4337 = (2 * 2168) + 1 ;
hence not 2 divides 4337 by NAT_4:9; :: thesis:
4337 = (3 * 1445) + 2 ;
hence not 3 divides 4337 by NAT_4:9; :: thesis:
4337 = (5 * 867) + 2 ;
hence not 5 divides 4337 by NAT_4:9; :: thesis:
4337 = (7 * 619) + 4 ;
hence not 7 divides 4337 by NAT_4:9; :: thesis:
4337 = (11 * 394) + 3 ;
hence not 11 divides 4337 by NAT_4:9; :: thesis:
4337 = (13 * 333) + 8 ;
hence not 13 divides 4337 by NAT_4:9; :: thesis:
4337 = (17 * 255) + 2 ;
hence not 17 divides 4337 by NAT_4:9; :: thesis:
4337 = (19 * 228) + 5 ;
hence not 19 divides 4337 by NAT_4:9; :: thesis:
4337 = (23 * 188) + 13 ;
hence not 23 divides 4337 by NAT_4:9; :: thesis:
4337 = (29 * 149) + 16 ;
hence not 29 divides 4337 by NAT_4:9; :: thesis:
4337 = (31 * 139) + 28 ;
hence not 31 divides 4337 by NAT_4:9; :: thesis:
4337 = (37 * 117) + 8 ;
hence not 37 divides 4337 by NAT_4:9; :: thesis:
4337 = (41 * 105) + 32 ;
hence not 41 divides 4337 by NAT_4:9; :: thesis:
4337 = (43 * 100) + 37 ;
hence not 43 divides 4337 by NAT_4:9; :: thesis:
4337 = (47 * 92) + 13 ;
hence not 47 divides 4337 by NAT_4:9; :: thesis:
4337 = (53 * 81) + 44 ;
hence not 53 divides 4337 by NAT_4:9; :: thesis:
4337 = (59 * 73) + 30 ;
hence not 59 divides 4337 by NAT_4:9; :: thesis:
4337 = (61 * 71) + 6 ;
hence not 61 divides 4337 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4337 & n is prime holds
not n divides 4337 by XPRIMET1:36;
hence 4337 is prime by NAT_4:14; :: thesis: