6131 = (2 * 3065) + 1 ;
hence not 2 divides 6131 by NAT_4:9; :: thesis:
6131 = (3 * 2043) + 2 ;
hence not 3 divides 6131 by NAT_4:9; :: thesis:
6131 = (5 * 1226) + 1 ;
hence not 5 divides 6131 by NAT_4:9; :: thesis:
6131 = (7 * 875) + 6 ;
hence not 7 divides 6131 by NAT_4:9; :: thesis:
6131 = (11 * 557) + 4 ;
hence not 11 divides 6131 by NAT_4:9; :: thesis:
6131 = (13 * 471) + 8 ;
hence not 13 divides 6131 by NAT_4:9; :: thesis:
6131 = (17 * 360) + 11 ;
hence not 17 divides 6131 by NAT_4:9; :: thesis:
6131 = (19 * 322) + 13 ;
hence not 19 divides 6131 by NAT_4:9; :: thesis:
6131 = (23 * 266) + 13 ;
hence not 23 divides 6131 by NAT_4:9; :: thesis:
6131 = (29 * 211) + 12 ;
hence not 29 divides 6131 by NAT_4:9; :: thesis:
6131 = (31 * 197) + 24 ;
hence not 31 divides 6131 by NAT_4:9; :: thesis:
6131 = (37 * 165) + 26 ;
hence not 37 divides 6131 by NAT_4:9; :: thesis:
6131 = (41 * 149) + 22 ;
hence not 41 divides 6131 by NAT_4:9; :: thesis:
6131 = (43 * 142) + 25 ;
hence not 43 divides 6131 by NAT_4:9; :: thesis:
6131 = (47 * 130) + 21 ;
hence not 47 divides 6131 by NAT_4:9; :: thesis:
6131 = (53 * 115) + 36 ;
hence not 53 divides 6131 by NAT_4:9; :: thesis:
6131 = (59 * 103) + 54 ;
hence not 59 divides 6131 by NAT_4:9; :: thesis:
6131 = (61 * 100) + 31 ;
hence not 61 divides 6131 by NAT_4:9; :: thesis:
6131 = (67 * 91) + 34 ;
hence not 67 divides 6131 by NAT_4:9; :: thesis:
6131 = (71 * 86) + 25 ;
hence not 71 divides 6131 by NAT_4:9; :: thesis:
6131 = (73 * 83) + 72 ;
hence not 73 divides 6131 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6131 & n is prime holds
not n divides 6131 by XPRIMET1:42;
hence 6131 is prime by NAT_4:14; :: thesis: