
theorem
  8861 is prime
proof
  now
    8861 = 2*4430 + 1; hence not 2 divides 8861 by NAT_4:9;
    8861 = 3*2953 + 2; hence not 3 divides 8861 by NAT_4:9;
    8861 = 5*1772 + 1; hence not 5 divides 8861 by NAT_4:9;
    8861 = 7*1265 + 6; hence not 7 divides 8861 by NAT_4:9;
    8861 = 11*805 + 6; hence not 11 divides 8861 by NAT_4:9;
    8861 = 13*681 + 8; hence not 13 divides 8861 by NAT_4:9;
    8861 = 17*521 + 4; hence not 17 divides 8861 by NAT_4:9;
    8861 = 19*466 + 7; hence not 19 divides 8861 by NAT_4:9;
    8861 = 23*385 + 6; hence not 23 divides 8861 by NAT_4:9;
    8861 = 29*305 + 16; hence not 29 divides 8861 by NAT_4:9;
    8861 = 31*285 + 26; hence not 31 divides 8861 by NAT_4:9;
    8861 = 37*239 + 18; hence not 37 divides 8861 by NAT_4:9;
    8861 = 41*216 + 5; hence not 41 divides 8861 by NAT_4:9;
    8861 = 43*206 + 3; hence not 43 divides 8861 by NAT_4:9;
    8861 = 47*188 + 25; hence not 47 divides 8861 by NAT_4:9;
    8861 = 53*167 + 10; hence not 53 divides 8861 by NAT_4:9;
    8861 = 59*150 + 11; hence not 59 divides 8861 by NAT_4:9;
    8861 = 61*145 + 16; hence not 61 divides 8861 by NAT_4:9;
    8861 = 67*132 + 17; hence not 67 divides 8861 by NAT_4:9;
    8861 = 71*124 + 57; hence not 71 divides 8861 by NAT_4:9;
    8861 = 73*121 + 28; hence not 73 divides 8861 by NAT_4:9;
    8861 = 79*112 + 13; hence not 79 divides 8861 by NAT_4:9;
    8861 = 83*106 + 63; hence not 83 divides 8861 by NAT_4:9;
    8861 = 89*99 + 50; hence not 89 divides 8861 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8861 & n is prime
  holds not n divides 8861 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
