function - Lambda calculus and church numerals confusion -

I am trying to understand the basics of Lambda calculus and church numbers. I am studying and practicing a lot, but I continue to keep stuck trying to see how some functions work.

The example I am caught on is as follows: Perhaps someone can explain to me where I went wrong.

1 Church number can be displayed as:

  λf λx. The exponentation function can be given in chunk numbers (M ^ N) in the form of: x  

  λm λn Nm  

What I have to do is show that after applying the exponentation function 1 and 1, I get back 1 after 1 1 1 = 1. I am doing this so I think better how these functions work. My job is as follows and every time I get stuck:

  // exp (1) (λm. Λn Nm) (λf1. Λx1.f1x1) (λf2. Λx2.f2x2) // option for m (λn. N (λf1. Λx1.f1x1)) (λf2. Λx2.f2x2) // space For option (λf2. Λx2.f2x2) (λf1. Λx1.f1x1) / / option for F2 (λx2. (Λf1. Λx1.f1x1) x2) // option for f1 λx2. (Λx1.x2x1)  

And I got stuck there. I've lost both 'F', left with X and I have not got back 1. Where am i going

Where am I wrong?

Nowhere! done. Remember, variable names are not important; This structure is important Name f or x2 are not meaningful; it only matters how they are used. Church number for 1

  λf λx. F x  

and you have

  λx2. (Λx1.x2x1)  

x2 to f and x1 to x and voice! You have

  λf (Λx. F x) = λf λx. FX