7393 = (2 * 3696) + 1 ;
hence not 2 divides 7393 by NAT_4:9; :: thesis:
7393 = (3 * 2464) + 1 ;
hence not 3 divides 7393 by NAT_4:9; :: thesis:
7393 = (5 * 1478) + 3 ;
hence not 5 divides 7393 by NAT_4:9; :: thesis:
7393 = (7 * 1056) + 1 ;
hence not 7 divides 7393 by NAT_4:9; :: thesis:
7393 = (11 * 672) + 1 ;
hence not 11 divides 7393 by NAT_4:9; :: thesis:
7393 = (13 * 568) + 9 ;
hence not 13 divides 7393 by NAT_4:9; :: thesis:
7393 = (17 * 434) + 15 ;
hence not 17 divides 7393 by NAT_4:9; :: thesis:
7393 = (19 * 389) + 2 ;
hence not 19 divides 7393 by NAT_4:9; :: thesis:
7393 = (23 * 321) + 10 ;
hence not 23 divides 7393 by NAT_4:9; :: thesis:
7393 = (29 * 254) + 27 ;
hence not 29 divides 7393 by NAT_4:9; :: thesis:
7393 = (31 * 238) + 15 ;
hence not 31 divides 7393 by NAT_4:9; :: thesis:
7393 = (37 * 199) + 30 ;
hence not 37 divides 7393 by NAT_4:9; :: thesis:
7393 = (41 * 180) + 13 ;
hence not 41 divides 7393 by NAT_4:9; :: thesis:
7393 = (43 * 171) + 40 ;
hence not 43 divides 7393 by NAT_4:9; :: thesis:
7393 = (47 * 157) + 14 ;
hence not 47 divides 7393 by NAT_4:9; :: thesis:
7393 = (53 * 139) + 26 ;
hence not 53 divides 7393 by NAT_4:9; :: thesis:
7393 = (59 * 125) + 18 ;
hence not 59 divides 7393 by NAT_4:9; :: thesis:
7393 = (61 * 121) + 12 ;
hence not 61 divides 7393 by NAT_4:9; :: thesis:
7393 = (67 * 110) + 23 ;
hence not 67 divides 7393 by NAT_4:9; :: thesis:
7393 = (71 * 104) + 9 ;
hence not 71 divides 7393 by NAT_4:9; :: thesis:
7393 = (73 * 101) + 20 ;
hence not 73 divides 7393 by NAT_4:9; :: thesis:
7393 = (79 * 93) + 46 ;
hence not 79 divides 7393 by NAT_4:9; :: thesis:
7393 = (83 * 89) + 6 ;
hence not 83 divides 7393 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7393 & n is prime holds
not n divides 7393 by XPRIMET1:46;
hence 7393 is prime by NAT_4:14; :: thesis: