3617 = (2 * 1808) + 1 ;
hence not 2 divides 3617 by NAT_4:9; :: thesis:
3617 = (3 * 1205) + 2 ;
hence not 3 divides 3617 by NAT_4:9; :: thesis:
3617 = (5 * 723) + 2 ;
hence not 5 divides 3617 by NAT_4:9; :: thesis:
3617 = (7 * 516) + 5 ;
hence not 7 divides 3617 by NAT_4:9; :: thesis:
3617 = (11 * 328) + 9 ;
hence not 11 divides 3617 by NAT_4:9; :: thesis:
3617 = (13 * 278) + 3 ;
hence not 13 divides 3617 by NAT_4:9; :: thesis:
3617 = (17 * 212) + 13 ;
hence not 17 divides 3617 by NAT_4:9; :: thesis:
3617 = (19 * 190) + 7 ;
hence not 19 divides 3617 by NAT_4:9; :: thesis:
3617 = (23 * 157) + 6 ;
hence not 23 divides 3617 by NAT_4:9; :: thesis:
3617 = (29 * 124) + 21 ;
hence not 29 divides 3617 by NAT_4:9; :: thesis:
3617 = (31 * 116) + 21 ;
hence not 31 divides 3617 by NAT_4:9; :: thesis:
3617 = (37 * 97) + 28 ;
hence not 37 divides 3617 by NAT_4:9; :: thesis:
3617 = (41 * 88) + 9 ;
hence not 41 divides 3617 by NAT_4:9; :: thesis:
3617 = (43 * 84) + 5 ;
hence not 43 divides 3617 by NAT_4:9; :: thesis:
3617 = (47 * 76) + 45 ;
hence not 47 divides 3617 by NAT_4:9; :: thesis:
3617 = (53 * 68) + 13 ;
hence not 53 divides 3617 by NAT_4:9; :: thesis:
3617 = (59 * 61) + 18 ;
hence not 59 divides 3617 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3617 & n is prime holds
not n divides 3617 by XPRIMET1:34;
hence 3617 is prime by NAT_4:14; :: thesis: