6361 = (2 * 3180) + 1 ;
hence not 2 divides 6361 by NAT_4:9; :: thesis:
6361 = (3 * 2120) + 1 ;
hence not 3 divides 6361 by NAT_4:9; :: thesis:
6361 = (5 * 1272) + 1 ;
hence not 5 divides 6361 by NAT_4:9; :: thesis:
6361 = (7 * 908) + 5 ;
hence not 7 divides 6361 by NAT_4:9; :: thesis:
6361 = (11 * 578) + 3 ;
hence not 11 divides 6361 by NAT_4:9; :: thesis:
6361 = (13 * 489) + 4 ;
hence not 13 divides 6361 by NAT_4:9; :: thesis:
6361 = (17 * 374) + 3 ;
hence not 17 divides 6361 by NAT_4:9; :: thesis:
6361 = (19 * 334) + 15 ;
hence not 19 divides 6361 by NAT_4:9; :: thesis:
6361 = (23 * 276) + 13 ;
hence not 23 divides 6361 by NAT_4:9; :: thesis:
6361 = (29 * 219) + 10 ;
hence not 29 divides 6361 by NAT_4:9; :: thesis:
6361 = (31 * 205) + 6 ;
hence not 31 divides 6361 by NAT_4:9; :: thesis:
6361 = (37 * 171) + 34 ;
hence not 37 divides 6361 by NAT_4:9; :: thesis:
6361 = (41 * 155) + 6 ;
hence not 41 divides 6361 by NAT_4:9; :: thesis:
6361 = (43 * 147) + 40 ;
hence not 43 divides 6361 by NAT_4:9; :: thesis:
6361 = (47 * 135) + 16 ;
hence not 47 divides 6361 by NAT_4:9; :: thesis:
6361 = (53 * 120) + 1 ;
hence not 53 divides 6361 by NAT_4:9; :: thesis:
6361 = (59 * 107) + 48 ;
hence not 59 divides 6361 by NAT_4:9; :: thesis:
6361 = (61 * 104) + 17 ;
hence not 61 divides 6361 by NAT_4:9; :: thesis:
6361 = (67 * 94) + 63 ;
hence not 67 divides 6361 by NAT_4:9; :: thesis:
6361 = (71 * 89) + 42 ;
hence not 71 divides 6361 by NAT_4:9; :: thesis:
6361 = (73 * 87) + 10 ;
hence not 73 divides 6361 by NAT_4:9; :: thesis:
6361 = (79 * 80) + 41 ;
hence not 79 divides 6361 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6361 & n is prime holds
not n divides 6361 by XPRIMET1:44;
hence 6361 is prime by NAT_4:14; :: thesis: