2843 = (2 * 1421) + 1 ;
hence not 2 divides 2843 by NAT_4:9; :: thesis:
2843 = (3 * 947) + 2 ;
hence not 3 divides 2843 by NAT_4:9; :: thesis:
2843 = (5 * 568) + 3 ;
hence not 5 divides 2843 by NAT_4:9; :: thesis:
2843 = (7 * 406) + 1 ;
hence not 7 divides 2843 by NAT_4:9; :: thesis:
2843 = (11 * 258) + 5 ;
hence not 11 divides 2843 by NAT_4:9; :: thesis:
2843 = (13 * 218) + 9 ;
hence not 13 divides 2843 by NAT_4:9; :: thesis:
2843 = (17 * 167) + 4 ;
hence not 17 divides 2843 by NAT_4:9; :: thesis:
2843 = (19 * 149) + 12 ;
hence not 19 divides 2843 by NAT_4:9; :: thesis:
2843 = (23 * 123) + 14 ;
hence not 23 divides 2843 by NAT_4:9; :: thesis:
2843 = (29 * 98) + 1 ;
hence not 29 divides 2843 by NAT_4:9; :: thesis:
2843 = (31 * 91) + 22 ;
hence not 31 divides 2843 by NAT_4:9; :: thesis:
2843 = (37 * 76) + 31 ;
hence not 37 divides 2843 by NAT_4:9; :: thesis:
2843 = (41 * 69) + 14 ;
hence not 41 divides 2843 by NAT_4:9; :: thesis:
2843 = (43 * 66) + 5 ;
hence not 43 divides 2843 by NAT_4:9; :: thesis:
2843 = (47 * 60) + 23 ;
hence not 47 divides 2843 by NAT_4:9; :: thesis:
2843 = (53 * 53) + 34 ;
hence not 53 divides 2843 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2843 & n is prime holds
not n divides 2843 by XPRIMET1:32;
hence 2843 is prime by NAT_4:14; :: thesis: