now :: thesis: ( not 2 divides 107 & not 3 divides 107 & not 5 divides 107 & not 7 divides 107 )

107 = (2 * 53) + 1 ;
hence not 2 divides 107 by NAT_4:9; :: thesis: ( not 3 divides 107 & not 5 divides 107 & not 7 divides 107 )
107 = (3 * 35) + 2 ;
hence not 3 divides 107 by NAT_4:9; :: thesis: ( not 5 divides 107 & not 7 divides 107 )
107 = (5 * 21) + 2 ;
hence not 5 divides 107 by NAT_4:9; :: thesis: not 7 divides 107
107 = (7 * 15) + 2 ;
hence not 7 divides 107 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 107 & n is prime holds
not n divides 107 by XPRIMET1:8;
hence 107 is prime by NAT_4:14; :: thesis: verum