859 = (2 * 429) + 1 ;
hence not 2 divides 859 by NAT_4:9; :: thesis:
859 = (3 * 286) + 1 ;
hence not 3 divides 859 by NAT_4:9; :: thesis:
859 = (5 * 171) + 4 ;
hence not 5 divides 859 by NAT_4:9; :: thesis:
859 = (7 * 122) + 5 ;
hence not 7 divides 859 by NAT_4:9; :: thesis:
859 = (11 * 78) + 1 ;
hence not 11 divides 859 by NAT_4:9; :: thesis:
859 = (13 * 66) + 1 ;
hence not 13 divides 859 by NAT_4:9; :: thesis:
859 = (17 * 50) + 9 ;
hence not 17 divides 859 by NAT_4:9; :: thesis:
859 = (19 * 45) + 4 ;
hence not 19 divides 859 by NAT_4:9; :: thesis:
859 = (23 * 37) + 8 ;
hence not 23 divides 859 by NAT_4:9; :: thesis:
859 = (29 * 29) + 18 ;
hence not 29 divides 859 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 859 & n is prime holds
not n divides 859 by XPRIMET1:20;
hence 859 is prime by NAT_4:14; :: thesis: