now :: thesis: ( not 2 divides 197 & not 3 divides 197 & not 5 divides 197 & not 7 divides 197 & not 11 divides 197 & not 13 divides 197 )

197 = (2 * 98) + 1 ;
hence not 2 divides 197 by NAT_4:9; :: thesis: ( not 3 divides 197 & not 5 divides 197 & not 7 divides 197 & not 11 divides 197 & not 13 divides 197 )
197 = (3 * 65) + 2 ;
hence not 3 divides 197 by NAT_4:9; :: thesis: ( not 5 divides 197 & not 7 divides 197 & not 11 divides 197 & not 13 divides 197 )
197 = (5 * 39) + 2 ;
hence not 5 divides 197 by NAT_4:9; :: thesis: ( not 7 divides 197 & not 11 divides 197 & not 13 divides 197 )
197 = (7 * 28) + 1 ;
hence not 7 divides 197 by NAT_4:9; :: thesis: ( not 11 divides 197 & not 13 divides 197 )
197 = (11 * 17) + 10 ;
hence not 11 divides 197 by NAT_4:9; :: thesis: not 13 divides 197
197 = (13 * 15) + 2 ;
hence not 13 divides 197 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 197 & n is prime holds
not n divides 197 by XPRIMET1:12;
hence 197 is prime by NAT_4:14; :: thesis: verum