
theorem
  4001 is prime
proof
  now
    4001 = 2*2000 + 1; hence not 2 divides 4001 by NAT_4:9;
    4001 = 3*1333 + 2; hence not 3 divides 4001 by NAT_4:9;
    4001 = 5*800 + 1; hence not 5 divides 4001 by NAT_4:9;
    4001 = 7*571 + 4; hence not 7 divides 4001 by NAT_4:9;
    4001 = 11*363 + 8; hence not 11 divides 4001 by NAT_4:9;
    4001 = 13*307 + 10; hence not 13 divides 4001 by NAT_4:9;
    4001 = 17*235 + 6; hence not 17 divides 4001 by NAT_4:9;
    4001 = 19*210 + 11; hence not 19 divides 4001 by NAT_4:9;
    4001 = 23*173 + 22; hence not 23 divides 4001 by NAT_4:9;
    4001 = 29*137 + 28; hence not 29 divides 4001 by NAT_4:9;
    4001 = 31*129 + 2; hence not 31 divides 4001 by NAT_4:9;
    4001 = 37*108 + 5; hence not 37 divides 4001 by NAT_4:9;
    4001 = 41*97 + 24; hence not 41 divides 4001 by NAT_4:9;
    4001 = 43*93 + 2; hence not 43 divides 4001 by NAT_4:9;
    4001 = 47*85 + 6; hence not 47 divides 4001 by NAT_4:9;
    4001 = 53*75 + 26; hence not 53 divides 4001 by NAT_4:9;
    4001 = 59*67 + 48; hence not 59 divides 4001 by NAT_4:9;
    4001 = 61*65 + 36; hence not 61 divides 4001 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 4001 & n is prime
  holds not n divides 4001 by XPRIMET1:36;
  hence thesis by NAT_4:14;
end;
