
theorem
  5557 is prime
proof
  now
    5557 = 2*2778 + 1; hence not 2 divides 5557 by NAT_4:9;
    5557 = 3*1852 + 1; hence not 3 divides 5557 by NAT_4:9;
    5557 = 5*1111 + 2; hence not 5 divides 5557 by NAT_4:9;
    5557 = 7*793 + 6; hence not 7 divides 5557 by NAT_4:9;
    5557 = 11*505 + 2; hence not 11 divides 5557 by NAT_4:9;
    5557 = 13*427 + 6; hence not 13 divides 5557 by NAT_4:9;
    5557 = 17*326 + 15; hence not 17 divides 5557 by NAT_4:9;
    5557 = 19*292 + 9; hence not 19 divides 5557 by NAT_4:9;
    5557 = 23*241 + 14; hence not 23 divides 5557 by NAT_4:9;
    5557 = 29*191 + 18; hence not 29 divides 5557 by NAT_4:9;
    5557 = 31*179 + 8; hence not 31 divides 5557 by NAT_4:9;
    5557 = 37*150 + 7; hence not 37 divides 5557 by NAT_4:9;
    5557 = 41*135 + 22; hence not 41 divides 5557 by NAT_4:9;
    5557 = 43*129 + 10; hence not 43 divides 5557 by NAT_4:9;
    5557 = 47*118 + 11; hence not 47 divides 5557 by NAT_4:9;
    5557 = 53*104 + 45; hence not 53 divides 5557 by NAT_4:9;
    5557 = 59*94 + 11; hence not 59 divides 5557 by NAT_4:9;
    5557 = 61*91 + 6; hence not 61 divides 5557 by NAT_4:9;
    5557 = 67*82 + 63; hence not 67 divides 5557 by NAT_4:9;
    5557 = 71*78 + 19; hence not 71 divides 5557 by NAT_4:9;
    5557 = 73*76 + 9; hence not 73 divides 5557 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 5557 & n is prime
  holds not n divides 5557 by XPRIMET1:42;
  hence thesis by NAT_4:14;
end;
