6221 = (2 * 3110) + 1 ;
hence not 2 divides 6221 by NAT_4:9; :: thesis:
6221 = (3 * 2073) + 2 ;
hence not 3 divides 6221 by NAT_4:9; :: thesis:
6221 = (5 * 1244) + 1 ;
hence not 5 divides 6221 by NAT_4:9; :: thesis:
6221 = (7 * 888) + 5 ;
hence not 7 divides 6221 by NAT_4:9; :: thesis:
6221 = (11 * 565) + 6 ;
hence not 11 divides 6221 by NAT_4:9; :: thesis:
6221 = (13 * 478) + 7 ;
hence not 13 divides 6221 by NAT_4:9; :: thesis:
6221 = (17 * 365) + 16 ;
hence not 17 divides 6221 by NAT_4:9; :: thesis:
6221 = (19 * 327) + 8 ;
hence not 19 divides 6221 by NAT_4:9; :: thesis:
6221 = (23 * 270) + 11 ;
hence not 23 divides 6221 by NAT_4:9; :: thesis:
6221 = (29 * 214) + 15 ;
hence not 29 divides 6221 by NAT_4:9; :: thesis:
6221 = (31 * 200) + 21 ;
hence not 31 divides 6221 by NAT_4:9; :: thesis:
6221 = (37 * 168) + 5 ;
hence not 37 divides 6221 by NAT_4:9; :: thesis:
6221 = (41 * 151) + 30 ;
hence not 41 divides 6221 by NAT_4:9; :: thesis:
6221 = (43 * 144) + 29 ;
hence not 43 divides 6221 by NAT_4:9; :: thesis:
6221 = (47 * 132) + 17 ;
hence not 47 divides 6221 by NAT_4:9; :: thesis:
6221 = (53 * 117) + 20 ;
hence not 53 divides 6221 by NAT_4:9; :: thesis:
6221 = (59 * 105) + 26 ;
hence not 59 divides 6221 by NAT_4:9; :: thesis:
6221 = (61 * 101) + 60 ;
hence not 61 divides 6221 by NAT_4:9; :: thesis:
6221 = (67 * 92) + 57 ;
hence not 67 divides 6221 by NAT_4:9; :: thesis:
6221 = (71 * 87) + 44 ;
hence not 71 divides 6221 by NAT_4:9; :: thesis:
6221 = (73 * 85) + 16 ;
hence not 73 divides 6221 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6221 & n is prime holds
not n divides 6221 by XPRIMET1:42;
hence 6221 is prime by NAT_4:14; :: thesis: