1259 = (2 * 629) + 1 ;
hence not 2 divides 1259 by NAT_4:9; :: thesis:
1259 = (3 * 419) + 2 ;
hence not 3 divides 1259 by NAT_4:9; :: thesis:
1259 = (5 * 251) + 4 ;
hence not 5 divides 1259 by NAT_4:9; :: thesis:
1259 = (7 * 179) + 6 ;
hence not 7 divides 1259 by NAT_4:9; :: thesis:
1259 = (11 * 114) + 5 ;
hence not 11 divides 1259 by NAT_4:9; :: thesis:
1259 = (13 * 96) + 11 ;
hence not 13 divides 1259 by NAT_4:9; :: thesis:
1259 = (17 * 74) + 1 ;
hence not 17 divides 1259 by NAT_4:9; :: thesis:
1259 = (19 * 66) + 5 ;
hence not 19 divides 1259 by NAT_4:9; :: thesis:
1259 = (23 * 54) + 17 ;
hence not 23 divides 1259 by NAT_4:9; :: thesis:
1259 = (29 * 43) + 12 ;
hence not 29 divides 1259 by NAT_4:9; :: thesis:
1259 = (31 * 40) + 19 ;
hence not 31 divides 1259 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1259 & n is prime holds
not n divides 1259 by XPRIMET1:22;
hence 1259 is prime by NAT_4:14; :: thesis: