5659 = (2 * 2829) + 1 ;
hence not 2 divides 5659 by NAT_4:9; :: thesis:
5659 = (3 * 1886) + 1 ;
hence not 3 divides 5659 by NAT_4:9; :: thesis:
5659 = (5 * 1131) + 4 ;
hence not 5 divides 5659 by NAT_4:9; :: thesis:
5659 = (7 * 808) + 3 ;
hence not 7 divides 5659 by NAT_4:9; :: thesis:
5659 = (11 * 514) + 5 ;
hence not 11 divides 5659 by NAT_4:9; :: thesis:
5659 = (13 * 435) + 4 ;
hence not 13 divides 5659 by NAT_4:9; :: thesis:
5659 = (17 * 332) + 15 ;
hence not 17 divides 5659 by NAT_4:9; :: thesis:
5659 = (19 * 297) + 16 ;
hence not 19 divides 5659 by NAT_4:9; :: thesis:
5659 = (23 * 246) + 1 ;
hence not 23 divides 5659 by NAT_4:9; :: thesis:
5659 = (29 * 195) + 4 ;
hence not 29 divides 5659 by NAT_4:9; :: thesis:
5659 = (31 * 182) + 17 ;
hence not 31 divides 5659 by NAT_4:9; :: thesis:
5659 = (37 * 152) + 35 ;
hence not 37 divides 5659 by NAT_4:9; :: thesis:
5659 = (41 * 138) + 1 ;
hence not 41 divides 5659 by NAT_4:9; :: thesis:
5659 = (43 * 131) + 26 ;
hence not 43 divides 5659 by NAT_4:9; :: thesis:
5659 = (47 * 120) + 19 ;
hence not 47 divides 5659 by NAT_4:9; :: thesis:
5659 = (53 * 106) + 41 ;
hence not 53 divides 5659 by NAT_4:9; :: thesis:
5659 = (59 * 95) + 54 ;
hence not 59 divides 5659 by NAT_4:9; :: thesis:
5659 = (61 * 92) + 47 ;
hence not 61 divides 5659 by NAT_4:9; :: thesis:
5659 = (67 * 84) + 31 ;
hence not 67 divides 5659 by NAT_4:9; :: thesis:
5659 = (71 * 79) + 50 ;
hence not 71 divides 5659 by NAT_4:9; :: thesis:
5659 = (73 * 77) + 38 ;
hence not 73 divides 5659 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5659 & n is prime holds
not n divides 5659 by XPRIMET1:42;
hence 5659 is prime by NAT_4:14; :: thesis: