6173 = (2 * 3086) + 1 ;
hence not 2 divides 6173 by NAT_4:9; :: thesis:
6173 = (3 * 2057) + 2 ;
hence not 3 divides 6173 by NAT_4:9; :: thesis:
6173 = (5 * 1234) + 3 ;
hence not 5 divides 6173 by NAT_4:9; :: thesis:
6173 = (7 * 881) + 6 ;
hence not 7 divides 6173 by NAT_4:9; :: thesis:
6173 = (11 * 561) + 2 ;
hence not 11 divides 6173 by NAT_4:9; :: thesis:
6173 = (13 * 474) + 11 ;
hence not 13 divides 6173 by NAT_4:9; :: thesis:
6173 = (17 * 363) + 2 ;
hence not 17 divides 6173 by NAT_4:9; :: thesis:
6173 = (19 * 324) + 17 ;
hence not 19 divides 6173 by NAT_4:9; :: thesis:
6173 = (23 * 268) + 9 ;
hence not 23 divides 6173 by NAT_4:9; :: thesis:
6173 = (29 * 212) + 25 ;
hence not 29 divides 6173 by NAT_4:9; :: thesis:
6173 = (31 * 199) + 4 ;
hence not 31 divides 6173 by NAT_4:9; :: thesis:
6173 = (37 * 166) + 31 ;
hence not 37 divides 6173 by NAT_4:9; :: thesis:
6173 = (41 * 150) + 23 ;
hence not 41 divides 6173 by NAT_4:9; :: thesis:
6173 = (43 * 143) + 24 ;
hence not 43 divides 6173 by NAT_4:9; :: thesis:
6173 = (47 * 131) + 16 ;
hence not 47 divides 6173 by NAT_4:9; :: thesis:
6173 = (53 * 116) + 25 ;
hence not 53 divides 6173 by NAT_4:9; :: thesis:
6173 = (59 * 104) + 37 ;
hence not 59 divides 6173 by NAT_4:9; :: thesis:
6173 = (61 * 101) + 12 ;
hence not 61 divides 6173 by NAT_4:9; :: thesis:
6173 = (67 * 92) + 9 ;
hence not 67 divides 6173 by NAT_4:9; :: thesis:
6173 = (71 * 86) + 67 ;
hence not 71 divides 6173 by NAT_4:9; :: thesis:
6173 = (73 * 84) + 41 ;
hence not 73 divides 6173 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6173 & n is prime holds
not n divides 6173 by XPRIMET1:42;
hence 6173 is prime by NAT_4:14; :: thesis: