
theorem
  7561 is prime
proof
  now
    7561 = 2*3780 + 1; hence not 2 divides 7561 by NAT_4:9;
    7561 = 3*2520 + 1; hence not 3 divides 7561 by NAT_4:9;
    7561 = 5*1512 + 1; hence not 5 divides 7561 by NAT_4:9;
    7561 = 7*1080 + 1; hence not 7 divides 7561 by NAT_4:9;
    7561 = 11*687 + 4; hence not 11 divides 7561 by NAT_4:9;
    7561 = 13*581 + 8; hence not 13 divides 7561 by NAT_4:9;
    7561 = 17*444 + 13; hence not 17 divides 7561 by NAT_4:9;
    7561 = 19*397 + 18; hence not 19 divides 7561 by NAT_4:9;
    7561 = 23*328 + 17; hence not 23 divides 7561 by NAT_4:9;
    7561 = 29*260 + 21; hence not 29 divides 7561 by NAT_4:9;
    7561 = 31*243 + 28; hence not 31 divides 7561 by NAT_4:9;
    7561 = 37*204 + 13; hence not 37 divides 7561 by NAT_4:9;
    7561 = 41*184 + 17; hence not 41 divides 7561 by NAT_4:9;
    7561 = 43*175 + 36; hence not 43 divides 7561 by NAT_4:9;
    7561 = 47*160 + 41; hence not 47 divides 7561 by NAT_4:9;
    7561 = 53*142 + 35; hence not 53 divides 7561 by NAT_4:9;
    7561 = 59*128 + 9; hence not 59 divides 7561 by NAT_4:9;
    7561 = 61*123 + 58; hence not 61 divides 7561 by NAT_4:9;
    7561 = 67*112 + 57; hence not 67 divides 7561 by NAT_4:9;
    7561 = 71*106 + 35; hence not 71 divides 7561 by NAT_4:9;
    7561 = 73*103 + 42; hence not 73 divides 7561 by NAT_4:9;
    7561 = 79*95 + 56; hence not 79 divides 7561 by NAT_4:9;
    7561 = 83*91 + 8; hence not 83 divides 7561 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 7561 & n is prime
  holds not n divides 7561 by XPRIMET1:46;
  hence thesis by NAT_4:14;
end;
