3163 = (2 * 1581) + 1 ;
hence not 2 divides 3163 by NAT_4:9; :: thesis:
3163 = (3 * 1054) + 1 ;
hence not 3 divides 3163 by NAT_4:9; :: thesis:
3163 = (5 * 632) + 3 ;
hence not 5 divides 3163 by NAT_4:9; :: thesis:
3163 = (7 * 451) + 6 ;
hence not 7 divides 3163 by NAT_4:9; :: thesis:
3163 = (11 * 287) + 6 ;
hence not 11 divides 3163 by NAT_4:9; :: thesis:
3163 = (13 * 243) + 4 ;
hence not 13 divides 3163 by NAT_4:9; :: thesis:
3163 = (17 * 186) + 1 ;
hence not 17 divides 3163 by NAT_4:9; :: thesis:
3163 = (19 * 166) + 9 ;
hence not 19 divides 3163 by NAT_4:9; :: thesis:
3163 = (23 * 137) + 12 ;
hence not 23 divides 3163 by NAT_4:9; :: thesis:
3163 = (29 * 109) + 2 ;
hence not 29 divides 3163 by NAT_4:9; :: thesis:
3163 = (31 * 102) + 1 ;
hence not 31 divides 3163 by NAT_4:9; :: thesis:
3163 = (37 * 85) + 18 ;
hence not 37 divides 3163 by NAT_4:9; :: thesis:
3163 = (41 * 77) + 6 ;
hence not 41 divides 3163 by NAT_4:9; :: thesis:
3163 = (43 * 73) + 24 ;
hence not 43 divides 3163 by NAT_4:9; :: thesis:
3163 = (47 * 67) + 14 ;
hence not 47 divides 3163 by NAT_4:9; :: thesis:
3163 = (53 * 59) + 36 ;
hence not 53 divides 3163 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3163 & n is prime holds
not n divides 3163 by XPRIMET1:32;
hence 3163 is prime by NAT_4:14; :: thesis: