program to an interface not implementation

[jss3426]

We all heard of this before from the the gang f four and other locations. Maybe lack of sleep is confusing my brain...here is my question

I have a c# user control. have 2 methods add and subtract.

case1:
they are declared as public methods so when the user control is instantiated, say in a form, one can do

usercontrol.add(1,1);

and

usercontrol.subtract(100,1);

The form where the control is instantiated does not know how the add and subtract methods are implemented. would that be a fair statement?

case2:
same as case1 except that add and subtracts methods are exposed via interfaces and one gets to them in usual manner

the question now is that in either of the 2 cases the client has no ideas on how the methods are implemented and in case of a user control why would i choose to go with interfaces and does this statement 'program to an interface not implementation' hold valid??