now :: thesis: ( not 2 divides 109 & not 3 divides 109 & not 5 divides 109 & not 7 divides 109 )

109 = (2 * 54) + 1 ;
hence not 2 divides 109 by NAT_4:9; :: thesis: ( not 3 divides 109 & not 5 divides 109 & not 7 divides 109 )
109 = (3 * 36) + 1 ;
hence not 3 divides 109 by NAT_4:9; :: thesis: ( not 5 divides 109 & not 7 divides 109 )
109 = (5 * 21) + 4 ;
hence not 5 divides 109 by NAT_4:9; :: thesis: not 7 divides 109
109 = (7 * 15) + 4 ;
hence not 7 divides 109 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 109 & n is prime holds
not n divides 109 by XPRIMET1:8;
hence 109 is prime by NAT_4:14; :: thesis: verum