now :: thesis: ( not 2 divides 191 & not 3 divides 191 & not 5 divides 191 & not 7 divides 191 & not 11 divides 191 & not 13 divides 191 )

191 = (2 * 95) + 1 ;
hence not 2 divides 191 by NAT_4:9; :: thesis: ( not 3 divides 191 & not 5 divides 191 & not 7 divides 191 & not 11 divides 191 & not 13 divides 191 )
191 = (3 * 63) + 2 ;
hence not 3 divides 191 by NAT_4:9; :: thesis: ( not 5 divides 191 & not 7 divides 191 & not 11 divides 191 & not 13 divides 191 )
191 = (5 * 38) + 1 ;
hence not 5 divides 191 by NAT_4:9; :: thesis: ( not 7 divides 191 & not 11 divides 191 & not 13 divides 191 )
191 = (7 * 27) + 2 ;
hence not 7 divides 191 by NAT_4:9; :: thesis: ( not 11 divides 191 & not 13 divides 191 )
191 = (11 * 17) + 4 ;
hence not 11 divides 191 by NAT_4:9; :: thesis: not 13 divides 191
191 = (13 * 14) + 9 ;
hence not 13 divides 191 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 191 & n is prime holds
not n divides 191 by XPRIMET1:12;
hence 191 is prime by NAT_4:14; :: thesis: verum