now :: thesis: ( not 2 divides 4099 & not 3 divides 4099 & not 5 divides 4099 & not 7 divides 4099 & not 11 divides 4099 & not 13 divides 4099 & not 17 divides 4099 & not 19 divides 4099 & not 23 divides 4099 & not 29 divides 4099 & not 31 divides 4099 & not 37 divides 4099 & not 41 divides 4099 & not 43 divides 4099 & not 47 divides 4099 & not 53 divides 4099 & not 59 divides 4099 & not 61 divides 4099 )

4099 = (2 * 2049) + 1 ;
hence not 2 divides 4099 by NAT_4:9; :: thesis: ( not 3 divides 4099 & not 5 divides 4099 & not 7 divides 4099 & not 11 divides 4099 & not 13 divides 4099 & not 17 divides 4099 & not 19 divides 4099 & not 23 divides 4099 & not 29 divides 4099 & not 31 divides 4099 & not 37 divides 4099 & not 41 divides 4099 & not 43 divides 4099 & not 47 divides 4099 & not 53 divides 4099 & not 59 divides 4099 & not 61 divides 4099 )
4099 = (3 * 1366) + 1 ;
hence not 3 divides 4099 by NAT_4:9; :: thesis: ( not 5 divides 4099 & not 7 divides 4099 & not 11 divides 4099 & not 13 divides 4099 & not 17 divides 4099 & not 19 divides 4099 & not 23 divides 4099 & not 29 divides 4099 & not 31 divides 4099 & not 37 divides 4099 & not 41 divides 4099 & not 43 divides 4099 & not 47 divides 4099 & not 53 divides 4099 & not 59 divides 4099 & not 61 divides 4099 )
4099 = (5 * 819) + 4 ;
hence not 5 divides 4099 by NAT_4:9; :: thesis: ( not 7 divides 4099 & not 11 divides 4099 & not 13 divides 4099 & not 17 divides 4099 & not 19 divides 4099 & not 23 divides 4099 & not 29 divides 4099 & not 31 divides 4099 & not 37 divides 4099 & not 41 divides 4099 & not 43 divides 4099 & not 47 divides 4099 & not 53 divides 4099 & not 59 divides 4099 & not 61 divides 4099 )
4099 = (7 * 585) + 4 ;
hence not 7 divides 4099 by NAT_4:9; :: thesis: ( not 11 divides 4099 & not 13 divides 4099 & not 17 divides 4099 & not 19 divides 4099 & not 23 divides 4099 & not 29 divides 4099 & not 31 divides 4099 & not 37 divides 4099 & not 41 divides 4099 & not 43 divides 4099 & not 47 divides 4099 & not 53 divides 4099 & not 59 divides 4099 & not 61 divides 4099 )
4099 = (11 * 372) + 7 ;
hence not 11 divides 4099 by NAT_4:9; :: thesis: ( not 13 divides 4099 & not 17 divides 4099 & not 19 divides 4099 & not 23 divides 4099 & not 29 divides 4099 & not 31 divides 4099 & not 37 divides 4099 & not 41 divides 4099 & not 43 divides 4099 & not 47 divides 4099 & not 53 divides 4099 & not 59 divides 4099 & not 61 divides 4099 )
4099 = (13 * 315) + 4 ;
hence not 13 divides 4099 by NAT_4:9; :: thesis: ( not 17 divides 4099 & not 19 divides 4099 & not 23 divides 4099 & not 29 divides 4099 & not 31 divides 4099 & not 37 divides 4099 & not 41 divides 4099 & not 43 divides 4099 & not 47 divides 4099 & not 53 divides 4099 & not 59 divides 4099 & not 61 divides 4099 )
4099 = (17 * 241) + 2 ;
hence not 17 divides 4099 by NAT_4:9; :: thesis: ( not 19 divides 4099 & not 23 divides 4099 & not 29 divides 4099 & not 31 divides 4099 & not 37 divides 4099 & not 41 divides 4099 & not 43 divides 4099 & not 47 divides 4099 & not 53 divides 4099 & not 59 divides 4099 & not 61 divides 4099 )
4099 = (19 * 215) + 14 ;
hence not 19 divides 4099 by NAT_4:9; :: thesis: ( not 23 divides 4099 & not 29 divides 4099 & not 31 divides 4099 & not 37 divides 4099 & not 41 divides 4099 & not 43 divides 4099 & not 47 divides 4099 & not 53 divides 4099 & not 59 divides 4099 & not 61 divides 4099 )
4099 = (23 * 178) + 5 ;
hence not 23 divides 4099 by NAT_4:9; :: thesis: ( not 29 divides 4099 & not 31 divides 4099 & not 37 divides 4099 & not 41 divides 4099 & not 43 divides 4099 & not 47 divides 4099 & not 53 divides 4099 & not 59 divides 4099 & not 61 divides 4099 )
4099 = (29 * 141) + 10 ;
hence not 29 divides 4099 by NAT_4:9; :: thesis: ( not 31 divides 4099 & not 37 divides 4099 & not 41 divides 4099 & not 43 divides 4099 & not 47 divides 4099 & not 53 divides 4099 & not 59 divides 4099 & not 61 divides 4099 )
4099 = (31 * 132) + 7 ;
hence not 31 divides 4099 by NAT_4:9; :: thesis: ( not 37 divides 4099 & not 41 divides 4099 & not 43 divides 4099 & not 47 divides 4099 & not 53 divides 4099 & not 59 divides 4099 & not 61 divides 4099 )
4099 = (37 * 110) + 29 ;
hence not 37 divides 4099 by NAT_4:9; :: thesis: ( not 41 divides 4099 & not 43 divides 4099 & not 47 divides 4099 & not 53 divides 4099 & not 59 divides 4099 & not 61 divides 4099 )
4099 = (41 * 99) + 40 ;
hence not 41 divides 4099 by NAT_4:9; :: thesis: ( not 43 divides 4099 & not 47 divides 4099 & not 53 divides 4099 & not 59 divides 4099 & not 61 divides 4099 )
4099 = (43 * 95) + 14 ;
hence not 43 divides 4099 by NAT_4:9; :: thesis: ( not 47 divides 4099 & not 53 divides 4099 & not 59 divides 4099 & not 61 divides 4099 )
4099 = (47 * 87) + 10 ;
hence not 47 divides 4099 by NAT_4:9; :: thesis: ( not 53 divides 4099 & not 59 divides 4099 & not 61 divides 4099 )
4099 = (53 * 77) + 18 ;
hence not 53 divides 4099 by NAT_4:9; :: thesis: ( not 59 divides 4099 & not 61 divides 4099 )
4099 = (59 * 69) + 28 ;
hence not 59 divides 4099 by NAT_4:9; :: thesis: not 61 divides 4099
4099 = (61 * 67) + 12 ;
hence not 61 divides 4099 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 4099 & n is prime holds
not n divides 4099 by XPRIMET1:36;
hence 4099 is prime by NAT_4:14; :: thesis: verum