
theorem
  2777 is prime
proof
  now
    2777 = 2*1388 + 1; hence not 2 divides 2777 by NAT_4:9;
    2777 = 3*925 + 2; hence not 3 divides 2777 by NAT_4:9;
    2777 = 5*555 + 2; hence not 5 divides 2777 by NAT_4:9;
    2777 = 7*396 + 5; hence not 7 divides 2777 by NAT_4:9;
    2777 = 11*252 + 5; hence not 11 divides 2777 by NAT_4:9;
    2777 = 13*213 + 8; hence not 13 divides 2777 by NAT_4:9;
    2777 = 17*163 + 6; hence not 17 divides 2777 by NAT_4:9;
    2777 = 19*146 + 3; hence not 19 divides 2777 by NAT_4:9;
    2777 = 23*120 + 17; hence not 23 divides 2777 by NAT_4:9;
    2777 = 29*95 + 22; hence not 29 divides 2777 by NAT_4:9;
    2777 = 31*89 + 18; hence not 31 divides 2777 by NAT_4:9;
    2777 = 37*75 + 2; hence not 37 divides 2777 by NAT_4:9;
    2777 = 41*67 + 30; hence not 41 divides 2777 by NAT_4:9;
    2777 = 43*64 + 25; hence not 43 divides 2777 by NAT_4:9;
    2777 = 47*59 + 4; hence not 47 divides 2777 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 2777 & n is prime
  holds not n divides 2777 by XPRIMET1:30;
  hence thesis by NAT_4:14;
end;
