now :: thesis: ( not 2 divides 149 & not 3 divides 149 & not 5 divides 149 & not 7 divides 149 & not 11 divides 149 )

149 = (2 * 74) + 1 ;
hence not 2 divides 149 by NAT_4:9; :: thesis: ( not 3 divides 149 & not 5 divides 149 & not 7 divides 149 & not 11 divides 149 )
149 = (3 * 49) + 2 ;
hence not 3 divides 149 by NAT_4:9; :: thesis: ( not 5 divides 149 & not 7 divides 149 & not 11 divides 149 )
149 = (5 * 29) + 4 ;
hence not 5 divides 149 by NAT_4:9; :: thesis: ( not 7 divides 149 & not 11 divides 149 )
149 = (7 * 21) + 2 ;
hence not 7 divides 149 by NAT_4:9; :: thesis: not 11 divides 149
149 = (11 * 13) + 6 ;
hence not 11 divides 149 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 149 & n is prime holds
not n divides 149 by XPRIMET1:10;
hence 149 is prime by NAT_4:14; :: thesis: verum