1319 = (2 * 659) + 1 ;
hence not 2 divides 1319 by NAT_4:9; :: thesis:
1319 = (3 * 439) + 2 ;
hence not 3 divides 1319 by NAT_4:9; :: thesis:
1319 = (5 * 263) + 4 ;
hence not 5 divides 1319 by NAT_4:9; :: thesis:
1319 = (7 * 188) + 3 ;
hence not 7 divides 1319 by NAT_4:9; :: thesis:
1319 = (11 * 119) + 10 ;
hence not 11 divides 1319 by NAT_4:9; :: thesis:
1319 = (13 * 101) + 6 ;
hence not 13 divides 1319 by NAT_4:9; :: thesis:
1319 = (17 * 77) + 10 ;
hence not 17 divides 1319 by NAT_4:9; :: thesis:
1319 = (19 * 69) + 8 ;
hence not 19 divides 1319 by NAT_4:9; :: thesis:
1319 = (23 * 57) + 8 ;
hence not 23 divides 1319 by NAT_4:9; :: thesis:
1319 = (29 * 45) + 14 ;
hence not 29 divides 1319 by NAT_4:9; :: thesis:
1319 = (31 * 42) + 17 ;
hence not 31 divides 1319 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1319 & n is prime holds
not n divides 1319 by XPRIMET1:22;
hence 1319 is prime by NAT_4:14; :: thesis: