7867 by NAT_4:9;
    7867 = 59*133 + 20; hence not 59 divides 7867 by NAT_4:9;
    7867 = 61*128 + 59; hence not 61 divides 7867 by NAT_4:9;
    7867 = 67*117 + 28; hence not 67 divides 7867 by NAT_4:9;
    7867 = 71*110 + 57; hence not 71 divides 7867 by NAT_4:9;
    7867 = 73*107 + 56; hence not 73 divides 7867 by NAT_4:9;
    7867 = 79*99 + 46; hence not 79 divides 7867 by NAT_4:9;
    7867 = 83*94 + 65; hence not 83 divides 7867 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 7867 & n is prime
  holds not n divides 7867 by XPRIMET1:46;
  hence thesis by NAT_4:14;
end;
