4591 = (2 * 2295) + 1 ;
hence not 2 divides 4591 by NAT_4:9; :: thesis:
4591 = (3 * 1530) + 1 ;
hence not 3 divides 4591 by NAT_4:9; :: thesis:
4591 = (5 * 918) + 1 ;
hence not 5 divides 4591 by NAT_4:9; :: thesis:
4591 = (7 * 655) + 6 ;
hence not 7 divides 4591 by NAT_4:9; :: thesis:
4591 = (11 * 417) + 4 ;
hence not 11 divides 4591 by NAT_4:9; :: thesis:
4591 = (13 * 353) + 2 ;
hence not 13 divides 4591 by NAT_4:9; :: thesis:
4591 = (17 * 270) + 1 ;
hence not 17 divides 4591 by NAT_4:9; :: thesis:
4591 = (19 * 241) + 12 ;
hence not 19 divides 4591 by NAT_4:9; :: thesis:
4591 = (23 * 199) + 14 ;
hence not 23 divides 4591 by NAT_4:9; :: thesis:
4591 = (29 * 158) + 9 ;
hence not 29 divides 4591 by NAT_4:9; :: thesis:
4591 = (31 * 148) + 3 ;
hence not 31 divides 4591 by NAT_4:9; :: thesis:
4591 = (37 * 124) + 3 ;
hence not 37 divides 4591 by NAT_4:9; :: thesis:
4591 = (41 * 111) + 40 ;
hence not 41 divides 4591 by NAT_4:9; :: thesis:
4591 = (43 * 106) + 33 ;
hence not 43 divides 4591 by NAT_4:9; :: thesis:
4591 = (47 * 97) + 32 ;
hence not 47 divides 4591 by NAT_4:9; :: thesis:
4591 = (53 * 86) + 33 ;
hence not 53 divides 4591 by NAT_4:9; :: thesis:
4591 = (59 * 77) + 48 ;
hence not 59 divides 4591 by NAT_4:9; :: thesis:
4591 = (61 * 75) + 16 ;
hence not 61 divides 4591 by NAT_4:9; :: thesis:
4591 = (67 * 68) + 35 ;
hence not 67 divides 4591 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4591 & n is prime holds
not n divides 4591 by XPRIMET1:38;
hence 4591 is prime by NAT_4:14; :: thesis: