9181 = (2 * 4590) + 1 ;
hence not 2 divides 9181 by NAT_4:9; :: thesis:
9181 = (3 * 3060) + 1 ;
hence not 3 divides 9181 by NAT_4:9; :: thesis:
9181 = (5 * 1836) + 1 ;
hence not 5 divides 9181 by NAT_4:9; :: thesis:
9181 = (7 * 1311) + 4 ;
hence not 7 divides 9181 by NAT_4:9; :: thesis:
9181 = (11 * 834) + 7 ;
hence not 11 divides 9181 by NAT_4:9; :: thesis:
9181 = (13 * 706) + 3 ;
hence not 13 divides 9181 by NAT_4:9; :: thesis:
9181 = (17 * 540) + 1 ;
hence not 17 divides 9181 by NAT_4:9; :: thesis:
9181 = (19 * 483) + 4 ;
hence not 19 divides 9181 by NAT_4:9; :: thesis:
9181 = (23 * 399) + 4 ;
hence not 23 divides 9181 by NAT_4:9; :: thesis:
9181 = (29 * 316) + 17 ;
hence not 29 divides 9181 by NAT_4:9; :: thesis:
9181 = (31 * 296) + 5 ;
hence not 31 divides 9181 by NAT_4:9; :: thesis:
9181 = (37 * 248) + 5 ;
hence not 37 divides 9181 by NAT_4:9; :: thesis:
9181 = (41 * 223) + 38 ;
hence not 41 divides 9181 by NAT_4:9; :: thesis:
9181 = (43 * 213) + 22 ;
hence not 43 divides 9181 by NAT_4:9; :: thesis:
9181 = (47 * 195) + 16 ;
hence not 47 divides 9181 by NAT_4:9; :: thesis:
9181 = (53 * 173) + 12 ;
hence not 53 divides 9181 by NAT_4:9; :: thesis:
9181 = (59 * 155) + 36 ;
hence not 59 divides 9181 by NAT_4:9; :: thesis:
9181 = (61 * 150) + 31 ;
hence not 61 divides 9181 by NAT_4:9; :: thesis:
9181 = (67 * 137) + 2 ;
hence not 67 divides 9181 by NAT_4:9; :: thesis:
9181 = (71 * 129) + 22 ;
hence not 71 divides 9181 by NAT_4:9; :: thesis:
9181 = (73 * 125) + 56 ;
hence not 73 divides 9181 by NAT_4:9; :: thesis:
9181 = (79 * 116) + 17 ;
hence not 79 divides 9181 by NAT_4:9; :: thesis:
9181 = (83 * 110) + 51 ;
hence not 83 divides 9181 by NAT_4:9; :: thesis:
9181 = (89 * 103) + 14 ;
hence not 89 divides 9181 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9181 & n is prime holds
not n divides 9181 by XPRIMET1:48;
hence 9181 is prime by NAT_4:14; :: thesis: