2423 = (2 * 1211) + 1 ;
hence not 2 divides 2423 by NAT_4:9; :: thesis:
2423 = (3 * 807) + 2 ;
hence not 3 divides 2423 by NAT_4:9; :: thesis:
2423 = (5 * 484) + 3 ;
hence not 5 divides 2423 by NAT_4:9; :: thesis:
2423 = (7 * 346) + 1 ;
hence not 7 divides 2423 by NAT_4:9; :: thesis:
2423 = (11 * 220) + 3 ;
hence not 11 divides 2423 by NAT_4:9; :: thesis:
2423 = (13 * 186) + 5 ;
hence not 13 divides 2423 by NAT_4:9; :: thesis:
2423 = (17 * 142) + 9 ;
hence not 17 divides 2423 by NAT_4:9; :: thesis:
2423 = (19 * 127) + 10 ;
hence not 19 divides 2423 by NAT_4:9; :: thesis:
2423 = (23 * 105) + 8 ;
hence not 23 divides 2423 by NAT_4:9; :: thesis:
2423 = (29 * 83) + 16 ;
hence not 29 divides 2423 by NAT_4:9; :: thesis:
2423 = (31 * 78) + 5 ;
hence not 31 divides 2423 by NAT_4:9; :: thesis:
2423 = (37 * 65) + 18 ;
hence not 37 divides 2423 by NAT_4:9; :: thesis:
2423 = (41 * 59) + 4 ;
hence not 41 divides 2423 by NAT_4:9; :: thesis:
2423 = (43 * 56) + 15 ;
hence not 43 divides 2423 by NAT_4:9; :: thesis:
2423 = (47 * 51) + 26 ;
hence not 47 divides 2423 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2423 & n is prime holds
not n divides 2423 by XPRIMET1:30;
hence 2423 is prime by NAT_4:14; :: thesis: