6257 = (2 * 3128) + 1 ;
hence not 2 divides 6257 by NAT_4:9; :: thesis:
6257 = (3 * 2085) + 2 ;
hence not 3 divides 6257 by NAT_4:9; :: thesis:
6257 = (5 * 1251) + 2 ;
hence not 5 divides 6257 by NAT_4:9; :: thesis:
6257 = (7 * 893) + 6 ;
hence not 7 divides 6257 by NAT_4:9; :: thesis:
6257 = (11 * 568) + 9 ;
hence not 11 divides 6257 by NAT_4:9; :: thesis:
6257 = (13 * 481) + 4 ;
hence not 13 divides 6257 by NAT_4:9; :: thesis:
6257 = (17 * 368) + 1 ;
hence not 17 divides 6257 by NAT_4:9; :: thesis:
6257 = (19 * 329) + 6 ;
hence not 19 divides 6257 by NAT_4:9; :: thesis:
6257 = (23 * 272) + 1 ;
hence not 23 divides 6257 by NAT_4:9; :: thesis:
6257 = (29 * 215) + 22 ;
hence not 29 divides 6257 by NAT_4:9; :: thesis:
6257 = (31 * 201) + 26 ;
hence not 31 divides 6257 by NAT_4:9; :: thesis:
6257 = (37 * 169) + 4 ;
hence not 37 divides 6257 by NAT_4:9; :: thesis:
6257 = (41 * 152) + 25 ;
hence not 41 divides 6257 by NAT_4:9; :: thesis:
6257 = (43 * 145) + 22 ;
hence not 43 divides 6257 by NAT_4:9; :: thesis:
6257 = (47 * 133) + 6 ;
hence not 47 divides 6257 by NAT_4:9; :: thesis:
6257 = (53 * 118) + 3 ;
hence not 53 divides 6257 by NAT_4:9; :: thesis:
6257 = (59 * 106) + 3 ;
hence not 59 divides 6257 by NAT_4:9; :: thesis:
6257 = (61 * 102) + 35 ;
hence not 61 divides 6257 by NAT_4:9; :: thesis:
6257 = (67 * 93) + 26 ;
hence not 67 divides 6257 by NAT_4:9; :: thesis:
6257 = (71 * 88) + 9 ;
hence not 71 divides 6257 by NAT_4:9; :: thesis:
6257 = (73 * 85) + 52 ;
hence not 73 divides 6257 by NAT_4:9; :: thesis:
6257 = (79 * 79) + 16 ;
hence not 79 divides 6257 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6257 & n is prime holds
not n divides 6257 by XPRIMET1:44;
hence 6257 is prime by NAT_4:14; :: thesis: