9091 = (2 * 4545) + 1 ;
hence not 2 divides 9091 by NAT_4:9; :: thesis:
9091 = (3 * 3030) + 1 ;
hence not 3 divides 9091 by NAT_4:9; :: thesis:
9091 = (5 * 1818) + 1 ;
hence not 5 divides 9091 by NAT_4:9; :: thesis:
9091 = (7 * 1298) + 5 ;
hence not 7 divides 9091 by NAT_4:9; :: thesis:
9091 = (11 * 826) + 5 ;
hence not 11 divides 9091 by NAT_4:9; :: thesis:
9091 = (13 * 699) + 4 ;
hence not 13 divides 9091 by NAT_4:9; :: thesis:
9091 = (17 * 534) + 13 ;
hence not 17 divides 9091 by NAT_4:9; :: thesis:
9091 = (19 * 478) + 9 ;
hence not 19 divides 9091 by NAT_4:9; :: thesis:
9091 = (23 * 395) + 6 ;
hence not 23 divides 9091 by NAT_4:9; :: thesis:
9091 = (29 * 313) + 14 ;
hence not 29 divides 9091 by NAT_4:9; :: thesis:
9091 = (31 * 293) + 8 ;
hence not 31 divides 9091 by NAT_4:9; :: thesis:
9091 = (37 * 245) + 26 ;
hence not 37 divides 9091 by NAT_4:9; :: thesis:
9091 = (41 * 221) + 30 ;
hence not 41 divides 9091 by NAT_4:9; :: thesis:
9091 = (43 * 211) + 18 ;
hence not 43 divides 9091 by NAT_4:9; :: thesis:
9091 = (47 * 193) + 20 ;
hence not 47 divides 9091 by NAT_4:9; :: thesis:
9091 = (53 * 171) + 28 ;
hence not 53 divides 9091 by NAT_4:9; :: thesis:
9091 = (59 * 154) + 5 ;
hence not 59 divides 9091 by NAT_4:9; :: thesis:
9091 = (61 * 149) + 2 ;
hence not 61 divides 9091 by NAT_4:9; :: thesis:
9091 = (67 * 135) + 46 ;
hence not 67 divides 9091 by NAT_4:9; :: thesis:
9091 = (71 * 128) + 3 ;
hence not 71 divides 9091 by NAT_4:9; :: thesis:
9091 = (73 * 124) + 39 ;
hence not 73 divides 9091 by NAT_4:9; :: thesis:
9091 = (79 * 115) + 6 ;
hence not 79 divides 9091 by NAT_4:9; :: thesis:
9091 = (83 * 109) + 44 ;
hence not 83 divides 9091 by NAT_4:9; :: thesis:
9091 = (89 * 102) + 13 ;
hence not 89 divides 9091 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9091 & n is prime holds
not n divides 9091 by XPRIMET1:48;
hence 9091 is prime by NAT_4:14; :: thesis: