  The formalism of the matching conditions between transverse connected Green
functions is extended to include the two next to leading corrections, namely
the two-loop $M_{H}^{2}$ and the one-loop $1/M_{H}^{2}$ contributions to the
coefficients of the electroweak chiral lagrangian which are relevant to the
LEP1 physics: $a_{0}$, ${\hat a}_{1}$ and ${\hat a}_{8}$. We derive general
expressions for these three coefficients in terms of just bare gauge boson
self-energies. By means of the screening theorem, it is shown that the same
expressions can be used to get directly from a MSM calculation, the leading
Higgs mass contribution to these coefficients at each loop order. In a more
general framework, we solve the problems concerning the loss of gauge
invariance and the inclusion of only gauge invariant operators by proposing a
new formulation of the matching conditions at two and higher loop order. As an
example of the usefulness of using an electroweak chiral lagrangian to
parametrize the MSM, we will give a simple proof of an extra screening for the
renormalized photon self-energy in the on-shell scheme at all orders. In
addition it is shown the automatic cancellation of the unphysical $M_{H}^4$
terms in the other gauge boson self-energies at two-loops in this scheme.
Finally we will apply the obtained electroweak chiral lagrangian to compute the
different Higgs mass contributions to the bosonic part of $\Delta \rho$,
$\Delta r$ and $\Delta \kappa$, analyzing carefully the hierarchy between
corrections.
