5591 = (2 * 2795) + 1 ;
hence not 2 divides 5591 by NAT_4:9; :: thesis:
5591 = (3 * 1863) + 2 ;
hence not 3 divides 5591 by NAT_4:9; :: thesis:
5591 = (5 * 1118) + 1 ;
hence not 5 divides 5591 by NAT_4:9; :: thesis:
5591 = (7 * 798) + 5 ;
hence not 7 divides 5591 by NAT_4:9; :: thesis:
5591 = (11 * 508) + 3 ;
hence not 11 divides 5591 by NAT_4:9; :: thesis:
5591 = (13 * 430) + 1 ;
hence not 13 divides 5591 by NAT_4:9; :: thesis:
5591 = (17 * 328) + 15 ;
hence not 17 divides 5591 by NAT_4:9; :: thesis:
5591 = (19 * 294) + 5 ;
hence not 19 divides 5591 by NAT_4:9; :: thesis:
5591 = (23 * 243) + 2 ;
hence not 23 divides 5591 by NAT_4:9; :: thesis:
5591 = (29 * 192) + 23 ;
hence not 29 divides 5591 by NAT_4:9; :: thesis:
5591 = (31 * 180) + 11 ;
hence not 31 divides 5591 by NAT_4:9; :: thesis:
5591 = (37 * 151) + 4 ;
hence not 37 divides 5591 by NAT_4:9; :: thesis:
5591 = (41 * 136) + 15 ;
hence not 41 divides 5591 by NAT_4:9; :: thesis:
5591 = (43 * 130) + 1 ;
hence not 43 divides 5591 by NAT_4:9; :: thesis:
5591 = (47 * 118) + 45 ;
hence not 47 divides 5591 by NAT_4:9; :: thesis:
5591 = (53 * 105) + 26 ;
hence not 53 divides 5591 by NAT_4:9; :: thesis:
5591 = (59 * 94) + 45 ;
hence not 59 divides 5591 by NAT_4:9; :: thesis:
5591 = (61 * 91) + 40 ;
hence not 61 divides 5591 by NAT_4:9; :: thesis:
5591 = (67 * 83) + 30 ;
hence not 67 divides 5591 by NAT_4:9; :: thesis:
5591 = (71 * 78) + 53 ;
hence not 71 divides 5591 by NAT_4:9; :: thesis:
5591 = (73 * 76) + 43 ;
hence not 73 divides 5591 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5591 & n is prime holds
not n divides 5591 by XPRIMET1:42;
hence 5591 is prime by NAT_4:14; :: thesis: