now :: thesis: ( not 2 divides 157 & not 3 divides 157 & not 5 divides 157 & not 7 divides 157 & not 11 divides 157 )

157 = (2 * 78) + 1 ;
hence not 2 divides 157 by NAT_4:9; :: thesis: ( not 3 divides 157 & not 5 divides 157 & not 7 divides 157 & not 11 divides 157 )
157 = (3 * 52) + 1 ;
hence not 3 divides 157 by NAT_4:9; :: thesis: ( not 5 divides 157 & not 7 divides 157 & not 11 divides 157 )
157 = (5 * 31) + 2 ;
hence not 5 divides 157 by NAT_4:9; :: thesis: ( not 7 divides 157 & not 11 divides 157 )
157 = (7 * 22) + 3 ;
hence not 7 divides 157 by NAT_4:9; :: thesis: not 11 divides 157
157 = (11 * 14) + 3 ;
hence not 11 divides 157 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 157 & n is prime holds
not n divides 157 by XPRIMET1:10;
hence 157 is prime by NAT_4:14; :: thesis: verum