now :: thesis: ( not 2 divides 41 & not 3 divides 41 & not 5 divides 41 )

41 = (2 * 20) + 1 ;
hence not 2 divides 41 by NAT_4:9; :: thesis: ( not 3 divides 41 & not 5 divides 41 )
41 = (3 * 13) + 2 ;
hence not 3 divides 41 by NAT_4:9; :: thesis: not 5 divides 41
41 = (5 * 8) + 1 ;
hence not 5 divides 41 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 41 & n is prime holds
not n divides 41 by XPRIMET1:6;
hence 41 is prime by NAT_4:14; :: thesis: verum