8737 = (2 * 4368) + 1 ;
hence not 2 divides 8737 by NAT_4:9; :: thesis:
8737 = (3 * 2912) + 1 ;
hence not 3 divides 8737 by NAT_4:9; :: thesis:
8737 = (5 * 1747) + 2 ;
hence not 5 divides 8737 by NAT_4:9; :: thesis:
8737 = (7 * 1248) + 1 ;
hence not 7 divides 8737 by NAT_4:9; :: thesis:
8737 = (11 * 794) + 3 ;
hence not 11 divides 8737 by NAT_4:9; :: thesis:
8737 = (13 * 672) + 1 ;
hence not 13 divides 8737 by NAT_4:9; :: thesis:
8737 = (17 * 513) + 16 ;
hence not 17 divides 8737 by NAT_4:9; :: thesis:
8737 = (19 * 459) + 16 ;
hence not 19 divides 8737 by NAT_4:9; :: thesis:
8737 = (23 * 379) + 20 ;
hence not 23 divides 8737 by NAT_4:9; :: thesis:
8737 = (29 * 301) + 8 ;
hence not 29 divides 8737 by NAT_4:9; :: thesis:
8737 = (31 * 281) + 26 ;
hence not 31 divides 8737 by NAT_4:9; :: thesis:
8737 = (37 * 236) + 5 ;
hence not 37 divides 8737 by NAT_4:9; :: thesis:
8737 = (41 * 213) + 4 ;
hence not 41 divides 8737 by NAT_4:9; :: thesis:
8737 = (43 * 203) + 8 ;
hence not 43 divides 8737 by NAT_4:9; :: thesis:
8737 = (47 * 185) + 42 ;
hence not 47 divides 8737 by NAT_4:9; :: thesis:
8737 = (53 * 164) + 45 ;
hence not 53 divides 8737 by NAT_4:9; :: thesis:
8737 = (59 * 148) + 5 ;
hence not 59 divides 8737 by NAT_4:9; :: thesis:
8737 = (61 * 143) + 14 ;
hence not 61 divides 8737 by NAT_4:9; :: thesis:
8737 = (67 * 130) + 27 ;
hence not 67 divides 8737 by NAT_4:9; :: thesis:
8737 = (71 * 123) + 4 ;
hence not 71 divides 8737 by NAT_4:9; :: thesis:
8737 = (73 * 119) + 50 ;
hence not 73 divides 8737 by NAT_4:9; :: thesis:
8737 = (79 * 110) + 47 ;
hence not 79 divides 8737 by NAT_4:9; :: thesis:
8737 = (83 * 105) + 22 ;
hence not 83 divides 8737 by NAT_4:9; :: thesis:
8737 = (89 * 98) + 15 ;
hence not 89 divides 8737 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8737 & n is prime holds
not n divides 8737 by XPRIMET1:48;
hence 8737 is prime by NAT_4:14; :: thesis: