
theorem
  6857 is prime
proof
  now
    6857 = 2*3428 + 1; hence not 2 divides 6857 by NAT_4:9;
    6857 = 3*2285 + 2; hence not 3 divides 6857 by NAT_4:9;
    6857 = 5*1371 + 2; hence not 5 divides 6857 by NAT_4:9;
    6857 = 7*979 + 4; hence not 7 divides 6857 by NAT_4:9;
    6857 = 11*623 + 4; hence not 11 divides 6857 by NAT_4:9;
    6857 = 13*527 + 6; hence not 13 divides 6857 by NAT_4:9;
    6857 = 17*403 + 6; hence not 17 divides 6857 by NAT_4:9;
    6857 = 19*360 + 17; hence not 19 divides 6857 by NAT_4:9;
    6857 = 23*298 + 3; hence not 23 divides 6857 by NAT_4:9;
    6857 = 29*236 + 13; hence not 29 divides 6857 by NAT_4:9;
    6857 = 31*221 + 6; hence not 31 divides 6857 by NAT_4:9;
    6857 = 37*185 + 12; hence not 37 divides 6857 by NAT_4:9;
    6857 = 41*167 + 10; hence not 41 divides 6857 by NAT_4:9;
    6857 = 43*159 + 20; hence not 43 divides 6857 by NAT_4:9;
    6857 = 47*145 + 42; hence not 47 divides 6857 by NAT_4:9;
    6857 = 53*129 + 20; hence not 53 divides 6857 by NAT_4:9;
    6857 = 59*116 + 13; hence not 59 divides 6857 by NAT_4:9;
    6857 = 61*112 + 25; hence not 61 divides 6857 by NAT_4:9;
    6857 = 67*102 + 23; hence not 67 divides 6857 by NAT_4:9;
    6857 = 71*96 + 41; hence not 71 divides 6857 by NAT_4:9;
    6857 = 73*93 + 68; hence not 73 divides 6857 by NAT_4:9;
    6857 = 79*86 + 63; hence not 79 divides 6857 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 6857 & n is prime
  holds not n divides 6857 by XPRIMET1:44;
  hence thesis by NAT_4:14;
end;
