2687 = (2 * 1343) + 1 ;
hence not 2 divides 2687 by NAT_4:9; :: thesis:
2687 = (3 * 895) + 2 ;
hence not 3 divides 2687 by NAT_4:9; :: thesis:
2687 = (5 * 537) + 2 ;
hence not 5 divides 2687 by NAT_4:9; :: thesis:
2687 = (7 * 383) + 6 ;
hence not 7 divides 2687 by NAT_4:9; :: thesis:
2687 = (11 * 244) + 3 ;
hence not 11 divides 2687 by NAT_4:9; :: thesis:
2687 = (13 * 206) + 9 ;
hence not 13 divides 2687 by NAT_4:9; :: thesis:
2687 = (17 * 158) + 1 ;
hence not 17 divides 2687 by NAT_4:9; :: thesis:
2687 = (19 * 141) + 8 ;
hence not 19 divides 2687 by NAT_4:9; :: thesis:
2687 = (23 * 116) + 19 ;
hence not 23 divides 2687 by NAT_4:9; :: thesis:
2687 = (29 * 92) + 19 ;
hence not 29 divides 2687 by NAT_4:9; :: thesis:
2687 = (31 * 86) + 21 ;
hence not 31 divides 2687 by NAT_4:9; :: thesis:
2687 = (37 * 72) + 23 ;
hence not 37 divides 2687 by NAT_4:9; :: thesis:
2687 = (41 * 65) + 22 ;
hence not 41 divides 2687 by NAT_4:9; :: thesis:
2687 = (43 * 62) + 21 ;
hence not 43 divides 2687 by NAT_4:9; :: thesis:
2687 = (47 * 57) + 8 ;
hence not 47 divides 2687 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2687 & n is prime holds
not n divides 2687 by XPRIMET1:30;
hence 2687 is prime by NAT_4:14; :: thesis: