9511 = (2 * 4755) + 1 ;
hence not 2 divides 9511 by NAT_4:9; :: thesis:
9511 = (3 * 3170) + 1 ;
hence not 3 divides 9511 by NAT_4:9; :: thesis:
9511 = (5 * 1902) + 1 ;
hence not 5 divides 9511 by NAT_4:9; :: thesis:
9511 = (7 * 1358) + 5 ;
hence not 7 divides 9511 by NAT_4:9; :: thesis:
9511 = (11 * 864) + 7 ;
hence not 11 divides 9511 by NAT_4:9; :: thesis:
9511 = (13 * 731) + 8 ;
hence not 13 divides 9511 by NAT_4:9; :: thesis:
9511 = (17 * 559) + 8 ;
hence not 17 divides 9511 by NAT_4:9; :: thesis:
9511 = (19 * 500) + 11 ;
hence not 19 divides 9511 by NAT_4:9; :: thesis:
9511 = (23 * 413) + 12 ;
hence not 23 divides 9511 by NAT_4:9; :: thesis:
9511 = (29 * 327) + 28 ;
hence not 29 divides 9511 by NAT_4:9; :: thesis:
9511 = (31 * 306) + 25 ;
hence not 31 divides 9511 by NAT_4:9; :: thesis:
9511 = (37 * 257) + 2 ;
hence not 37 divides 9511 by NAT_4:9; :: thesis:
9511 = (41 * 231) + 40 ;
hence not 41 divides 9511 by NAT_4:9; :: thesis:
9511 = (43 * 221) + 8 ;
hence not 43 divides 9511 by NAT_4:9; :: thesis:
9511 = (47 * 202) + 17 ;
hence not 47 divides 9511 by NAT_4:9; :: thesis:
9511 = (53 * 179) + 24 ;
hence not 53 divides 9511 by NAT_4:9; :: thesis:
9511 = (59 * 161) + 12 ;
hence not 59 divides 9511 by NAT_4:9; :: thesis:
9511 = (61 * 155) + 56 ;
hence not 61 divides 9511 by NAT_4:9; :: thesis:
9511 = (67 * 141) + 64 ;
hence not 67 divides 9511 by NAT_4:9; :: thesis:
9511 = (71 * 133) + 68 ;
hence not 71 divides 9511 by NAT_4:9; :: thesis:
9511 = (73 * 130) + 21 ;
hence not 73 divides 9511 by NAT_4:9; :: thesis:
9511 = (79 * 120) + 31 ;
hence not 79 divides 9511 by NAT_4:9; :: thesis:
9511 = (83 * 114) + 49 ;
hence not 83 divides 9511 by NAT_4:9; :: thesis:
9511 = (89 * 106) + 77 ;
hence not 89 divides 9511 by NAT_4:9; :: thesis:
9511 = (97 * 98) + 5 ;
hence not 97 divides 9511 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9511 & n is prime holds
not n divides 9511 by XPRIMET1:50;
hence 9511 is prime by NAT_4:14; :: thesis: