
theorem
  6581 is prime
proof
  now
    6581 = 2*3290 + 1; hence not 2 divides 6581 by NAT_4:9;
    6581 = 3*2193 + 2; hence not 3 divides 6581 by NAT_4:9;
    6581 = 5*1316 + 1; hence not 5 divides 6581 by NAT_4:9;
    6581 = 7*940 + 1; hence not 7 divides 6581 by NAT_4:9;
    6581 = 11*598 + 3; hence not 11 divides 6581 by NAT_4:9;
    6581 = 13*506 + 3; hence not 13 divides 6581 by NAT_4:9;
    6581 = 17*387 + 2; hence not 17 divides 6581 by NAT_4:9;
    6581 = 19*346 + 7; hence not 19 divides 6581 by NAT_4:9;
    6581 = 23*286 + 3; hence not 23 divides 6581 by NAT_4:9;
    6581 = 29*226 + 27; hence not 29 divides 6581 by NAT_4:9;
    6581 = 31*212 + 9; hence not 31 divides 6581 by NAT_4:9;
    6581 = 37*177 + 32; hence not 37 divides 6581 by NAT_4:9;
    6581 = 41*160 + 21; hence not 41 divides 6581 by NAT_4:9;
    6581 = 43*153 + 2; hence not 43 divides 6581 by NAT_4:9;
    6581 = 47*140 + 1; hence not 47 divides 6581 by NAT_4:9;
    6581 = 53*124 + 9; hence not 53 divides 6581 by NAT_4:9;
    6581 = 59*111 + 32; hence not 59 divides 6581 by NAT_4:9;
    6581 = 61*107 + 54; hence not 61 divides 6581 by NAT_4:9;
    6581 = 67*98 + 15; hence not 67 divides 6581 by NAT_4:9;
    6581 = 71*92 + 49; hence not 71 divides 6581 by NAT_4:9;
    6581 = 73*90 + 11; hence not 73 divides 6581 by NAT_4:9;
    6581 = 79*83 + 24; hence not 79 divides 6581 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 6581 & n is prime
  holds not n divides 6581 by XPRIMET1:44;
  hence thesis by NAT_4:14;
end;
