1229 = (2 * 614) + 1 ;
hence not 2 divides 1229 by NAT_4:9; :: thesis:
1229 = (3 * 409) + 2 ;
hence not 3 divides 1229 by NAT_4:9; :: thesis:
1229 = (5 * 245) + 4 ;
hence not 5 divides 1229 by NAT_4:9; :: thesis:
1229 = (7 * 175) + 4 ;
hence not 7 divides 1229 by NAT_4:9; :: thesis:
1229 = (11 * 111) + 8 ;
hence not 11 divides 1229 by NAT_4:9; :: thesis:
1229 = (13 * 94) + 7 ;
hence not 13 divides 1229 by NAT_4:9; :: thesis:
1229 = (17 * 72) + 5 ;
hence not 17 divides 1229 by NAT_4:9; :: thesis:
1229 = (19 * 64) + 13 ;
hence not 19 divides 1229 by NAT_4:9; :: thesis:
1229 = (23 * 53) + 10 ;
hence not 23 divides 1229 by NAT_4:9; :: thesis:
1229 = (29 * 42) + 11 ;
hence not 29 divides 1229 by NAT_4:9; :: thesis:
1229 = (31 * 39) + 20 ;
hence not 31 divides 1229 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1229 & n is prime holds
not n divides 1229 by XPRIMET1:22;
hence 1229 is prime by NAT_4:14; :: thesis: