
theorem
  7757 is prime
proof
  now
    7757 = 2*3878 + 1; hence not 2 divides 7757 by NAT_4:9;
    7757 = 3*2585 + 2; hence not 3 divides 7757 by NAT_4:9;
    7757 = 5*1551 + 2; hence not 5 divides 7757 by NAT_4:9;
    7757 = 7*1108 + 1; hence not 7 divides 7757 by NAT_4:9;
    7757 = 11*705 + 2; hence not 11 divides 7757 by NAT_4:9;
    7757 = 13*596 + 9; hence not 13 divides 7757 by NAT_4:9;
    7757 = 17*456 + 5; hence not 17 divides 7757 by NAT_4:9;
    7757 = 19*408 + 5; hence not 19 divides 7757 by NAT_4:9;
    7757 = 23*337 + 6; hence not 23 divides 7757 by NAT_4:9;
    7757 = 29*267 + 14; hence not 29 divides 7757 by NAT_4:9;
    7757 = 31*250 + 7; hence not 31 divides 7757 by NAT_4:9;
    7757 = 37*209 + 24; hence not 37 divides 7757 by NAT_4:9;
    7757 = 41*189 + 8; hence not 41 divides 7757 by NAT_4:9;
    7757 = 43*180 + 17; hence not 43 divides 7757 by NAT_4:9;
    7757 = 47*165 + 2; hence not 47 divides 7757 by NAT_4:9;
    7757 = 53*146 + 19; hence not 53 divides 7757 by NAT_4:9;
    7757 = 59*131 + 28; hence not 59 divides 7757 by NAT_4:9;
    7757 = 61*127 + 10; hence not 61 divides 7757 by NAT_4:9;
    7757 = 67*115 + 52; hence not 67 divides 7757 by NAT_4:9;
    7757 = 71*109 + 18; hence not 71 divides 7757 by NAT_4:9;
    7757 = 73*106 + 19; hence not 73 divides 7757 by NAT_4:9;
    7757 = 79*98 + 15; hence not 79 divides 7757 by NAT_4:9;
    7757 = 83*93 + 38; hence not 83 divides 7757 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 7757 & n is prime
  holds not n divides 7757 by XPRIMET1:46;
  hence thesis by NAT_4:14;
end;
