1297 = (2 * 648) + 1 ;
hence not 2 divides 1297 by NAT_4:9; :: thesis:
1297 = (3 * 432) + 1 ;
hence not 3 divides 1297 by NAT_4:9; :: thesis:
1297 = (5 * 259) + 2 ;
hence not 5 divides 1297 by NAT_4:9; :: thesis:
1297 = (7 * 185) + 2 ;
hence not 7 divides 1297 by NAT_4:9; :: thesis:
1297 = (11 * 117) + 10 ;
hence not 11 divides 1297 by NAT_4:9; :: thesis:
1297 = (13 * 99) + 10 ;
hence not 13 divides 1297 by NAT_4:9; :: thesis:
1297 = (17 * 76) + 5 ;
hence not 17 divides 1297 by NAT_4:9; :: thesis:
1297 = (19 * 68) + 5 ;
hence not 19 divides 1297 by NAT_4:9; :: thesis:
1297 = (23 * 56) + 9 ;
hence not 23 divides 1297 by NAT_4:9; :: thesis:
1297 = (29 * 44) + 21 ;
hence not 29 divides 1297 by NAT_4:9; :: thesis:
1297 = (31 * 41) + 26 ;
hence not 31 divides 1297 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1297 & n is prime holds
not n divides 1297 by XPRIMET1:22;
hence 1297 is prime by NAT_4:14; :: thesis: