2677 = (2 * 1338) + 1 ;
hence not 2 divides 2677 by NAT_4:9; :: thesis:
2677 = (3 * 892) + 1 ;
hence not 3 divides 2677 by NAT_4:9; :: thesis:
2677 = (5 * 535) + 2 ;
hence not 5 divides 2677 by NAT_4:9; :: thesis:
2677 = (7 * 382) + 3 ;
hence not 7 divides 2677 by NAT_4:9; :: thesis:
2677 = (11 * 243) + 4 ;
hence not 11 divides 2677 by NAT_4:9; :: thesis:
2677 = (13 * 205) + 12 ;
hence not 13 divides 2677 by NAT_4:9; :: thesis:
2677 = (17 * 157) + 8 ;
hence not 17 divides 2677 by NAT_4:9; :: thesis:
2677 = (19 * 140) + 17 ;
hence not 19 divides 2677 by NAT_4:9; :: thesis:
2677 = (23 * 116) + 9 ;
hence not 23 divides 2677 by NAT_4:9; :: thesis:
2677 = (29 * 92) + 9 ;
hence not 29 divides 2677 by NAT_4:9; :: thesis:
2677 = (31 * 86) + 11 ;
hence not 31 divides 2677 by NAT_4:9; :: thesis:
2677 = (37 * 72) + 13 ;
hence not 37 divides 2677 by NAT_4:9; :: thesis:
2677 = (41 * 65) + 12 ;
hence not 41 divides 2677 by NAT_4:9; :: thesis:
2677 = (43 * 62) + 11 ;
hence not 43 divides 2677 by NAT_4:9; :: thesis:
2677 = (47 * 56) + 45 ;
hence not 47 divides 2677 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2677 & n is prime holds
not n divides 2677 by XPRIMET1:30;
hence 2677 is prime by NAT_4:14; :: thesis: