2851 = (2 * 1425) + 1 ;
hence not 2 divides 2851 by NAT_4:9; :: thesis:
2851 = (3 * 950) + 1 ;
hence not 3 divides 2851 by NAT_4:9; :: thesis:
2851 = (5 * 570) + 1 ;
hence not 5 divides 2851 by NAT_4:9; :: thesis:
2851 = (7 * 407) + 2 ;
hence not 7 divides 2851 by NAT_4:9; :: thesis:
2851 = (11 * 259) + 2 ;
hence not 11 divides 2851 by NAT_4:9; :: thesis:
2851 = (13 * 219) + 4 ;
hence not 13 divides 2851 by NAT_4:9; :: thesis:
2851 = (17 * 167) + 12 ;
hence not 17 divides 2851 by NAT_4:9; :: thesis:
2851 = (19 * 150) + 1 ;
hence not 19 divides 2851 by NAT_4:9; :: thesis:
2851 = (23 * 123) + 22 ;
hence not 23 divides 2851 by NAT_4:9; :: thesis:
2851 = (29 * 98) + 9 ;
hence not 29 divides 2851 by NAT_4:9; :: thesis:
2851 = (31 * 91) + 30 ;
hence not 31 divides 2851 by NAT_4:9; :: thesis:
2851 = (37 * 77) + 2 ;
hence not 37 divides 2851 by NAT_4:9; :: thesis:
2851 = (41 * 69) + 22 ;
hence not 41 divides 2851 by NAT_4:9; :: thesis:
2851 = (43 * 66) + 13 ;
hence not 43 divides 2851 by NAT_4:9; :: thesis:
2851 = (47 * 60) + 31 ;
hence not 47 divides 2851 by NAT_4:9; :: thesis:
2851 = (53 * 53) + 42 ;
hence not 53 divides 2851 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2851 & n is prime holds
not n divides 2851 by XPRIMET1:32;
hence 2851 is prime by NAT_4:14; :: thesis: