2699 = (2 * 1349) + 1 ;
hence not 2 divides 2699 by NAT_4:9; :: thesis:
2699 = (3 * 899) + 2 ;
hence not 3 divides 2699 by NAT_4:9; :: thesis:
2699 = (5 * 539) + 4 ;
hence not 5 divides 2699 by NAT_4:9; :: thesis:
2699 = (7 * 385) + 4 ;
hence not 7 divides 2699 by NAT_4:9; :: thesis:
2699 = (11 * 245) + 4 ;
hence not 11 divides 2699 by NAT_4:9; :: thesis:
2699 = (13 * 207) + 8 ;
hence not 13 divides 2699 by NAT_4:9; :: thesis:
2699 = (17 * 158) + 13 ;
hence not 17 divides 2699 by NAT_4:9; :: thesis:
2699 = (19 * 142) + 1 ;
hence not 19 divides 2699 by NAT_4:9; :: thesis:
2699 = (23 * 117) + 8 ;
hence not 23 divides 2699 by NAT_4:9; :: thesis:
2699 = (29 * 93) + 2 ;
hence not 29 divides 2699 by NAT_4:9; :: thesis:
2699 = (31 * 87) + 2 ;
hence not 31 divides 2699 by NAT_4:9; :: thesis:
2699 = (37 * 72) + 35 ;
hence not 37 divides 2699 by NAT_4:9; :: thesis:
2699 = (41 * 65) + 34 ;
hence not 41 divides 2699 by NAT_4:9; :: thesis:
2699 = (43 * 62) + 33 ;
hence not 43 divides 2699 by NAT_4:9; :: thesis:
2699 = (47 * 57) + 20 ;
hence not 47 divides 2699 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2699 & n is prime holds
not n divides 2699 by XPRIMET1:30;
hence 2699 is prime by NAT_4:14; :: thesis: