853 = (2 * 426) + 1 ;
hence not 2 divides 853 by NAT_4:9; :: thesis:
853 = (3 * 284) + 1 ;
hence not 3 divides 853 by NAT_4:9; :: thesis:
853 = (5 * 170) + 3 ;
hence not 5 divides 853 by NAT_4:9; :: thesis:
853 = (7 * 121) + 6 ;
hence not 7 divides 853 by NAT_4:9; :: thesis:
853 = (11 * 77) + 6 ;
hence not 11 divides 853 by NAT_4:9; :: thesis:
853 = (13 * 65) + 8 ;
hence not 13 divides 853 by NAT_4:9; :: thesis:
853 = (17 * 50) + 3 ;
hence not 17 divides 853 by NAT_4:9; :: thesis:
853 = (19 * 44) + 17 ;
hence not 19 divides 853 by NAT_4:9; :: thesis:
853 = (23 * 37) + 2 ;
hence not 23 divides 853 by NAT_4:9; :: thesis:
853 = (29 * 29) + 12 ;
hence not 29 divides 853 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 853 & n is prime holds
not n divides 853 by XPRIMET1:20;
hence 853 is prime by NAT_4:14; :: thesis: