now :: thesis: ( not 2 divides 101 & not 3 divides 101 & not 5 divides 101 & not 7 divides 101 )

101 = (2 * 50) + 1 ;
hence not 2 divides 101 by NAT_4:9; :: thesis: ( not 3 divides 101 & not 5 divides 101 & not 7 divides 101 )
101 = (3 * 33) + 2 ;
hence not 3 divides 101 by NAT_4:9; :: thesis: ( not 5 divides 101 & not 7 divides 101 )
101 = (5 * 20) + 1 ;
hence not 5 divides 101 by NAT_4:9; :: thesis: not 7 divides 101
101 = (7 * 14) + 3 ;
hence not 7 divides 101 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 101 & n is prime holds
not n divides 101 by XPRIMET1:8;
hence 101 is prime by NAT_4:14; :: thesis: verum