6599 = (2 * 3299) + 1 ;
hence not 2 divides 6599 by NAT_4:9; :: thesis:
6599 = (3 * 2199) + 2 ;
hence not 3 divides 6599 by NAT_4:9; :: thesis:
6599 = (5 * 1319) + 4 ;
hence not 5 divides 6599 by NAT_4:9; :: thesis:
6599 = (7 * 942) + 5 ;
hence not 7 divides 6599 by NAT_4:9; :: thesis:
6599 = (11 * 599) + 10 ;
hence not 11 divides 6599 by NAT_4:9; :: thesis:
6599 = (13 * 507) + 8 ;
hence not 13 divides 6599 by NAT_4:9; :: thesis:
6599 = (17 * 388) + 3 ;
hence not 17 divides 6599 by NAT_4:9; :: thesis:
6599 = (19 * 347) + 6 ;
hence not 19 divides 6599 by NAT_4:9; :: thesis:
6599 = (23 * 286) + 21 ;
hence not 23 divides 6599 by NAT_4:9; :: thesis:
6599 = (29 * 227) + 16 ;
hence not 29 divides 6599 by NAT_4:9; :: thesis:
6599 = (31 * 212) + 27 ;
hence not 31 divides 6599 by NAT_4:9; :: thesis:
6599 = (37 * 178) + 13 ;
hence not 37 divides 6599 by NAT_4:9; :: thesis:
6599 = (41 * 160) + 39 ;
hence not 41 divides 6599 by NAT_4:9; :: thesis:
6599 = (43 * 153) + 20 ;
hence not 43 divides 6599 by NAT_4:9; :: thesis:
6599 = (47 * 140) + 19 ;
hence not 47 divides 6599 by NAT_4:9; :: thesis:
6599 = (53 * 124) + 27 ;
hence not 53 divides 6599 by NAT_4:9; :: thesis:
6599 = (59 * 111) + 50 ;
hence not 59 divides 6599 by NAT_4:9; :: thesis:
6599 = (61 * 108) + 11 ;
hence not 61 divides 6599 by NAT_4:9; :: thesis:
6599 = (67 * 98) + 33 ;
hence not 67 divides 6599 by NAT_4:9; :: thesis:
6599 = (71 * 92) + 67 ;
hence not 71 divides 6599 by NAT_4:9; :: thesis:
6599 = (73 * 90) + 29 ;
hence not 73 divides 6599 by NAT_4:9; :: thesis:
6599 = (79 * 83) + 42 ;
hence not 79 divides 6599 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6599 & n is prime holds
not n divides 6599 by XPRIMET1:44;
hence 6599 is prime by NAT_4:14; :: thesis: