
theorem
  863 is prime
proof
  now
    863 = 2*431 + 1; hence not 2 divides 863 by NAT_4:9;
    863 = 3*287 + 2; hence not 3 divides 863 by NAT_4:9;
    863 = 5*172 + 3; hence not 5 divides 863 by NAT_4:9;
    863 = 7*123 + 2; hence not 7 divides 863 by NAT_4:9;
    863 = 11*78 + 5; hence not 11 divides 863 by NAT_4:9;
    863 = 13*66 + 5; hence not 13 divides 863 by NAT_4:9;
    863 = 17*50 + 13; hence not 17 divides 863 by NAT_4:9;
    863 = 19*45 + 8; hence not 19 divides 863 by NAT_4:9;
    863 = 23*37 + 12; hence not 23 divides 863 by NAT_4:9;
    863 = 29*29 + 22; hence not 29 divides 863 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 863 & n is prime
  holds not n divides 863 by XPRIMET1:20;
  hence thesis by NAT_4:14;
end;
