Deriving Y combinator in Scheme

y.scm

(define fact0
  (lambda (n)
    (if (= n 0)
        1
        (* n (fact0 (- n 1))))))

(define fact1
  (let
      ((g
        (lambda (h)
          (lambda (n)
            (if (= n 0)
                1
                (* n ((h h) (- n 1))))))))
    (g g)))


(define fact2
  (let
      ((g
        (lambda (h)
          (lambda (n)
            (let
                ((f
                  (lambda (q)
                    (lambda (n)
                      (if (= n 0)
                          1
                          (* n (q (- n 1))))))))
              ((f (h h)) n))))))
    (g g)))

(define fact3
  (let
      ((y
        (lambda (f)
          (let
              ((g
                (lambda (h)
                  (lambda (n)
                    ((f (h h)) n)))))
          (g g)))))
    (y
     (lambda (q)
       (lambda (n)
         (if (= n 0)
             1
             (* n (q (- n 1)))))))))

(define Y
  (lambda (h)
    (let
        ((g
          (lambda (f)
            (lambda (n)
              ((h (f f)) n)))))
      (g g))))

(define fact4
  (Y
   (lambda (f)
     (lambda (n)
       (if (= n 0)
           1
           (* n (f (- n 1))))))))

(define fib
  (Y
   (lambda (f)
     (lambda (n)
       (if (<= n 1)
           1
           (+ (f (- n 1)) (f (- n 2))))))))