now :: thesis: ( not 2 divides 8431 & not 3 divides 8431 & not 5 divides 8431 & not 7 divides 8431 & not 11 divides 8431 & not 13 divides 8431 & not 17 divides 8431 & not 19 divides 8431 & not 23 divides 8431 & not 29 divides 8431 & not 31 divides 8431 & not 37 divides 8431 & not 41 divides 8431 & not 43 divides 8431 & not 47 divides 8431 & not 53 divides 8431 & not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )

8431 = (2 * 4215) + 1 ;
hence not 2 divides 8431 by NAT_4:9; :: thesis: ( not 3 divides 8431 & not 5 divides 8431 & not 7 divides 8431 & not 11 divides 8431 & not 13 divides 8431 & not 17 divides 8431 & not 19 divides 8431 & not 23 divides 8431 & not 29 divides 8431 & not 31 divides 8431 & not 37 divides 8431 & not 41 divides 8431 & not 43 divides 8431 & not 47 divides 8431 & not 53 divides 8431 & not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (3 * 2810) + 1 ;
hence not 3 divides 8431 by NAT_4:9; :: thesis: ( not 5 divides 8431 & not 7 divides 8431 & not 11 divides 8431 & not 13 divides 8431 & not 17 divides 8431 & not 19 divides 8431 & not 23 divides 8431 & not 29 divides 8431 & not 31 divides 8431 & not 37 divides 8431 & not 41 divides 8431 & not 43 divides 8431 & not 47 divides 8431 & not 53 divides 8431 & not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (5 * 1686) + 1 ;
hence not 5 divides 8431 by NAT_4:9; :: thesis: ( not 7 divides 8431 & not 11 divides 8431 & not 13 divides 8431 & not 17 divides 8431 & not 19 divides 8431 & not 23 divides 8431 & not 29 divides 8431 & not 31 divides 8431 & not 37 divides 8431 & not 41 divides 8431 & not 43 divides 8431 & not 47 divides 8431 & not 53 divides 8431 & not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (7 * 1204) + 3 ;
hence not 7 divides 8431 by NAT_4:9; :: thesis: ( not 11 divides 8431 & not 13 divides 8431 & not 17 divides 8431 & not 19 divides 8431 & not 23 divides 8431 & not 29 divides 8431 & not 31 divides 8431 & not 37 divides 8431 & not 41 divides 8431 & not 43 divides 8431 & not 47 divides 8431 & not 53 divides 8431 & not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (11 * 766) + 5 ;
hence not 11 divides 8431 by NAT_4:9; :: thesis: ( not 13 divides 8431 & not 17 divides 8431 & not 19 divides 8431 & not 23 divides 8431 & not 29 divides 8431 & not 31 divides 8431 & not 37 divides 8431 & not 41 divides 8431 & not 43 divides 8431 & not 47 divides 8431 & not 53 divides 8431 & not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (13 * 648) + 7 ;
hence not 13 divides 8431 by NAT_4:9; :: thesis: ( not 17 divides 8431 & not 19 divides 8431 & not 23 divides 8431 & not 29 divides 8431 & not 31 divides 8431 & not 37 divides 8431 & not 41 divides 8431 & not 43 divides 8431 & not 47 divides 8431 & not 53 divides 8431 & not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (17 * 495) + 16 ;
hence not 17 divides 8431 by NAT_4:9; :: thesis: ( not 19 divides 8431 & not 23 divides 8431 & not 29 divides 8431 & not 31 divides 8431 & not 37 divides 8431 & not 41 divides 8431 & not 43 divides 8431 & not 47 divides 8431 & not 53 divides 8431 & not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (19 * 443) + 14 ;
hence not 19 divides 8431 by NAT_4:9; :: thesis: ( not 23 divides 8431 & not 29 divides 8431 & not 31 divides 8431 & not 37 divides 8431 & not 41 divides 8431 & not 43 divides 8431 & not 47 divides 8431 & not 53 divides 8431 & not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (23 * 366) + 13 ;
hence not 23 divides 8431 by NAT_4:9; :: thesis: ( not 29 divides 8431 & not 31 divides 8431 & not 37 divides 8431 & not 41 divides 8431 & not 43 divides 8431 & not 47 divides 8431 & not 53 divides 8431 & not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (29 * 290) + 21 ;
hence not 29 divides 8431 by NAT_4:9; :: thesis: ( not 31 divides 8431 & not 37 divides 8431 & not 41 divides 8431 & not 43 divides 8431 & not 47 divides 8431 & not 53 divides 8431 & not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (31 * 271) + 30 ;
hence not 31 divides 8431 by NAT_4:9; :: thesis: ( not 37 divides 8431 & not 41 divides 8431 & not 43 divides 8431 & not 47 divides 8431 & not 53 divides 8431 & not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (37 * 227) + 32 ;
hence not 37 divides 8431 by NAT_4:9; :: thesis: ( not 41 divides 8431 & not 43 divides 8431 & not 47 divides 8431 & not 53 divides 8431 & not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (41 * 205) + 26 ;
hence not 41 divides 8431 by NAT_4:9; :: thesis: ( not 43 divides 8431 & not 47 divides 8431 & not 53 divides 8431 & not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (43 * 196) + 3 ;
hence not 43 divides 8431 by NAT_4:9; :: thesis: ( not 47 divides 8431 & not 53 divides 8431 & not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (47 * 179) + 18 ;
hence not 47 divides 8431 by NAT_4:9; :: thesis: ( not 53 divides 8431 & not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (53 * 159) + 4 ;
hence not 53 divides 8431 by NAT_4:9; :: thesis: ( not 59 divides 8431 & not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (59 * 142) + 53 ;
hence not 59 divides 8431 by NAT_4:9; :: thesis: ( not 61 divides 8431 & not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (61 * 138) + 13 ;
hence not 61 divides 8431 by NAT_4:9; :: thesis: ( not 67 divides 8431 & not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (67 * 125) + 56 ;
hence not 67 divides 8431 by NAT_4:9; :: thesis: ( not 71 divides 8431 & not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (71 * 118) + 53 ;
hence not 71 divides 8431 by NAT_4:9; :: thesis: ( not 73 divides 8431 & not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (73 * 115) + 36 ;
hence not 73 divides 8431 by NAT_4:9; :: thesis: ( not 79 divides 8431 & not 83 divides 8431 & not 89 divides 8431 )
8431 = (79 * 106) + 57 ;
hence not 79 divides 8431 by NAT_4:9; :: thesis: ( not 83 divides 8431 & not 89 divides 8431 )
8431 = (83 * 101) + 48 ;
hence not 83 divides 8431 by NAT_4:9; :: thesis: not 89 divides 8431
8431 = (89 * 94) + 65 ;
hence not 89 divides 8431 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 8431 & n is prime holds
not n divides 8431 by XPRIMET1:48;
hence 8431 is prime by NAT_4:14; :: thesis: verum