  We present the detailed formalism of the extremely correlated Fermi liquid
theory, developed for treating the physics of the t-J model. We start from the
exact Schwinger equation of motion for the Greens function for projected
electrons, and develop a systematic expansion in a parameter \lambda, relating
to the double occupancy. The resulting Greens function has a canonical part
arising from an effective Hamiltonian of the auxiliary electrons, and a
caparison part, playing the role of a frequency dependent adaptive spectral
weight. This adaptive weight balances the requirement at low \omega, of the
invariance of the Fermi volume, and at high \omega, of decaying as c_0/(i
\omega), with a correlation depleted c_0 <1. The effective Hamiltonian H_{eff}
describing the auxiliary Fermions is given a natural interpretation with an
effective interaction V_{eff} containing both the exchange J(ij), and the
hopping parameters t(ij). It is made Hermitian by adding suitable terms that
ultimately vanish, in the symmetrized theory developed in this paper. Simple
but important shift invariances of the t-J model are noted with respect to
translating its parameters uniformly. These play a crucial role in constraining
the form of V_{eff} and also provide checks for further approximations. The
auxiliary and physical Greens function satisfy two sum rules, and the Lagrange
multipliers for these are identified. A complete set of expressions for the
Greens functions to second order in \lambda is given, satisfying various
invariances. A systematic iterative procedure for higher order approximations
is detailed. A superconducting instability of the theory is noted at the
simplest level with a high transition temperature.
