now :: thesis: ( not 2 divides 9403 & not 3 divides 9403 & not 5 divides 9403 & not 7 divides 9403 & not 11 divides 9403 & not 13 divides 9403 & not 17 divides 9403 & not 19 divides 9403 & not 23 divides 9403 & not 29 divides 9403 & not 31 divides 9403 & not 37 divides 9403 & not 41 divides 9403 & not 43 divides 9403 & not 47 divides 9403 & not 53 divides 9403 & not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )

9403 = (2 * 4701) + 1 ;
hence not 2 divides 9403 by NAT_4:9; :: thesis: ( not 3 divides 9403 & not 5 divides 9403 & not 7 divides 9403 & not 11 divides 9403 & not 13 divides 9403 & not 17 divides 9403 & not 19 divides 9403 & not 23 divides 9403 & not 29 divides 9403 & not 31 divides 9403 & not 37 divides 9403 & not 41 divides 9403 & not 43 divides 9403 & not 47 divides 9403 & not 53 divides 9403 & not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (3 * 3134) + 1 ;
hence not 3 divides 9403 by NAT_4:9; :: thesis: ( not 5 divides 9403 & not 7 divides 9403 & not 11 divides 9403 & not 13 divides 9403 & not 17 divides 9403 & not 19 divides 9403 & not 23 divides 9403 & not 29 divides 9403 & not 31 divides 9403 & not 37 divides 9403 & not 41 divides 9403 & not 43 divides 9403 & not 47 divides 9403 & not 53 divides 9403 & not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (5 * 1880) + 3 ;
hence not 5 divides 9403 by NAT_4:9; :: thesis: ( not 7 divides 9403 & not 11 divides 9403 & not 13 divides 9403 & not 17 divides 9403 & not 19 divides 9403 & not 23 divides 9403 & not 29 divides 9403 & not 31 divides 9403 & not 37 divides 9403 & not 41 divides 9403 & not 43 divides 9403 & not 47 divides 9403 & not 53 divides 9403 & not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (7 * 1343) + 2 ;
hence not 7 divides 9403 by NAT_4:9; :: thesis: ( not 11 divides 9403 & not 13 divides 9403 & not 17 divides 9403 & not 19 divides 9403 & not 23 divides 9403 & not 29 divides 9403 & not 31 divides 9403 & not 37 divides 9403 & not 41 divides 9403 & not 43 divides 9403 & not 47 divides 9403 & not 53 divides 9403 & not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (11 * 854) + 9 ;
hence not 11 divides 9403 by NAT_4:9; :: thesis: ( not 13 divides 9403 & not 17 divides 9403 & not 19 divides 9403 & not 23 divides 9403 & not 29 divides 9403 & not 31 divides 9403 & not 37 divides 9403 & not 41 divides 9403 & not 43 divides 9403 & not 47 divides 9403 & not 53 divides 9403 & not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (13 * 723) + 4 ;
hence not 13 divides 9403 by NAT_4:9; :: thesis: ( not 17 divides 9403 & not 19 divides 9403 & not 23 divides 9403 & not 29 divides 9403 & not 31 divides 9403 & not 37 divides 9403 & not 41 divides 9403 & not 43 divides 9403 & not 47 divides 9403 & not 53 divides 9403 & not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (17 * 553) + 2 ;
hence not 17 divides 9403 by NAT_4:9; :: thesis: ( not 19 divides 9403 & not 23 divides 9403 & not 29 divides 9403 & not 31 divides 9403 & not 37 divides 9403 & not 41 divides 9403 & not 43 divides 9403 & not 47 divides 9403 & not 53 divides 9403 & not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (19 * 494) + 17 ;
hence not 19 divides 9403 by NAT_4:9; :: thesis: ( not 23 divides 9403 & not 29 divides 9403 & not 31 divides 9403 & not 37 divides 9403 & not 41 divides 9403 & not 43 divides 9403 & not 47 divides 9403 & not 53 divides 9403 & not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (23 * 408) + 19 ;
hence not 23 divides 9403 by NAT_4:9; :: thesis: ( not 29 divides 9403 & not 31 divides 9403 & not 37 divides 9403 & not 41 divides 9403 & not 43 divides 9403 & not 47 divides 9403 & not 53 divides 9403 & not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (29 * 324) + 7 ;
hence not 29 divides 9403 by NAT_4:9; :: thesis: ( not 31 divides 9403 & not 37 divides 9403 & not 41 divides 9403 & not 43 divides 9403 & not 47 divides 9403 & not 53 divides 9403 & not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (31 * 303) + 10 ;
hence not 31 divides 9403 by NAT_4:9; :: thesis: ( not 37 divides 9403 & not 41 divides 9403 & not 43 divides 9403 & not 47 divides 9403 & not 53 divides 9403 & not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (37 * 254) + 5 ;
hence not 37 divides 9403 by NAT_4:9; :: thesis: ( not 41 divides 9403 & not 43 divides 9403 & not 47 divides 9403 & not 53 divides 9403 & not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (41 * 229) + 14 ;
hence not 41 divides 9403 by NAT_4:9; :: thesis: ( not 43 divides 9403 & not 47 divides 9403 & not 53 divides 9403 & not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (43 * 218) + 29 ;
hence not 43 divides 9403 by NAT_4:9; :: thesis: ( not 47 divides 9403 & not 53 divides 9403 & not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (47 * 200) + 3 ;
hence not 47 divides 9403 by NAT_4:9; :: thesis: ( not 53 divides 9403 & not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (53 * 177) + 22 ;
hence not 53 divides 9403 by NAT_4:9; :: thesis: ( not 59 divides 9403 & not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (59 * 159) + 22 ;
hence not 59 divides 9403 by NAT_4:9; :: thesis: ( not 61 divides 9403 & not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (61 * 154) + 9 ;
hence not 61 divides 9403 by NAT_4:9; :: thesis: ( not 67 divides 9403 & not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (67 * 140) + 23 ;
hence not 67 divides 9403 by NAT_4:9; :: thesis: ( not 71 divides 9403 & not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (71 * 132) + 31 ;
hence not 71 divides 9403 by NAT_4:9; :: thesis: ( not 73 divides 9403 & not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (73 * 128) + 59 ;
hence not 73 divides 9403 by NAT_4:9; :: thesis: ( not 79 divides 9403 & not 83 divides 9403 & not 89 divides 9403 )
9403 = (79 * 119) + 2 ;
hence not 79 divides 9403 by NAT_4:9; :: thesis: ( not 83 divides 9403 & not 89 divides 9403 )
9403 = (83 * 113) + 24 ;
hence not 83 divides 9403 by NAT_4:9; :: thesis: not 89 divides 9403
9403 = (89 * 105) + 58 ;
hence not 89 divides 9403 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 9403 & n is prime holds
not n divides 9403 by XPRIMET1:48;
hence 9403 is prime by NAT_4:14; :: thesis: verum