now :: thesis: ( not 2 divides 43 & not 3 divides 43 & not 5 divides 43 )

43 = (2 * 21) + 1 ;
hence not 2 divides 43 by NAT_4:9; :: thesis: ( not 3 divides 43 & not 5 divides 43 )
43 = (3 * 14) + 1 ;
hence not 3 divides 43 by NAT_4:9; :: thesis: not 5 divides 43
43 = (5 * 8) + 3 ;
hence not 5 divides 43 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 43 & n is prime holds
not n divides 43 by XPRIMET1:6;
hence 43 is prime by NAT_4:14; :: thesis: verum