
theorem
  6781 is prime
proof
  now
    6781 = 2*3390 + 1; hence not 2 divides 6781 by NAT_4:9;
    6781 = 3*2260 + 1; hence not 3 divides 6781 by NAT_4:9;
    6781 = 5*1356 + 1; hence not 5 divides 6781 by NAT_4:9;
    6781 = 7*968 + 5; hence not 7 divides 6781 by NAT_4:9;
    6781 = 11*616 + 5; hence not 11 divides 6781 by NAT_4:9;
    6781 = 13*521 + 8; hence not 13 divides 6781 by NAT_4:9;
    6781 = 17*398 + 15; hence not 17 divides 6781 by NAT_4:9;
    6781 = 19*356 + 17; hence not 19 divides 6781 by NAT_4:9;
    6781 = 23*294 + 19; hence not 23 divides 6781 by NAT_4:9;
    6781 = 29*233 + 24; hence not 29 divides 6781 by NAT_4:9;
    6781 = 31*218 + 23; hence not 31 divides 6781 by NAT_4:9;
    6781 = 37*183 + 10; hence not 37 divides 6781 by NAT_4:9;
    6781 = 41*165 + 16; hence not 41 divides 6781 by NAT_4:9;
    6781 = 43*157 + 30; hence not 43 divides 6781 by NAT_4:9;
    6781 = 47*144 + 13; hence not 47 divides 6781 by NAT_4:9;
    6781 = 53*127 + 50; hence not 53 divides 6781 by NAT_4:9;
    6781 = 59*114 + 55; hence not 59 divides 6781 by NAT_4:9;
    6781 = 61*111 + 10; hence not 61 divides 6781 by NAT_4:9;
    6781 = 67*101 + 14; hence not 67 divides 6781 by NAT_4:9;
    6781 = 71*95 + 36; hence not 71 divides 6781 by NAT_4:9;
    6781 = 73*92 + 65; hence not 73 divides 6781 by NAT_4:9;
    6781 = 79*85 + 66; hence not 79 divides 6781 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 6781 & n is prime
  holds not n divides 6781 by XPRIMET1:44;
  hence thesis by NAT_4:14;
end;
