now :: thesis: ( not 2 divides 8269 & not 3 divides 8269 & not 5 divides 8269 & not 7 divides 8269 & not 11 divides 8269 & not 13 divides 8269 & not 17 divides 8269 & not 19 divides 8269 & not 23 divides 8269 & not 29 divides 8269 & not 31 divides 8269 & not 37 divides 8269 & not 41 divides 8269 & not 43 divides 8269 & not 47 divides 8269 & not 53 divides 8269 & not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )

8269 = (2 * 4134) + 1 ;
hence not 2 divides 8269 by NAT_4:9; :: thesis: ( not 3 divides 8269 & not 5 divides 8269 & not 7 divides 8269 & not 11 divides 8269 & not 13 divides 8269 & not 17 divides 8269 & not 19 divides 8269 & not 23 divides 8269 & not 29 divides 8269 & not 31 divides 8269 & not 37 divides 8269 & not 41 divides 8269 & not 43 divides 8269 & not 47 divides 8269 & not 53 divides 8269 & not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (3 * 2756) + 1 ;
hence not 3 divides 8269 by NAT_4:9; :: thesis: ( not 5 divides 8269 & not 7 divides 8269 & not 11 divides 8269 & not 13 divides 8269 & not 17 divides 8269 & not 19 divides 8269 & not 23 divides 8269 & not 29 divides 8269 & not 31 divides 8269 & not 37 divides 8269 & not 41 divides 8269 & not 43 divides 8269 & not 47 divides 8269 & not 53 divides 8269 & not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (5 * 1653) + 4 ;
hence not 5 divides 8269 by NAT_4:9; :: thesis: ( not 7 divides 8269 & not 11 divides 8269 & not 13 divides 8269 & not 17 divides 8269 & not 19 divides 8269 & not 23 divides 8269 & not 29 divides 8269 & not 31 divides 8269 & not 37 divides 8269 & not 41 divides 8269 & not 43 divides 8269 & not 47 divides 8269 & not 53 divides 8269 & not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (7 * 1181) + 2 ;
hence not 7 divides 8269 by NAT_4:9; :: thesis: ( not 11 divides 8269 & not 13 divides 8269 & not 17 divides 8269 & not 19 divides 8269 & not 23 divides 8269 & not 29 divides 8269 & not 31 divides 8269 & not 37 divides 8269 & not 41 divides 8269 & not 43 divides 8269 & not 47 divides 8269 & not 53 divides 8269 & not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (11 * 751) + 8 ;
hence not 11 divides 8269 by NAT_4:9; :: thesis: ( not 13 divides 8269 & not 17 divides 8269 & not 19 divides 8269 & not 23 divides 8269 & not 29 divides 8269 & not 31 divides 8269 & not 37 divides 8269 & not 41 divides 8269 & not 43 divides 8269 & not 47 divides 8269 & not 53 divides 8269 & not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (13 * 636) + 1 ;
hence not 13 divides 8269 by NAT_4:9; :: thesis: ( not 17 divides 8269 & not 19 divides 8269 & not 23 divides 8269 & not 29 divides 8269 & not 31 divides 8269 & not 37 divides 8269 & not 41 divides 8269 & not 43 divides 8269 & not 47 divides 8269 & not 53 divides 8269 & not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (17 * 486) + 7 ;
hence not 17 divides 8269 by NAT_4:9; :: thesis: ( not 19 divides 8269 & not 23 divides 8269 & not 29 divides 8269 & not 31 divides 8269 & not 37 divides 8269 & not 41 divides 8269 & not 43 divides 8269 & not 47 divides 8269 & not 53 divides 8269 & not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (19 * 435) + 4 ;
hence not 19 divides 8269 by NAT_4:9; :: thesis: ( not 23 divides 8269 & not 29 divides 8269 & not 31 divides 8269 & not 37 divides 8269 & not 41 divides 8269 & not 43 divides 8269 & not 47 divides 8269 & not 53 divides 8269 & not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (23 * 359) + 12 ;
hence not 23 divides 8269 by NAT_4:9; :: thesis: ( not 29 divides 8269 & not 31 divides 8269 & not 37 divides 8269 & not 41 divides 8269 & not 43 divides 8269 & not 47 divides 8269 & not 53 divides 8269 & not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (29 * 285) + 4 ;
hence not 29 divides 8269 by NAT_4:9; :: thesis: ( not 31 divides 8269 & not 37 divides 8269 & not 41 divides 8269 & not 43 divides 8269 & not 47 divides 8269 & not 53 divides 8269 & not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (31 * 266) + 23 ;
hence not 31 divides 8269 by NAT_4:9; :: thesis: ( not 37 divides 8269 & not 41 divides 8269 & not 43 divides 8269 & not 47 divides 8269 & not 53 divides 8269 & not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (37 * 223) + 18 ;
hence not 37 divides 8269 by NAT_4:9; :: thesis: ( not 41 divides 8269 & not 43 divides 8269 & not 47 divides 8269 & not 53 divides 8269 & not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (41 * 201) + 28 ;
hence not 41 divides 8269 by NAT_4:9; :: thesis: ( not 43 divides 8269 & not 47 divides 8269 & not 53 divides 8269 & not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (43 * 192) + 13 ;
hence not 43 divides 8269 by NAT_4:9; :: thesis: ( not 47 divides 8269 & not 53 divides 8269 & not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (47 * 175) + 44 ;
hence not 47 divides 8269 by NAT_4:9; :: thesis: ( not 53 divides 8269 & not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (53 * 156) + 1 ;
hence not 53 divides 8269 by NAT_4:9; :: thesis: ( not 59 divides 8269 & not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (59 * 140) + 9 ;
hence not 59 divides 8269 by NAT_4:9; :: thesis: ( not 61 divides 8269 & not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (61 * 135) + 34 ;
hence not 61 divides 8269 by NAT_4:9; :: thesis: ( not 67 divides 8269 & not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (67 * 123) + 28 ;
hence not 67 divides 8269 by NAT_4:9; :: thesis: ( not 71 divides 8269 & not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (71 * 116) + 33 ;
hence not 71 divides 8269 by NAT_4:9; :: thesis: ( not 73 divides 8269 & not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (73 * 113) + 20 ;
hence not 73 divides 8269 by NAT_4:9; :: thesis: ( not 79 divides 8269 & not 83 divides 8269 & not 89 divides 8269 )
8269 = (79 * 104) + 53 ;
hence not 79 divides 8269 by NAT_4:9; :: thesis: ( not 83 divides 8269 & not 89 divides 8269 )
8269 = (83 * 99) + 52 ;
hence not 83 divides 8269 by NAT_4:9; :: thesis: not 89 divides 8269
8269 = (89 * 92) + 81 ;
hence not 89 divides 8269 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 8269 & n is prime holds
not n divides 8269 by XPRIMET1:48;
hence 8269 is prime by NAT_4:14; :: thesis: verum