In reference to the formula 1-(-1) = 2 you posted on my talk page as being proof of addition by subtraction. Fine in theory (as I have yet to see -1 of anything), but I can prove that nothing is gained in your equation.  
Observe:

A=B.

Multiply both sides by A and you get:

A2=AB.

Add A2 to both sides to get:

A2+A2=A2+AB.

Using addition we get:

2A2=A2+AB.

Subtract 2AB from both sides:

2A2-2AB=A2+AB-2AB.

Thus:

2A2-2AB=A2-AB.

Combining terms on the LHS:

2(A2-AB)= A2-AB.

Divide both sides by  (A2-AB):

2(A2-AB)/(A2-AB)= (A2-AB)/(A2-AB).

Cancel out like terms and Viola!  We now see:

2 = 1.

So nothing is gained from addition by subtraction.

fin