
theorem
  7577 is prime
proof
  now
    7577 = 2*3788 + 1; hence not 2 divides 7577 by NAT_4:9;
    7577 = 3*2525 + 2; hence not 3 divides 7577 by NAT_4:9;
    7577 = 5*1515 + 2; hence not 5 divides 7577 by NAT_4:9;
    7577 = 7*1082 + 3; hence not 7 divides 7577 by NAT_4:9;
    7577 = 11*688 + 9; hence not 11 divides 7577 by NAT_4:9;
    7577 = 13*582 + 11; hence not 13 divides 7577 by NAT_4:9;
    7577 = 17*445 + 12; hence not 17 divides 7577 by NAT_4:9;
    7577 = 19*398 + 15; hence not 19 divides 7577 by NAT_4:9;
    7577 = 23*329 + 10; hence not 23 divides 7577 by NAT_4:9;
    7577 = 29*261 + 8; hence not 29 divides 7577 by NAT_4:9;
    7577 = 31*244 + 13; hence not 31 divides 7577 by NAT_4:9;
    7577 = 37*204 + 29; hence not 37 divides 7577 by NAT_4:9;
    7577 = 41*184 + 33; hence not 41 divides 7577 by NAT_4:9;
    7577 = 43*176 + 9; hence not 43 divides 7577 by NAT_4:9;
    7577 = 47*161 + 10; hence not 47 divides 7577 by NAT_4:9;
    7577 = 53*142 + 51; hence not 53 divides 7577 by NAT_4:9;
    7577 = 59*128 + 25; hence not 59 divides 7577 by NAT_4:9;
    7577 = 61*124 + 13; hence not 61 divides 7577 by NAT_4:9;
    7577 = 67*113 + 6; hence not 67 divides 7577 by NAT_4:9;
    7577 = 71*106 + 51; hence not 71 divides 7577 by NAT_4:9;
    7577 = 73*103 + 58; hence not 73 divides 7577 by NAT_4:9;
    7577 = 79*95 + 72; hence not 79 divides 7577 by NAT_4:9;
    7577 = 83*91 + 24; hence not 83 divides 7577 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 7577 & n is prime
  holds not n divides 7577 by XPRIMET1:46;
  hence thesis by NAT_4:14;
end;
