2777 = (2 * 1388) + 1 ;
hence not 2 divides 2777 by NAT_4:9; :: thesis:
2777 = (3 * 925) + 2 ;
hence not 3 divides 2777 by NAT_4:9; :: thesis:
2777 = (5 * 555) + 2 ;
hence not 5 divides 2777 by NAT_4:9; :: thesis:
2777 = (7 * 396) + 5 ;
hence not 7 divides 2777 by NAT_4:9; :: thesis:
2777 = (11 * 252) + 5 ;
hence not 11 divides 2777 by NAT_4:9; :: thesis:
2777 = (13 * 213) + 8 ;
hence not 13 divides 2777 by NAT_4:9; :: thesis:
2777 = (17 * 163) + 6 ;
hence not 17 divides 2777 by NAT_4:9; :: thesis:
2777 = (19 * 146) + 3 ;
hence not 19 divides 2777 by NAT_4:9; :: thesis:
2777 = (23 * 120) + 17 ;
hence not 23 divides 2777 by NAT_4:9; :: thesis:
2777 = (29 * 95) + 22 ;
hence not 29 divides 2777 by NAT_4:9; :: thesis:
2777 = (31 * 89) + 18 ;
hence not 31 divides 2777 by NAT_4:9; :: thesis:
2777 = (37 * 75) + 2 ;
hence not 37 divides 2777 by NAT_4:9; :: thesis:
2777 = (41 * 67) + 30 ;
hence not 41 divides 2777 by NAT_4:9; :: thesis:
2777 = (43 * 64) + 25 ;
hence not 43 divides 2777 by NAT_4:9; :: thesis:
2777 = (47 * 59) + 4 ;
hence not 47 divides 2777 by NAT_4:9; :: thesis:

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 2777 is prime by NAT_4:14; :: thesis: