8741 = (2 * 4370) + 1 ;
hence not 2 divides 8741 by NAT_4:9; :: thesis:
8741 = (3 * 2913) + 2 ;
hence not 3 divides 8741 by NAT_4:9; :: thesis:
8741 = (5 * 1748) + 1 ;
hence not 5 divides 8741 by NAT_4:9; :: thesis:
8741 = (7 * 1248) + 5 ;
hence not 7 divides 8741 by NAT_4:9; :: thesis:
8741 = (11 * 794) + 7 ;
hence not 11 divides 8741 by NAT_4:9; :: thesis:
8741 = (13 * 672) + 5 ;
hence not 13 divides 8741 by NAT_4:9; :: thesis:
8741 = (17 * 514) + 3 ;
hence not 17 divides 8741 by NAT_4:9; :: thesis:
8741 = (19 * 460) + 1 ;
hence not 19 divides 8741 by NAT_4:9; :: thesis:
8741 = (23 * 380) + 1 ;
hence not 23 divides 8741 by NAT_4:9; :: thesis:
8741 = (29 * 301) + 12 ;
hence not 29 divides 8741 by NAT_4:9; :: thesis:
8741 = (31 * 281) + 30 ;
hence not 31 divides 8741 by NAT_4:9; :: thesis:
8741 = (37 * 236) + 9 ;
hence not 37 divides 8741 by NAT_4:9; :: thesis:
8741 = (41 * 213) + 8 ;
hence not 41 divides 8741 by NAT_4:9; :: thesis:
8741 = (43 * 203) + 12 ;
hence not 43 divides 8741 by NAT_4:9; :: thesis:
8741 = (47 * 185) + 46 ;
hence not 47 divides 8741 by NAT_4:9; :: thesis:
8741 = (53 * 164) + 49 ;
hence not 53 divides 8741 by NAT_4:9; :: thesis:
8741 = (59 * 148) + 9 ;
hence not 59 divides 8741 by NAT_4:9; :: thesis:
8741 = (61 * 143) + 18 ;
hence not 61 divides 8741 by NAT_4:9; :: thesis:
8741 = (67 * 130) + 31 ;
hence not 67 divides 8741 by NAT_4:9; :: thesis:
8741 = (71 * 123) + 8 ;
hence not 71 divides 8741 by NAT_4:9; :: thesis:
8741 = (73 * 119) + 54 ;
hence not 73 divides 8741 by NAT_4:9; :: thesis:
8741 = (79 * 110) + 51 ;
hence not 79 divides 8741 by NAT_4:9; :: thesis:
8741 = (83 * 105) + 26 ;
hence not 83 divides 8741 by NAT_4:9; :: thesis:
8741 = (89 * 98) + 19 ;
hence not 89 divides 8741 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8741 & n is prime holds
not n divides 8741 by XPRIMET1:48;
hence 8741 is prime by NAT_4:14; :: thesis: