now :: thesis: ( not 2 divides 151 & not 3 divides 151 & not 5 divides 151 & not 7 divides 151 & not 11 divides 151 )

151 = (2 * 75) + 1 ;
hence not 2 divides 151 by NAT_4:9; :: thesis: ( not 3 divides 151 & not 5 divides 151 & not 7 divides 151 & not 11 divides 151 )
151 = (3 * 50) + 1 ;
hence not 3 divides 151 by NAT_4:9; :: thesis: ( not 5 divides 151 & not 7 divides 151 & not 11 divides 151 )
151 = (5 * 30) + 1 ;
hence not 5 divides 151 by NAT_4:9; :: thesis: ( not 7 divides 151 & not 11 divides 151 )
151 = (7 * 21) + 4 ;
hence not 7 divides 151 by NAT_4:9; :: thesis: not 11 divides 151
151 = (11 * 13) + 8 ;
hence not 11 divides 151 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 151 & n is prime holds
not n divides 151 by XPRIMET1:10;
hence 151 is prime by NAT_4:14; :: thesis: verum