857 = (2 * 428) + 1 ;
hence not 2 divides 857 by NAT_4:9; :: thesis:
857 = (3 * 285) + 2 ;
hence not 3 divides 857 by NAT_4:9; :: thesis:
857 = (5 * 171) + 2 ;
hence not 5 divides 857 by NAT_4:9; :: thesis:
857 = (7 * 122) + 3 ;
hence not 7 divides 857 by NAT_4:9; :: thesis:
857 = (11 * 77) + 10 ;
hence not 11 divides 857 by NAT_4:9; :: thesis:
857 = (13 * 65) + 12 ;
hence not 13 divides 857 by NAT_4:9; :: thesis:
857 = (17 * 50) + 7 ;
hence not 17 divides 857 by NAT_4:9; :: thesis:
857 = (19 * 45) + 2 ;
hence not 19 divides 857 by NAT_4:9; :: thesis:
857 = (23 * 37) + 6 ;
hence not 23 divides 857 by NAT_4:9; :: thesis:
857 = (29 * 29) + 16 ;
hence not 29 divides 857 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 857 & n is prime holds
not n divides 857 by XPRIMET1:20;
hence 857 is prime by NAT_4:14; :: thesis: