8887 = (2 * 4443) + 1 ;
hence not 2 divides 8887 by NAT_4:9; :: thesis:
8887 = (3 * 2962) + 1 ;
hence not 3 divides 8887 by NAT_4:9; :: thesis:
8887 = (5 * 1777) + 2 ;
hence not 5 divides 8887 by NAT_4:9; :: thesis:
8887 = (7 * 1269) + 4 ;
hence not 7 divides 8887 by NAT_4:9; :: thesis:
8887 = (11 * 807) + 10 ;
hence not 11 divides 8887 by NAT_4:9; :: thesis:
8887 = (13 * 683) + 8 ;
hence not 13 divides 8887 by NAT_4:9; :: thesis:
8887 = (17 * 522) + 13 ;
hence not 17 divides 8887 by NAT_4:9; :: thesis:
8887 = (19 * 467) + 14 ;
hence not 19 divides 8887 by NAT_4:9; :: thesis:
8887 = (23 * 386) + 9 ;
hence not 23 divides 8887 by NAT_4:9; :: thesis:
8887 = (29 * 306) + 13 ;
hence not 29 divides 8887 by NAT_4:9; :: thesis:
8887 = (31 * 286) + 21 ;
hence not 31 divides 8887 by NAT_4:9; :: thesis:
8887 = (37 * 240) + 7 ;
hence not 37 divides 8887 by NAT_4:9; :: thesis:
8887 = (41 * 216) + 31 ;
hence not 41 divides 8887 by NAT_4:9; :: thesis:
8887 = (43 * 206) + 29 ;
hence not 43 divides 8887 by NAT_4:9; :: thesis:
8887 = (47 * 189) + 4 ;
hence not 47 divides 8887 by NAT_4:9; :: thesis:
8887 = (53 * 167) + 36 ;
hence not 53 divides 8887 by NAT_4:9; :: thesis:
8887 = (59 * 150) + 37 ;
hence not 59 divides 8887 by NAT_4:9; :: thesis:
8887 = (61 * 145) + 42 ;
hence not 61 divides 8887 by NAT_4:9; :: thesis:
8887 = (67 * 132) + 43 ;
hence not 67 divides 8887 by NAT_4:9; :: thesis:
8887 = (71 * 125) + 12 ;
hence not 71 divides 8887 by NAT_4:9; :: thesis:
8887 = (73 * 121) + 54 ;
hence not 73 divides 8887 by NAT_4:9; :: thesis:
8887 = (79 * 112) + 39 ;
hence not 79 divides 8887 by NAT_4:9; :: thesis:
8887 = (83 * 107) + 6 ;
hence not 83 divides 8887 by NAT_4:9; :: thesis:
8887 = (89 * 99) + 76 ;
hence not 89 divides 8887 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8887 & n is prime holds
not n divides 8887 by XPRIMET1:48;
hence 8887 is prime by NAT_4:14; :: thesis: