now :: thesis: ( not 2 divides 71 & not 3 divides 71 & not 5 divides 71 & not 7 divides 71 )

71 = (2 * 35) + 1 ;
hence not 2 divides 71 by NAT_4:9; :: thesis: ( not 3 divides 71 & not 5 divides 71 & not 7 divides 71 )
71 = (3 * 23) + 2 ;
hence not 3 divides 71 by NAT_4:9; :: thesis: ( not 5 divides 71 & not 7 divides 71 )
71 = (5 * 14) + 1 ;
hence not 5 divides 71 by NAT_4:9; :: thesis: not 7 divides 71
71 = (7 * 10) + 1 ;
hence not 7 divides 71 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 71 & n is prime holds
not n divides 71 by XPRIMET1:8;
hence 71 is prime by NAT_4:14; :: thesis: verum