Here's the first one:

Here's an example so you can believe it:

Find this sum:

Now, find these sums and add them:

Do you believe it?  (You should have gotten the same thing for both.)

This may not seem like a big deal, but you'll need it to prove some important calculus stuff, so let's get used to it.

Here's the proof:

Just expand it out...

Now, use the commutative and associative properties...  Gather up the a's...  Gather up the b's...

Here are the other two properties -- I'll let you do the proofs: