now :: thesis: ( not 2 divides 163 & not 3 divides 163 & not 5 divides 163 & not 7 divides 163 & not 11 divides 163 )

163 = (2 * 81) + 1 ;
hence not 2 divides 163 by NAT_4:9; :: thesis: ( not 3 divides 163 & not 5 divides 163 & not 7 divides 163 & not 11 divides 163 )
163 = (3 * 54) + 1 ;
hence not 3 divides 163 by NAT_4:9; :: thesis: ( not 5 divides 163 & not 7 divides 163 & not 11 divides 163 )
163 = (5 * 32) + 3 ;
hence not 5 divides 163 by NAT_4:9; :: thesis: ( not 7 divides 163 & not 11 divides 163 )
163 = (7 * 23) + 2 ;
hence not 7 divides 163 by NAT_4:9; :: thesis: not 11 divides 163
163 = (11 * 14) + 9 ;
hence not 11 divides 163 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 163 & n is prime holds
not n divides 163 by XPRIMET1:10;
hence 163 is prime by NAT_4:14; :: thesis: verum