719 = (2 * 359) + 1 ;
hence not 2 divides 719 by NAT_4:9; :: thesis:
719 = (3 * 239) + 2 ;
hence not 3 divides 719 by NAT_4:9; :: thesis:
719 = (5 * 143) + 4 ;
hence not 5 divides 719 by NAT_4:9; :: thesis:
719 = (7 * 102) + 5 ;
hence not 7 divides 719 by NAT_4:9; :: thesis:
719 = (11 * 65) + 4 ;
hence not 11 divides 719 by NAT_4:9; :: thesis:
719 = (13 * 55) + 4 ;
hence not 13 divides 719 by NAT_4:9; :: thesis:
719 = (17 * 42) + 5 ;
hence not 17 divides 719 by NAT_4:9; :: thesis:
719 = (19 * 37) + 16 ;
hence not 19 divides 719 by NAT_4:9; :: thesis:
719 = (23 * 31) + 6 ;
hence not 23 divides 719 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 719 & n is prime holds
not n divides 719 by XPRIMET1:18;
hence 719 is prime by NAT_4:14; :: thesis: