5717 = (2 * 2858) + 1 ;
hence not 2 divides 5717 by NAT_4:9; :: thesis:
5717 = (3 * 1905) + 2 ;
hence not 3 divides 5717 by NAT_4:9; :: thesis:
5717 = (5 * 1143) + 2 ;
hence not 5 divides 5717 by NAT_4:9; :: thesis:
5717 = (7 * 816) + 5 ;
hence not 7 divides 5717 by NAT_4:9; :: thesis:
5717 = (11 * 519) + 8 ;
hence not 11 divides 5717 by NAT_4:9; :: thesis:
5717 = (13 * 439) + 10 ;
hence not 13 divides 5717 by NAT_4:9; :: thesis:
5717 = (17 * 336) + 5 ;
hence not 17 divides 5717 by NAT_4:9; :: thesis:
5717 = (19 * 300) + 17 ;
hence not 19 divides 5717 by NAT_4:9; :: thesis:
5717 = (23 * 248) + 13 ;
hence not 23 divides 5717 by NAT_4:9; :: thesis:
5717 = (29 * 197) + 4 ;
hence not 29 divides 5717 by NAT_4:9; :: thesis:
5717 = (31 * 184) + 13 ;
hence not 31 divides 5717 by NAT_4:9; :: thesis:
5717 = (37 * 154) + 19 ;
hence not 37 divides 5717 by NAT_4:9; :: thesis:
5717 = (41 * 139) + 18 ;
hence not 41 divides 5717 by NAT_4:9; :: thesis:
5717 = (43 * 132) + 41 ;
hence not 43 divides 5717 by NAT_4:9; :: thesis:
5717 = (47 * 121) + 30 ;
hence not 47 divides 5717 by NAT_4:9; :: thesis:
5717 = (53 * 107) + 46 ;
hence not 53 divides 5717 by NAT_4:9; :: thesis:
5717 = (59 * 96) + 53 ;
hence not 59 divides 5717 by NAT_4:9; :: thesis:
5717 = (61 * 93) + 44 ;
hence not 61 divides 5717 by NAT_4:9; :: thesis:
5717 = (67 * 85) + 22 ;
hence not 67 divides 5717 by NAT_4:9; :: thesis:
5717 = (71 * 80) + 37 ;
hence not 71 divides 5717 by NAT_4:9; :: thesis:
5717 = (73 * 78) + 23 ;
hence not 73 divides 5717 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5717 & n is prime holds
not n divides 5717 by XPRIMET1:42;
hence 5717 is prime by NAT_4:14; :: thesis: