Takeuti’s method of clearing away implicit second-order
inferences:

If we are wanting to prove by transfinite induction that
cuts can be eliminated from a derivation containing at least
one operational inference, then we might as well assume, as
part of the I.H., that they can be eliminated from the
derivation formed by omitting that operational inference

If a derivation of    contains two mated implicit
second-order inferences, then, e.g.,   , F() can be
derived by omitting the right one

When a derivation of    is so turned into a derivation
of   , F(), then, by the I.H., the last sequent has a
cut-free derivation

If ordinal numbers can be so assigned that cut-free
derivations have finite ordinal numbers, then we can
assume there is a derivation of   , F() with a finite
ordinal number

For any predicate V, we can assume that a derivation of
  , F(V) exists and has a finite ordinal number
F(V), F(),    
F(),   
Let this left second-order  inference in a derivation be
mated with a right one and let the conclusion of the
derivation be   ; then, by the reasoning on the last
slide,   , F(V) has a derivation with ordinal < , the
inference above can be omitted from the derivation and
the formula F(V) removed by a cut instead
The trick now is to assign ordinal numbers so that the
inference shown above increases the ordinal number of
the whole derivation more than the cut which replaces it
Solution: define “power” for implicit second-order 
(left) inferences similarly to cuts and let the conclusion of
one have the ordinal 2(O #), O the ordinal of premiss
But this works only when a quite stringent limitation is
imposed on the derivation as a whole