now :: thesis: ( not 2 divides 97 & not 3 divides 97 & not 5 divides 97 & not 7 divides 97 )

97 = (2 * 48) + 1 ;
hence not 2 divides 97 by NAT_4:9; :: thesis: ( not 3 divides 97 & not 5 divides 97 & not 7 divides 97 )
97 = (3 * 32) + 1 ;
hence not 3 divides 97 by NAT_4:9; :: thesis: ( not 5 divides 97 & not 7 divides 97 )
97 = (5 * 19) + 2 ;
hence not 5 divides 97 by NAT_4:9; :: thesis: not 7 divides 97
97 = (7 * 13) + 6 ;
hence not 7 divides 97 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 97 & n is prime holds
not n divides 97 by XPRIMET1:8;
hence 97 is prime by NAT_4:14; :: thesis: verum