now :: thesis: ( not 2 divides 193 & not 3 divides 193 & not 5 divides 193 & not 7 divides 193 & not 11 divides 193 & not 13 divides 193 )

193 = (2 * 96) + 1 ;
hence not 2 divides 193 by NAT_4:9; :: thesis: ( not 3 divides 193 & not 5 divides 193 & not 7 divides 193 & not 11 divides 193 & not 13 divides 193 )
193 = (3 * 64) + 1 ;
hence not 3 divides 193 by NAT_4:9; :: thesis: ( not 5 divides 193 & not 7 divides 193 & not 11 divides 193 & not 13 divides 193 )
193 = (5 * 38) + 3 ;
hence not 5 divides 193 by NAT_4:9; :: thesis: ( not 7 divides 193 & not 11 divides 193 & not 13 divides 193 )
193 = (7 * 27) + 4 ;
hence not 7 divides 193 by NAT_4:9; :: thesis: ( not 11 divides 193 & not 13 divides 193 )
193 = (11 * 17) + 6 ;
hence not 11 divides 193 by NAT_4:9; :: thesis: not 13 divides 193
193 = (13 * 14) + 11 ;
hence not 13 divides 193 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 193 & n is prime holds
not n divides 193 by XPRIMET1:12;
hence 193 is prime by NAT_4:14; :: thesis: verum