7559 = (2 * 3779) + 1 ;
hence not 2 divides 7559 by NAT_4:9; :: thesis:
7559 = (3 * 2519) + 2 ;
hence not 3 divides 7559 by NAT_4:9; :: thesis:
7559 = (5 * 1511) + 4 ;
hence not 5 divides 7559 by NAT_4:9; :: thesis:
7559 = (7 * 1079) + 6 ;
hence not 7 divides 7559 by NAT_4:9; :: thesis:
7559 = (11 * 687) + 2 ;
hence not 11 divides 7559 by NAT_4:9; :: thesis:
7559 = (13 * 581) + 6 ;
hence not 13 divides 7559 by NAT_4:9; :: thesis:
7559 = (17 * 444) + 11 ;
hence not 17 divides 7559 by NAT_4:9; :: thesis:
7559 = (19 * 397) + 16 ;
hence not 19 divides 7559 by NAT_4:9; :: thesis:
7559 = (23 * 328) + 15 ;
hence not 23 divides 7559 by NAT_4:9; :: thesis:
7559 = (29 * 260) + 19 ;
hence not 29 divides 7559 by NAT_4:9; :: thesis:
7559 = (31 * 243) + 26 ;
hence not 31 divides 7559 by NAT_4:9; :: thesis:
7559 = (37 * 204) + 11 ;
hence not 37 divides 7559 by NAT_4:9; :: thesis:
7559 = (41 * 184) + 15 ;
hence not 41 divides 7559 by NAT_4:9; :: thesis:
7559 = (43 * 175) + 34 ;
hence not 43 divides 7559 by NAT_4:9; :: thesis:
7559 = (47 * 160) + 39 ;
hence not 47 divides 7559 by NAT_4:9; :: thesis:
7559 = (53 * 142) + 33 ;
hence not 53 divides 7559 by NAT_4:9; :: thesis:
7559 = (59 * 128) + 7 ;
hence not 59 divides 7559 by NAT_4:9; :: thesis:
7559 = (61 * 123) + 56 ;
hence not 61 divides 7559 by NAT_4:9; :: thesis:
7559 = (67 * 112) + 55 ;
hence not 67 divides 7559 by NAT_4:9; :: thesis:
7559 = (71 * 106) + 33 ;
hence not 71 divides 7559 by NAT_4:9; :: thesis:
7559 = (73 * 103) + 40 ;
hence not 73 divides 7559 by NAT_4:9; :: thesis:
7559 = (79 * 95) + 54 ;
hence not 79 divides 7559 by NAT_4:9; :: thesis:
7559 = (83 * 91) + 6 ;
hence not 83 divides 7559 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7559 & n is prime holds
not n divides 7559 by XPRIMET1:46;
hence 7559 is prime by NAT_4:14; :: thesis: