6163 = (2 * 3081) + 1 ;
hence not 2 divides 6163 by NAT_4:9; :: thesis:
6163 = (3 * 2054) + 1 ;
hence not 3 divides 6163 by NAT_4:9; :: thesis:
6163 = (5 * 1232) + 3 ;
hence not 5 divides 6163 by NAT_4:9; :: thesis:
6163 = (7 * 880) + 3 ;
hence not 7 divides 6163 by NAT_4:9; :: thesis:
6163 = (11 * 560) + 3 ;
hence not 11 divides 6163 by NAT_4:9; :: thesis:
6163 = (13 * 474) + 1 ;
hence not 13 divides 6163 by NAT_4:9; :: thesis:
6163 = (17 * 362) + 9 ;
hence not 17 divides 6163 by NAT_4:9; :: thesis:
6163 = (19 * 324) + 7 ;
hence not 19 divides 6163 by NAT_4:9; :: thesis:
6163 = (23 * 267) + 22 ;
hence not 23 divides 6163 by NAT_4:9; :: thesis:
6163 = (29 * 212) + 15 ;
hence not 29 divides 6163 by NAT_4:9; :: thesis:
6163 = (31 * 198) + 25 ;
hence not 31 divides 6163 by NAT_4:9; :: thesis:
6163 = (37 * 166) + 21 ;
hence not 37 divides 6163 by NAT_4:9; :: thesis:
6163 = (41 * 150) + 13 ;
hence not 41 divides 6163 by NAT_4:9; :: thesis:
6163 = (43 * 143) + 14 ;
hence not 43 divides 6163 by NAT_4:9; :: thesis:
6163 = (47 * 131) + 6 ;
hence not 47 divides 6163 by NAT_4:9; :: thesis:
6163 = (53 * 116) + 15 ;
hence not 53 divides 6163 by NAT_4:9; :: thesis:
6163 = (59 * 104) + 27 ;
hence not 59 divides 6163 by NAT_4:9; :: thesis:
6163 = (61 * 101) + 2 ;
hence not 61 divides 6163 by NAT_4:9; :: thesis:
6163 = (67 * 91) + 66 ;
hence not 67 divides 6163 by NAT_4:9; :: thesis:
6163 = (71 * 86) + 57 ;
hence not 71 divides 6163 by NAT_4:9; :: thesis:
6163 = (73 * 84) + 31 ;
hence not 73 divides 6163 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6163 & n is prime holds
not n divides 6163 by XPRIMET1:42;
hence 6163 is prime by NAT_4:14; :: thesis: