now :: thesis: ( not 2 divides 5393 & not 3 divides 5393 & not 5 divides 5393 & not 7 divides 5393 & not 11 divides 5393 & not 13 divides 5393 & not 17 divides 5393 & not 19 divides 5393 & not 23 divides 5393 & not 29 divides 5393 & not 31 divides 5393 & not 37 divides 5393 & not 41 divides 5393 & not 43 divides 5393 & not 47 divides 5393 & not 53 divides 5393 & not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )

5393 = (2 * 2696) + 1 ;
hence not 2 divides 5393 by NAT_4:9; :: thesis: ( not 3 divides 5393 & not 5 divides 5393 & not 7 divides 5393 & not 11 divides 5393 & not 13 divides 5393 & not 17 divides 5393 & not 19 divides 5393 & not 23 divides 5393 & not 29 divides 5393 & not 31 divides 5393 & not 37 divides 5393 & not 41 divides 5393 & not 43 divides 5393 & not 47 divides 5393 & not 53 divides 5393 & not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (3 * 1797) + 2 ;
hence not 3 divides 5393 by NAT_4:9; :: thesis: ( not 5 divides 5393 & not 7 divides 5393 & not 11 divides 5393 & not 13 divides 5393 & not 17 divides 5393 & not 19 divides 5393 & not 23 divides 5393 & not 29 divides 5393 & not 31 divides 5393 & not 37 divides 5393 & not 41 divides 5393 & not 43 divides 5393 & not 47 divides 5393 & not 53 divides 5393 & not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (5 * 1078) + 3 ;
hence not 5 divides 5393 by NAT_4:9; :: thesis: ( not 7 divides 5393 & not 11 divides 5393 & not 13 divides 5393 & not 17 divides 5393 & not 19 divides 5393 & not 23 divides 5393 & not 29 divides 5393 & not 31 divides 5393 & not 37 divides 5393 & not 41 divides 5393 & not 43 divides 5393 & not 47 divides 5393 & not 53 divides 5393 & not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (7 * 770) + 3 ;
hence not 7 divides 5393 by NAT_4:9; :: thesis: ( not 11 divides 5393 & not 13 divides 5393 & not 17 divides 5393 & not 19 divides 5393 & not 23 divides 5393 & not 29 divides 5393 & not 31 divides 5393 & not 37 divides 5393 & not 41 divides 5393 & not 43 divides 5393 & not 47 divides 5393 & not 53 divides 5393 & not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (11 * 490) + 3 ;
hence not 11 divides 5393 by NAT_4:9; :: thesis: ( not 13 divides 5393 & not 17 divides 5393 & not 19 divides 5393 & not 23 divides 5393 & not 29 divides 5393 & not 31 divides 5393 & not 37 divides 5393 & not 41 divides 5393 & not 43 divides 5393 & not 47 divides 5393 & not 53 divides 5393 & not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (13 * 414) + 11 ;
hence not 13 divides 5393 by NAT_4:9; :: thesis: ( not 17 divides 5393 & not 19 divides 5393 & not 23 divides 5393 & not 29 divides 5393 & not 31 divides 5393 & not 37 divides 5393 & not 41 divides 5393 & not 43 divides 5393 & not 47 divides 5393 & not 53 divides 5393 & not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (17 * 317) + 4 ;
hence not 17 divides 5393 by NAT_4:9; :: thesis: ( not 19 divides 5393 & not 23 divides 5393 & not 29 divides 5393 & not 31 divides 5393 & not 37 divides 5393 & not 41 divides 5393 & not 43 divides 5393 & not 47 divides 5393 & not 53 divides 5393 & not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (19 * 283) + 16 ;
hence not 19 divides 5393 by NAT_4:9; :: thesis: ( not 23 divides 5393 & not 29 divides 5393 & not 31 divides 5393 & not 37 divides 5393 & not 41 divides 5393 & not 43 divides 5393 & not 47 divides 5393 & not 53 divides 5393 & not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (23 * 234) + 11 ;
hence not 23 divides 5393 by NAT_4:9; :: thesis: ( not 29 divides 5393 & not 31 divides 5393 & not 37 divides 5393 & not 41 divides 5393 & not 43 divides 5393 & not 47 divides 5393 & not 53 divides 5393 & not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (29 * 185) + 28 ;
hence not 29 divides 5393 by NAT_4:9; :: thesis: ( not 31 divides 5393 & not 37 divides 5393 & not 41 divides 5393 & not 43 divides 5393 & not 47 divides 5393 & not 53 divides 5393 & not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (31 * 173) + 30 ;
hence not 31 divides 5393 by NAT_4:9; :: thesis: ( not 37 divides 5393 & not 41 divides 5393 & not 43 divides 5393 & not 47 divides 5393 & not 53 divides 5393 & not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (37 * 145) + 28 ;
hence not 37 divides 5393 by NAT_4:9; :: thesis: ( not 41 divides 5393 & not 43 divides 5393 & not 47 divides 5393 & not 53 divides 5393 & not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (41 * 131) + 22 ;
hence not 41 divides 5393 by NAT_4:9; :: thesis: ( not 43 divides 5393 & not 47 divides 5393 & not 53 divides 5393 & not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (43 * 125) + 18 ;
hence not 43 divides 5393 by NAT_4:9; :: thesis: ( not 47 divides 5393 & not 53 divides 5393 & not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (47 * 114) + 35 ;
hence not 47 divides 5393 by NAT_4:9; :: thesis: ( not 53 divides 5393 & not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (53 * 101) + 40 ;
hence not 53 divides 5393 by NAT_4:9; :: thesis: ( not 59 divides 5393 & not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (59 * 91) + 24 ;
hence not 59 divides 5393 by NAT_4:9; :: thesis: ( not 61 divides 5393 & not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (61 * 88) + 25 ;
hence not 61 divides 5393 by NAT_4:9; :: thesis: ( not 67 divides 5393 & not 71 divides 5393 & not 73 divides 5393 )
5393 = (67 * 80) + 33 ;
hence not 67 divides 5393 by NAT_4:9; :: thesis: ( not 71 divides 5393 & not 73 divides 5393 )
5393 = (71 * 75) + 68 ;
hence not 71 divides 5393 by NAT_4:9; :: thesis: not 73 divides 5393
5393 = (73 * 73) + 64 ;
hence not 73 divides 5393 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 5393 & n is prime holds
not n divides 5393 by XPRIMET1:42;
hence 5393 is prime by NAT_4:14; :: thesis: verum