Proving property of a term using Induction
A term is defined recursively.  You can do induction on recursive types, using structural induction.

For instance, suppose we say that a term is something of the form


$c$ (a constant), or
$t_1+t_2$, where $t_1,t_2$ are terms.


Then structural induction is (loosely) similar to doing a proof by (strong) induction on the size of the tree.  You prove that the property holds if the term has the form $c$ (that's the base case).  Then, you prove that if the property holds for terms $t_1$ and $t_2$, then the property holds for the term $t_1+t_2$.  You can probably see how to generalize this to terms whose syntax has a more complicated definition.