8761 = (2 * 4380) + 1 ;
hence not 2 divides 8761 by NAT_4:9; :: thesis:
8761 = (3 * 2920) + 1 ;
hence not 3 divides 8761 by NAT_4:9; :: thesis:
8761 = (5 * 1752) + 1 ;
hence not 5 divides 8761 by NAT_4:9; :: thesis:
8761 = (7 * 1251) + 4 ;
hence not 7 divides 8761 by NAT_4:9; :: thesis:
8761 = (11 * 796) + 5 ;
hence not 11 divides 8761 by NAT_4:9; :: thesis:
8761 = (13 * 673) + 12 ;
hence not 13 divides 8761 by NAT_4:9; :: thesis:
8761 = (17 * 515) + 6 ;
hence not 17 divides 8761 by NAT_4:9; :: thesis:
8761 = (19 * 461) + 2 ;
hence not 19 divides 8761 by NAT_4:9; :: thesis:
8761 = (23 * 380) + 21 ;
hence not 23 divides 8761 by NAT_4:9; :: thesis:
8761 = (29 * 302) + 3 ;
hence not 29 divides 8761 by NAT_4:9; :: thesis:
8761 = (31 * 282) + 19 ;
hence not 31 divides 8761 by NAT_4:9; :: thesis:
8761 = (37 * 236) + 29 ;
hence not 37 divides 8761 by NAT_4:9; :: thesis:
8761 = (41 * 213) + 28 ;
hence not 41 divides 8761 by NAT_4:9; :: thesis:
8761 = (43 * 203) + 32 ;
hence not 43 divides 8761 by NAT_4:9; :: thesis:
8761 = (47 * 186) + 19 ;
hence not 47 divides 8761 by NAT_4:9; :: thesis:
8761 = (53 * 165) + 16 ;
hence not 53 divides 8761 by NAT_4:9; :: thesis:
8761 = (59 * 148) + 29 ;
hence not 59 divides 8761 by NAT_4:9; :: thesis:
8761 = (61 * 143) + 38 ;
hence not 61 divides 8761 by NAT_4:9; :: thesis:
8761 = (67 * 130) + 51 ;
hence not 67 divides 8761 by NAT_4:9; :: thesis:
8761 = (71 * 123) + 28 ;
hence not 71 divides 8761 by NAT_4:9; :: thesis:
8761 = (73 * 120) + 1 ;
hence not 73 divides 8761 by NAT_4:9; :: thesis:
8761 = (79 * 110) + 71 ;
hence not 79 divides 8761 by NAT_4:9; :: thesis:
8761 = (83 * 105) + 46 ;
hence not 83 divides 8761 by NAT_4:9; :: thesis:
8761 = (89 * 98) + 39 ;
hence not 89 divides 8761 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8761 & n is prime holds
not n divides 8761 by XPRIMET1:48;
hence 8761 is prime by NAT_4:14; :: thesis: