now :: thesis: ( not 2 divides 6359 & not 3 divides 6359 & not 5 divides 6359 & not 7 divides 6359 & not 11 divides 6359 & not 13 divides 6359 & not 17 divides 6359 & not 19 divides 6359 & not 23 divides 6359 & not 29 divides 6359 & not 31 divides 6359 & not 37 divides 6359 & not 41 divides 6359 & not 43 divides 6359 & not 47 divides 6359 & not 53 divides 6359 & not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )

6359 = (2 * 3179) + 1 ;
hence not 2 divides 6359 by NAT_4:9; :: thesis: ( not 3 divides 6359 & not 5 divides 6359 & not 7 divides 6359 & not 11 divides 6359 & not 13 divides 6359 & not 17 divides 6359 & not 19 divides 6359 & not 23 divides 6359 & not 29 divides 6359 & not 31 divides 6359 & not 37 divides 6359 & not 41 divides 6359 & not 43 divides 6359 & not 47 divides 6359 & not 53 divides 6359 & not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (3 * 2119) + 2 ;
hence not 3 divides 6359 by NAT_4:9; :: thesis: ( not 5 divides 6359 & not 7 divides 6359 & not 11 divides 6359 & not 13 divides 6359 & not 17 divides 6359 & not 19 divides 6359 & not 23 divides 6359 & not 29 divides 6359 & not 31 divides 6359 & not 37 divides 6359 & not 41 divides 6359 & not 43 divides 6359 & not 47 divides 6359 & not 53 divides 6359 & not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (5 * 1271) + 4 ;
hence not 5 divides 6359 by NAT_4:9; :: thesis: ( not 7 divides 6359 & not 11 divides 6359 & not 13 divides 6359 & not 17 divides 6359 & not 19 divides 6359 & not 23 divides 6359 & not 29 divides 6359 & not 31 divides 6359 & not 37 divides 6359 & not 41 divides 6359 & not 43 divides 6359 & not 47 divides 6359 & not 53 divides 6359 & not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (7 * 908) + 3 ;
hence not 7 divides 6359 by NAT_4:9; :: thesis: ( not 11 divides 6359 & not 13 divides 6359 & not 17 divides 6359 & not 19 divides 6359 & not 23 divides 6359 & not 29 divides 6359 & not 31 divides 6359 & not 37 divides 6359 & not 41 divides 6359 & not 43 divides 6359 & not 47 divides 6359 & not 53 divides 6359 & not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (11 * 578) + 1 ;
hence not 11 divides 6359 by NAT_4:9; :: thesis: ( not 13 divides 6359 & not 17 divides 6359 & not 19 divides 6359 & not 23 divides 6359 & not 29 divides 6359 & not 31 divides 6359 & not 37 divides 6359 & not 41 divides 6359 & not 43 divides 6359 & not 47 divides 6359 & not 53 divides 6359 & not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (13 * 489) + 2 ;
hence not 13 divides 6359 by NAT_4:9; :: thesis: ( not 17 divides 6359 & not 19 divides 6359 & not 23 divides 6359 & not 29 divides 6359 & not 31 divides 6359 & not 37 divides 6359 & not 41 divides 6359 & not 43 divides 6359 & not 47 divides 6359 & not 53 divides 6359 & not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (17 * 374) + 1 ;
hence not 17 divides 6359 by NAT_4:9; :: thesis: ( not 19 divides 6359 & not 23 divides 6359 & not 29 divides 6359 & not 31 divides 6359 & not 37 divides 6359 & not 41 divides 6359 & not 43 divides 6359 & not 47 divides 6359 & not 53 divides 6359 & not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (19 * 334) + 13 ;
hence not 19 divides 6359 by NAT_4:9; :: thesis: ( not 23 divides 6359 & not 29 divides 6359 & not 31 divides 6359 & not 37 divides 6359 & not 41 divides 6359 & not 43 divides 6359 & not 47 divides 6359 & not 53 divides 6359 & not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (23 * 276) + 11 ;
hence not 23 divides 6359 by NAT_4:9; :: thesis: ( not 29 divides 6359 & not 31 divides 6359 & not 37 divides 6359 & not 41 divides 6359 & not 43 divides 6359 & not 47 divides 6359 & not 53 divides 6359 & not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (29 * 219) + 8 ;
hence not 29 divides 6359 by NAT_4:9; :: thesis: ( not 31 divides 6359 & not 37 divides 6359 & not 41 divides 6359 & not 43 divides 6359 & not 47 divides 6359 & not 53 divides 6359 & not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (31 * 205) + 4 ;
hence not 31 divides 6359 by NAT_4:9; :: thesis: ( not 37 divides 6359 & not 41 divides 6359 & not 43 divides 6359 & not 47 divides 6359 & not 53 divides 6359 & not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (37 * 171) + 32 ;
hence not 37 divides 6359 by NAT_4:9; :: thesis: ( not 41 divides 6359 & not 43 divides 6359 & not 47 divides 6359 & not 53 divides 6359 & not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (41 * 155) + 4 ;
hence not 41 divides 6359 by NAT_4:9; :: thesis: ( not 43 divides 6359 & not 47 divides 6359 & not 53 divides 6359 & not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (43 * 147) + 38 ;
hence not 43 divides 6359 by NAT_4:9; :: thesis: ( not 47 divides 6359 & not 53 divides 6359 & not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (47 * 135) + 14 ;
hence not 47 divides 6359 by NAT_4:9; :: thesis: ( not 53 divides 6359 & not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (53 * 119) + 52 ;
hence not 53 divides 6359 by NAT_4:9; :: thesis: ( not 59 divides 6359 & not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (59 * 107) + 46 ;
hence not 59 divides 6359 by NAT_4:9; :: thesis: ( not 61 divides 6359 & not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (61 * 104) + 15 ;
hence not 61 divides 6359 by NAT_4:9; :: thesis: ( not 67 divides 6359 & not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (67 * 94) + 61 ;
hence not 67 divides 6359 by NAT_4:9; :: thesis: ( not 71 divides 6359 & not 73 divides 6359 & not 79 divides 6359 )
6359 = (71 * 89) + 40 ;
hence not 71 divides 6359 by NAT_4:9; :: thesis: ( not 73 divides 6359 & not 79 divides 6359 )
6359 = (73 * 87) + 8 ;
hence not 73 divides 6359 by NAT_4:9; :: thesis: not 79 divides 6359
6359 = (79 * 80) + 39 ;
hence not 79 divides 6359 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 6359 & n is prime holds
not n divides 6359 by XPRIMET1:44;
hence 6359 is prime by NAT_4:14; :: thesis: verum