4517 = (2 * 2258) + 1 ;
hence not 2 divides 4517 by NAT_4:9; :: thesis:
4517 = (3 * 1505) + 2 ;
hence not 3 divides 4517 by NAT_4:9; :: thesis:
4517 = (5 * 903) + 2 ;
hence not 5 divides 4517 by NAT_4:9; :: thesis:
4517 = (7 * 645) + 2 ;
hence not 7 divides 4517 by NAT_4:9; :: thesis:
4517 = (11 * 410) + 7 ;
hence not 11 divides 4517 by NAT_4:9; :: thesis:
4517 = (13 * 347) + 6 ;
hence not 13 divides 4517 by NAT_4:9; :: thesis:
4517 = (17 * 265) + 12 ;
hence not 17 divides 4517 by NAT_4:9; :: thesis:
4517 = (19 * 237) + 14 ;
hence not 19 divides 4517 by NAT_4:9; :: thesis:
4517 = (23 * 196) + 9 ;
hence not 23 divides 4517 by NAT_4:9; :: thesis:
4517 = (29 * 155) + 22 ;
hence not 29 divides 4517 by NAT_4:9; :: thesis:
4517 = (31 * 145) + 22 ;
hence not 31 divides 4517 by NAT_4:9; :: thesis:
4517 = (37 * 122) + 3 ;
hence not 37 divides 4517 by NAT_4:9; :: thesis:
4517 = (41 * 110) + 7 ;
hence not 41 divides 4517 by NAT_4:9; :: thesis:
4517 = (43 * 105) + 2 ;
hence not 43 divides 4517 by NAT_4:9; :: thesis:
4517 = (47 * 96) + 5 ;
hence not 47 divides 4517 by NAT_4:9; :: thesis:
4517 = (53 * 85) + 12 ;
hence not 53 divides 4517 by NAT_4:9; :: thesis:
4517 = (59 * 76) + 33 ;
hence not 59 divides 4517 by NAT_4:9; :: thesis:
4517 = (61 * 74) + 3 ;
hence not 61 divides 4517 by NAT_4:9; :: thesis:
4517 = (67 * 67) + 28 ;
hence not 67 divides 4517 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4517 & n is prime holds
not n divides 4517 by XPRIMET1:38;
hence 4517 is prime by NAT_4:14; :: thesis: