2269 = (2 * 1134) + 1 ;
hence not 2 divides 2269 by NAT_4:9; :: thesis:
2269 = (3 * 756) + 1 ;
hence not 3 divides 2269 by NAT_4:9; :: thesis:
2269 = (5 * 453) + 4 ;
hence not 5 divides 2269 by NAT_4:9; :: thesis:
2269 = (7 * 324) + 1 ;
hence not 7 divides 2269 by NAT_4:9; :: thesis:
2269 = (11 * 206) + 3 ;
hence not 11 divides 2269 by NAT_4:9; :: thesis:
2269 = (13 * 174) + 7 ;
hence not 13 divides 2269 by NAT_4:9; :: thesis:
2269 = (17 * 133) + 8 ;
hence not 17 divides 2269 by NAT_4:9; :: thesis:
2269 = (19 * 119) + 8 ;
hence not 19 divides 2269 by NAT_4:9; :: thesis:
2269 = (23 * 98) + 15 ;
hence not 23 divides 2269 by NAT_4:9; :: thesis:
2269 = (29 * 78) + 7 ;
hence not 29 divides 2269 by NAT_4:9; :: thesis:
2269 = (31 * 73) + 6 ;
hence not 31 divides 2269 by NAT_4:9; :: thesis:
2269 = (37 * 61) + 12 ;
hence not 37 divides 2269 by NAT_4:9; :: thesis:
2269 = (41 * 55) + 14 ;
hence not 41 divides 2269 by NAT_4:9; :: thesis:
2269 = (43 * 52) + 33 ;
hence not 43 divides 2269 by NAT_4:9; :: thesis:
2269 = (47 * 48) + 13 ;
hence not 47 divides 2269 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2269 & n is prime holds
not n divides 2269 by XPRIMET1:30;
hence 2269 is prime by NAT_4:14; :: thesis: