now :: thesis: ( not 2 divides 179 & not 3 divides 179 & not 5 divides 179 & not 7 divides 179 & not 11 divides 179 & not 13 divides 179 )

179 = (2 * 89) + 1 ;
hence not 2 divides 179 by NAT_4:9; :: thesis: ( not 3 divides 179 & not 5 divides 179 & not 7 divides 179 & not 11 divides 179 & not 13 divides 179 )
179 = (3 * 59) + 2 ;
hence not 3 divides 179 by NAT_4:9; :: thesis: ( not 5 divides 179 & not 7 divides 179 & not 11 divides 179 & not 13 divides 179 )
179 = (5 * 35) + 4 ;
hence not 5 divides 179 by NAT_4:9; :: thesis: ( not 7 divides 179 & not 11 divides 179 & not 13 divides 179 )
179 = (7 * 25) + 4 ;
hence not 7 divides 179 by NAT_4:9; :: thesis: ( not 11 divides 179 & not 13 divides 179 )
179 = (11 * 16) + 3 ;
hence not 11 divides 179 by NAT_4:9; :: thesis: not 13 divides 179
179 = (13 * 13) + 10 ;
hence not 13 divides 179 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 179 & n is prime holds
not n divides 179 by XPRIMET1:12;
hence 179 is prime by NAT_4:14; :: thesis: verum