9391 = (2 * 4695) + 1 ;
hence not 2 divides 9391 by NAT_4:9; :: thesis:
9391 = (3 * 3130) + 1 ;
hence not 3 divides 9391 by NAT_4:9; :: thesis:
9391 = (5 * 1878) + 1 ;
hence not 5 divides 9391 by NAT_4:9; :: thesis:
9391 = (7 * 1341) + 4 ;
hence not 7 divides 9391 by NAT_4:9; :: thesis:
9391 = (11 * 853) + 8 ;
hence not 11 divides 9391 by NAT_4:9; :: thesis:
9391 = (13 * 722) + 5 ;
hence not 13 divides 9391 by NAT_4:9; :: thesis:
9391 = (17 * 552) + 7 ;
hence not 17 divides 9391 by NAT_4:9; :: thesis:
9391 = (19 * 494) + 5 ;
hence not 19 divides 9391 by NAT_4:9; :: thesis:
9391 = (23 * 408) + 7 ;
hence not 23 divides 9391 by NAT_4:9; :: thesis:
9391 = (29 * 323) + 24 ;
hence not 29 divides 9391 by NAT_4:9; :: thesis:
9391 = (31 * 302) + 29 ;
hence not 31 divides 9391 by NAT_4:9; :: thesis:
9391 = (37 * 253) + 30 ;
hence not 37 divides 9391 by NAT_4:9; :: thesis:
9391 = (41 * 229) + 2 ;
hence not 41 divides 9391 by NAT_4:9; :: thesis:
9391 = (43 * 218) + 17 ;
hence not 43 divides 9391 by NAT_4:9; :: thesis:
9391 = (47 * 199) + 38 ;
hence not 47 divides 9391 by NAT_4:9; :: thesis:
9391 = (53 * 177) + 10 ;
hence not 53 divides 9391 by NAT_4:9; :: thesis:
9391 = (59 * 159) + 10 ;
hence not 59 divides 9391 by NAT_4:9; :: thesis:
9391 = (61 * 153) + 58 ;
hence not 61 divides 9391 by NAT_4:9; :: thesis:
9391 = (67 * 140) + 11 ;
hence not 67 divides 9391 by NAT_4:9; :: thesis:
9391 = (71 * 132) + 19 ;
hence not 71 divides 9391 by NAT_4:9; :: thesis:
9391 = (73 * 128) + 47 ;
hence not 73 divides 9391 by NAT_4:9; :: thesis:
9391 = (79 * 118) + 69 ;
hence not 79 divides 9391 by NAT_4:9; :: thesis:
9391 = (83 * 113) + 12 ;
hence not 83 divides 9391 by NAT_4:9; :: thesis:
9391 = (89 * 105) + 46 ;
hence not 89 divides 9391 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9391 & n is prime holds
not n divides 9391 by XPRIMET1:48;
hence 9391 is prime by NAT_4:14; :: thesis: