6353 = (2 * 3176) + 1 ;
hence not 2 divides 6353 by NAT_4:9; :: thesis:
6353 = (3 * 2117) + 2 ;
hence not 3 divides 6353 by NAT_4:9; :: thesis:
6353 = (5 * 1270) + 3 ;
hence not 5 divides 6353 by NAT_4:9; :: thesis:
6353 = (7 * 907) + 4 ;
hence not 7 divides 6353 by NAT_4:9; :: thesis:
6353 = (11 * 577) + 6 ;
hence not 11 divides 6353 by NAT_4:9; :: thesis:
6353 = (13 * 488) + 9 ;
hence not 13 divides 6353 by NAT_4:9; :: thesis:
6353 = (17 * 373) + 12 ;
hence not 17 divides 6353 by NAT_4:9; :: thesis:
6353 = (19 * 334) + 7 ;
hence not 19 divides 6353 by NAT_4:9; :: thesis:
6353 = (23 * 276) + 5 ;
hence not 23 divides 6353 by NAT_4:9; :: thesis:
6353 = (29 * 219) + 2 ;
hence not 29 divides 6353 by NAT_4:9; :: thesis:
6353 = (31 * 204) + 29 ;
hence not 31 divides 6353 by NAT_4:9; :: thesis:
6353 = (37 * 171) + 26 ;
hence not 37 divides 6353 by NAT_4:9; :: thesis:
6353 = (41 * 154) + 39 ;
hence not 41 divides 6353 by NAT_4:9; :: thesis:
6353 = (43 * 147) + 32 ;
hence not 43 divides 6353 by NAT_4:9; :: thesis:
6353 = (47 * 135) + 8 ;
hence not 47 divides 6353 by NAT_4:9; :: thesis:
6353 = (53 * 119) + 46 ;
hence not 53 divides 6353 by NAT_4:9; :: thesis:
6353 = (59 * 107) + 40 ;
hence not 59 divides 6353 by NAT_4:9; :: thesis:
6353 = (61 * 104) + 9 ;
hence not 61 divides 6353 by NAT_4:9; :: thesis:
6353 = (67 * 94) + 55 ;
hence not 67 divides 6353 by NAT_4:9; :: thesis:
6353 = (71 * 89) + 34 ;
hence not 71 divides 6353 by NAT_4:9; :: thesis:
6353 = (73 * 87) + 2 ;
hence not 73 divides 6353 by NAT_4:9; :: thesis:
6353 = (79 * 80) + 33 ;
hence not 79 divides 6353 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6353 & n is prime holds
not n divides 6353 by XPRIMET1:44;
hence 6353 is prime by NAT_4:14; :: thesis: