now :: thesis: ( not 2 divides 199 & not 3 divides 199 & not 5 divides 199 & not 7 divides 199 & not 11 divides 199 & not 13 divides 199 )

199 = (2 * 99) + 1 ;
hence not 2 divides 199 by NAT_4:9; :: thesis: ( not 3 divides 199 & not 5 divides 199 & not 7 divides 199 & not 11 divides 199 & not 13 divides 199 )
199 = (3 * 66) + 1 ;
hence not 3 divides 199 by NAT_4:9; :: thesis: ( not 5 divides 199 & not 7 divides 199 & not 11 divides 199 & not 13 divides 199 )
199 = (5 * 39) + 4 ;
hence not 5 divides 199 by NAT_4:9; :: thesis: ( not 7 divides 199 & not 11 divides 199 & not 13 divides 199 )
199 = (7 * 28) + 3 ;
hence not 7 divides 199 by NAT_4:9; :: thesis: ( not 11 divides 199 & not 13 divides 199 )
199 = (11 * 18) + 1 ;
hence not 11 divides 199 by NAT_4:9; :: thesis: not 13 divides 199
199 = (13 * 15) + 4 ;
hence not 13 divides 199 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 199 & n is prime holds
not n divides 199 by XPRIMET1:12;
hence 199 is prime by NAT_4:14; :: thesis: verum