1607 = (2 * 803) + 1 ;
hence not 2 divides 1607 by NAT_4:9; :: thesis:
1607 = (3 * 535) + 2 ;
hence not 3 divides 1607 by NAT_4:9; :: thesis:
1607 = (5 * 321) + 2 ;
hence not 5 divides 1607 by NAT_4:9; :: thesis:
1607 = (7 * 229) + 4 ;
hence not 7 divides 1607 by NAT_4:9; :: thesis:
1607 = (11 * 146) + 1 ;
hence not 11 divides 1607 by NAT_4:9; :: thesis:
1607 = (13 * 123) + 8 ;
hence not 13 divides 1607 by NAT_4:9; :: thesis:
1607 = (17 * 94) + 9 ;
hence not 17 divides 1607 by NAT_4:9; :: thesis:
1607 = (19 * 84) + 11 ;
hence not 19 divides 1607 by NAT_4:9; :: thesis:
1607 = (23 * 69) + 20 ;
hence not 23 divides 1607 by NAT_4:9; :: thesis:
1607 = (29 * 55) + 12 ;
hence not 29 divides 1607 by NAT_4:9; :: thesis:
1607 = (31 * 51) + 26 ;
hence not 31 divides 1607 by NAT_4:9; :: thesis:
1607 = (37 * 43) + 16 ;
hence not 37 divides 1607 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1607 & n is prime holds
not n divides 1607 by XPRIMET1:24;
hence 1607 is prime by NAT_4:14; :: thesis: