
theorem
  6569 is prime
proof
  now
    6569 = 2*3284 + 1; hence not 2 divides 6569 by NAT_4:9;
    6569 = 3*2189 + 2; hence not 3 divides 6569 by NAT_4:9;
    6569 = 5*1313 + 4; hence not 5 divides 6569 by NAT_4:9;
    6569 = 7*938 + 3; hence not 7 divides 6569 by NAT_4:9;
    6569 = 11*597 + 2; hence not 11 divides 6569 by NAT_4:9;
    6569 = 13*505 + 4; hence not 13 divides 6569 by NAT_4:9;
    6569 = 17*386 + 7; hence not 17 divides 6569 by NAT_4:9;
    6569 = 19*345 + 14; hence not 19 divides 6569 by NAT_4:9;
    6569 = 23*285 + 14; hence not 23 divides 6569 by NAT_4:9;
    6569 = 29*226 + 15; hence not 29 divides 6569 by NAT_4:9;
    6569 = 31*211 + 28; hence not 31 divides 6569 by NAT_4:9;
    6569 = 37*177 + 20; hence not 37 divides 6569 by NAT_4:9;
    6569 = 41*160 + 9; hence not 41 divides 6569 by NAT_4:9;
    6569 = 43*152 + 33; hence not 43 divides 6569 by NAT_4:9;
    6569 = 47*139 + 36; hence not 47 divides 6569 by NAT_4:9;
    6569 = 53*123 + 50; hence not 53 divides 6569 by NAT_4:9;
    6569 = 59*111 + 20; hence not 59 divides 6569 by NAT_4:9;
    6569 = 61*107 + 42; hence not 61 divides 6569 by NAT_4:9;
    6569 = 67*98 + 3; hence not 67 divides 6569 by NAT_4:9;
    6569 = 71*92 + 37; hence not 71 divides 6569 by NAT_4:9;
    6569 = 73*89 + 72; hence not 73 divides 6569 by NAT_4:9;
    6569 = 79*83 + 12; hence not 79 divides 6569 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 6569 & n is prime
  holds not n divides 6569 by XPRIMET1:44;
  hence thesis by NAT_4:14;
end;
