1423 = (2 * 711) + 1 ;
hence not 2 divides 1423 by NAT_4:9; :: thesis:
1423 = (3 * 474) + 1 ;
hence not 3 divides 1423 by NAT_4:9; :: thesis:
1423 = (5 * 284) + 3 ;
hence not 5 divides 1423 by NAT_4:9; :: thesis:
1423 = (7 * 203) + 2 ;
hence not 7 divides 1423 by NAT_4:9; :: thesis:
1423 = (11 * 129) + 4 ;
hence not 11 divides 1423 by NAT_4:9; :: thesis:
1423 = (13 * 109) + 6 ;
hence not 13 divides 1423 by NAT_4:9; :: thesis:
1423 = (17 * 83) + 12 ;
hence not 17 divides 1423 by NAT_4:9; :: thesis:
1423 = (19 * 74) + 17 ;
hence not 19 divides 1423 by NAT_4:9; :: thesis:
1423 = (23 * 61) + 20 ;
hence not 23 divides 1423 by NAT_4:9; :: thesis:
1423 = (29 * 49) + 2 ;
hence not 29 divides 1423 by NAT_4:9; :: thesis:
1423 = (31 * 45) + 28 ;
hence not 31 divides 1423 by NAT_4:9; :: thesis:
1423 = (37 * 38) + 17 ;
hence not 37 divides 1423 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1423 & n is prime holds
not n divides 1423 by XPRIMET1:24;
hence 1423 is prime by NAT_4:14; :: thesis: