1597 = (2 * 798) + 1 ;
hence not 2 divides 1597 by NAT_4:9; :: thesis:
1597 = (3 * 532) + 1 ;
hence not 3 divides 1597 by NAT_4:9; :: thesis:
1597 = (5 * 319) + 2 ;
hence not 5 divides 1597 by NAT_4:9; :: thesis:
1597 = (7 * 228) + 1 ;
hence not 7 divides 1597 by NAT_4:9; :: thesis:
1597 = (11 * 145) + 2 ;
hence not 11 divides 1597 by NAT_4:9; :: thesis:
1597 = (13 * 122) + 11 ;
hence not 13 divides 1597 by NAT_4:9; :: thesis:
1597 = (17 * 93) + 16 ;
hence not 17 divides 1597 by NAT_4:9; :: thesis:
1597 = (19 * 84) + 1 ;
hence not 19 divides 1597 by NAT_4:9; :: thesis:
1597 = (23 * 69) + 10 ;
hence not 23 divides 1597 by NAT_4:9; :: thesis:
1597 = (29 * 55) + 2 ;
hence not 29 divides 1597 by NAT_4:9; :: thesis:
1597 = (31 * 51) + 16 ;
hence not 31 divides 1597 by NAT_4:9; :: thesis:
1597 = (37 * 43) + 6 ;
hence not 37 divides 1597 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1597 & n is prime holds
not n divides 1597 by XPRIMET1:24;
hence 1597 is prime by NAT_4:14; :: thesis: