1307 = (2 * 653) + 1 ;
hence not 2 divides 1307 by NAT_4:9; :: thesis:
1307 = (3 * 435) + 2 ;
hence not 3 divides 1307 by NAT_4:9; :: thesis:
1307 = (5 * 261) + 2 ;
hence not 5 divides 1307 by NAT_4:9; :: thesis:
1307 = (7 * 186) + 5 ;
hence not 7 divides 1307 by NAT_4:9; :: thesis:
1307 = (11 * 118) + 9 ;
hence not 11 divides 1307 by NAT_4:9; :: thesis:
1307 = (13 * 100) + 7 ;
hence not 13 divides 1307 by NAT_4:9; :: thesis:
1307 = (17 * 76) + 15 ;
hence not 17 divides 1307 by NAT_4:9; :: thesis:
1307 = (19 * 68) + 15 ;
hence not 19 divides 1307 by NAT_4:9; :: thesis:
1307 = (23 * 56) + 19 ;
hence not 23 divides 1307 by NAT_4:9; :: thesis:
1307 = (29 * 45) + 2 ;
hence not 29 divides 1307 by NAT_4:9; :: thesis:
1307 = (31 * 42) + 5 ;
hence not 31 divides 1307 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1307 & n is prime holds
not n divides 1307 by XPRIMET1:22;
hence 1307 is prime by NAT_4:14; :: thesis: