7591 = (2 * 3795) + 1 ;
hence not 2 divides 7591 by NAT_4:9; :: thesis:
7591 = (3 * 2530) + 1 ;
hence not 3 divides 7591 by NAT_4:9; :: thesis:
7591 = (5 * 1518) + 1 ;
hence not 5 divides 7591 by NAT_4:9; :: thesis:
7591 = (7 * 1084) + 3 ;
hence not 7 divides 7591 by NAT_4:9; :: thesis:
7591 = (11 * 690) + 1 ;
hence not 11 divides 7591 by NAT_4:9; :: thesis:
7591 = (13 * 583) + 12 ;
hence not 13 divides 7591 by NAT_4:9; :: thesis:
7591 = (17 * 446) + 9 ;
hence not 17 divides 7591 by NAT_4:9; :: thesis:
7591 = (19 * 399) + 10 ;
hence not 19 divides 7591 by NAT_4:9; :: thesis:
7591 = (23 * 330) + 1 ;
hence not 23 divides 7591 by NAT_4:9; :: thesis:
7591 = (29 * 261) + 22 ;
hence not 29 divides 7591 by NAT_4:9; :: thesis:
7591 = (31 * 244) + 27 ;
hence not 31 divides 7591 by NAT_4:9; :: thesis:
7591 = (37 * 205) + 6 ;
hence not 37 divides 7591 by NAT_4:9; :: thesis:
7591 = (41 * 185) + 6 ;
hence not 41 divides 7591 by NAT_4:9; :: thesis:
7591 = (43 * 176) + 23 ;
hence not 43 divides 7591 by NAT_4:9; :: thesis:
7591 = (47 * 161) + 24 ;
hence not 47 divides 7591 by NAT_4:9; :: thesis:
7591 = (53 * 143) + 12 ;
hence not 53 divides 7591 by NAT_4:9; :: thesis:
7591 = (59 * 128) + 39 ;
hence not 59 divides 7591 by NAT_4:9; :: thesis:
7591 = (61 * 124) + 27 ;
hence not 61 divides 7591 by NAT_4:9; :: thesis:
7591 = (67 * 113) + 20 ;
hence not 67 divides 7591 by NAT_4:9; :: thesis:
7591 = (71 * 106) + 65 ;
hence not 71 divides 7591 by NAT_4:9; :: thesis:
7591 = (73 * 103) + 72 ;
hence not 73 divides 7591 by NAT_4:9; :: thesis:
7591 = (79 * 96) + 7 ;
hence not 79 divides 7591 by NAT_4:9; :: thesis:
7591 = (83 * 91) + 38 ;
hence not 83 divides 7591 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7591 & n is prime holds
not n divides 7591 by XPRIMET1:46;
hence 7591 is prime by NAT_4:14; :: thesis: