newtonian mechanics - Work-energy theorem and Conservation of energy formula

We have $$W_{net}=\Delta K \quad(work-energy\; theorem)\tag{1}$$ And also $$W_{net}=\Delta K +\Delta U \quad(Conservation\;of \;energy \;formula)\tag{2}$$ How's that happening?

In proof for bernoulli's equation there's a place which they say:$\Delta W=\Delta E $which$ \Delta W$ does not contain work of gravity and however it's related to external or internal energy but I can't understand? You can see the proof here: http://www.4physics.com/phy_demo/bernoulli-effect-equation.html

Answer

This is a nice example of why notation matters.

You know there are two types of forces: non-conservative and conservative forces.

Let's call $W_{cons}$ the work done by conservative forces. Let's call $W_{NC}$ the work done by the rest.

The work-energy theorem states that

$$W=\Delta E_k $$

However, this work is the total work. This can be splitted in two parts: $W=W_{cons}+W_{nc}$. So

$$ \Delta E_k = W_{cons}+W_{nc} $$

Now, we define a quantity called $E_p$ such taht $W_cons=-\Delta E_p$. The minus sign is a convention, but it is important to keep it in mind. Hence

$$ \Delta E_k = W_{cons}+W_{nc} $$ $$ \Delta E_k = -\Delta E_p +W_{nc} $$ $$ \boxed{ \Delta E_k + \Delta E_p = W_{nc} } $$

So your "net" work refers only to non conservative forces in your second equation.