now :: thesis: ( not 2 divides 3167 & not 3 divides 3167 & not 5 divides 3167 & not 7 divides 3167 & not 11 divides 3167 & not 13 divides 3167 & not 17 divides 3167 & not 19 divides 3167 & not 23 divides 3167 & not 29 divides 3167 & not 31 divides 3167 & not 37 divides 3167 & not 41 divides 3167 & not 43 divides 3167 & not 47 divides 3167 & not 53 divides 3167 )

3167 = (2 * 1583) + 1 ;
hence not 2 divides 3167 by NAT_4:9; :: thesis: ( not 3 divides 3167 & not 5 divides 3167 & not 7 divides 3167 & not 11 divides 3167 & not 13 divides 3167 & not 17 divides 3167 & not 19 divides 3167 & not 23 divides 3167 & not 29 divides 3167 & not 31 divides 3167 & not 37 divides 3167 & not 41 divides 3167 & not 43 divides 3167 & not 47 divides 3167 & not 53 divides 3167 )
3167 = (3 * 1055) + 2 ;
hence not 3 divides 3167 by NAT_4:9; :: thesis: ( not 5 divides 3167 & not 7 divides 3167 & not 11 divides 3167 & not 13 divides 3167 & not 17 divides 3167 & not 19 divides 3167 & not 23 divides 3167 & not 29 divides 3167 & not 31 divides 3167 & not 37 divides 3167 & not 41 divides 3167 & not 43 divides 3167 & not 47 divides 3167 & not 53 divides 3167 )
3167 = (5 * 633) + 2 ;
hence not 5 divides 3167 by NAT_4:9; :: thesis: ( not 7 divides 3167 & not 11 divides 3167 & not 13 divides 3167 & not 17 divides 3167 & not 19 divides 3167 & not 23 divides 3167 & not 29 divides 3167 & not 31 divides 3167 & not 37 divides 3167 & not 41 divides 3167 & not 43 divides 3167 & not 47 divides 3167 & not 53 divides 3167 )
3167 = (7 * 452) + 3 ;
hence not 7 divides 3167 by NAT_4:9; :: thesis: ( not 11 divides 3167 & not 13 divides 3167 & not 17 divides 3167 & not 19 divides 3167 & not 23 divides 3167 & not 29 divides 3167 & not 31 divides 3167 & not 37 divides 3167 & not 41 divides 3167 & not 43 divides 3167 & not 47 divides 3167 & not 53 divides 3167 )
3167 = (11 * 287) + 10 ;
hence not 11 divides 3167 by NAT_4:9; :: thesis: ( not 13 divides 3167 & not 17 divides 3167 & not 19 divides 3167 & not 23 divides 3167 & not 29 divides 3167 & not 31 divides 3167 & not 37 divides 3167 & not 41 divides 3167 & not 43 divides 3167 & not 47 divides 3167 & not 53 divides 3167 )
3167 = (13 * 243) + 8 ;
hence not 13 divides 3167 by NAT_4:9; :: thesis: ( not 17 divides 3167 & not 19 divides 3167 & not 23 divides 3167 & not 29 divides 3167 & not 31 divides 3167 & not 37 divides 3167 & not 41 divides 3167 & not 43 divides 3167 & not 47 divides 3167 & not 53 divides 3167 )
3167 = (17 * 186) + 5 ;
hence not 17 divides 3167 by NAT_4:9; :: thesis: ( not 19 divides 3167 & not 23 divides 3167 & not 29 divides 3167 & not 31 divides 3167 & not 37 divides 3167 & not 41 divides 3167 & not 43 divides 3167 & not 47 divides 3167 & not 53 divides 3167 )
3167 = (19 * 166) + 13 ;
hence not 19 divides 3167 by NAT_4:9; :: thesis: ( not 23 divides 3167 & not 29 divides 3167 & not 31 divides 3167 & not 37 divides 3167 & not 41 divides 3167 & not 43 divides 3167 & not 47 divides 3167 & not 53 divides 3167 )
3167 = (23 * 137) + 16 ;
hence not 23 divides 3167 by NAT_4:9; :: thesis: ( not 29 divides 3167 & not 31 divides 3167 & not 37 divides 3167 & not 41 divides 3167 & not 43 divides 3167 & not 47 divides 3167 & not 53 divides 3167 )
3167 = (29 * 109) + 6 ;
hence not 29 divides 3167 by NAT_4:9; :: thesis: ( not 31 divides 3167 & not 37 divides 3167 & not 41 divides 3167 & not 43 divides 3167 & not 47 divides 3167 & not 53 divides 3167 )
3167 = (31 * 102) + 5 ;
hence not 31 divides 3167 by NAT_4:9; :: thesis: ( not 37 divides 3167 & not 41 divides 3167 & not 43 divides 3167 & not 47 divides 3167 & not 53 divides 3167 )
3167 = (37 * 85) + 22 ;
hence not 37 divides 3167 by NAT_4:9; :: thesis: ( not 41 divides 3167 & not 43 divides 3167 & not 47 divides 3167 & not 53 divides 3167 )
3167 = (41 * 77) + 10 ;
hence not 41 divides 3167 by NAT_4:9; :: thesis: ( not 43 divides 3167 & not 47 divides 3167 & not 53 divides 3167 )
3167 = (43 * 73) + 28 ;
hence not 43 divides 3167 by NAT_4:9; :: thesis: ( not 47 divides 3167 & not 53 divides 3167 )
3167 = (47 * 67) + 18 ;
hence not 47 divides 3167 by NAT_4:9; :: thesis: not 53 divides 3167
3167 = (53 * 59) + 40 ;
hence not 53 divides 3167 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 3167 & n is prime holds
not n divides 3167 by XPRIMET1:32;
hence 3167 is prime by NAT_4:14; :: thesis: verum