now :: thesis: ( not 2 divides 47 & not 3 divides 47 & not 5 divides 47 )

47 = (2 * 23) + 1 ;
hence not 2 divides 47 by NAT_4:9; :: thesis: ( not 3 divides 47 & not 5 divides 47 )
47 = (3 * 15) + 2 ;
hence not 3 divides 47 by NAT_4:9; :: thesis: not 5 divides 47
47 = (5 * 9) + 2 ;
hence not 5 divides 47 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 47 & n is prime holds
not n divides 47 by XPRIMET1:6;
hence 47 is prime by NAT_4:14; :: thesis: verum