1699 = (2 * 849) + 1 ;
hence not 2 divides 1699 by NAT_4:9; :: thesis:
1699 = (3 * 566) + 1 ;
hence not 3 divides 1699 by NAT_4:9; :: thesis:
1699 = (5 * 339) + 4 ;
hence not 5 divides 1699 by NAT_4:9; :: thesis:
1699 = (7 * 242) + 5 ;
hence not 7 divides 1699 by NAT_4:9; :: thesis:
1699 = (11 * 154) + 5 ;
hence not 11 divides 1699 by NAT_4:9; :: thesis:
1699 = (13 * 130) + 9 ;
hence not 13 divides 1699 by NAT_4:9; :: thesis:
1699 = (17 * 99) + 16 ;
hence not 17 divides 1699 by NAT_4:9; :: thesis:
1699 = (19 * 89) + 8 ;
hence not 19 divides 1699 by NAT_4:9; :: thesis:
1699 = (23 * 73) + 20 ;
hence not 23 divides 1699 by NAT_4:9; :: thesis:
1699 = (29 * 58) + 17 ;
hence not 29 divides 1699 by NAT_4:9; :: thesis:
1699 = (31 * 54) + 25 ;
hence not 31 divides 1699 by NAT_4:9; :: thesis:
1699 = (37 * 45) + 34 ;
hence not 37 divides 1699 by NAT_4:9; :: thesis:
1699 = (41 * 41) + 18 ;
hence not 41 divides 1699 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1699 & n is prime holds
not n divides 1699 by XPRIMET1:26;
hence 1699 is prime by NAT_4:14; :: thesis: