919 = (2 * 459) + 1 ;
hence not 2 divides 919 by NAT_4:9; :: thesis:
919 = (3 * 306) + 1 ;
hence not 3 divides 919 by NAT_4:9; :: thesis:
919 = (5 * 183) + 4 ;
hence not 5 divides 919 by NAT_4:9; :: thesis:
919 = (7 * 131) + 2 ;
hence not 7 divides 919 by NAT_4:9; :: thesis:
919 = (11 * 83) + 6 ;
hence not 11 divides 919 by NAT_4:9; :: thesis:
919 = (13 * 70) + 9 ;
hence not 13 divides 919 by NAT_4:9; :: thesis:
919 = (17 * 54) + 1 ;
hence not 17 divides 919 by NAT_4:9; :: thesis:
919 = (19 * 48) + 7 ;
hence not 19 divides 919 by NAT_4:9; :: thesis:
919 = (23 * 39) + 22 ;
hence not 23 divides 919 by NAT_4:9; :: thesis:
919 = (29 * 31) + 20 ;
hence not 29 divides 919 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 919 & n is prime holds
not n divides 919 by XPRIMET1:20;
hence 919 is prime by NAT_4:14; :: thesis: