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conductivity electrons spread on the entire width W of the band. ϵ$_{f}$ is the energy of localized electrons. Both these energies are evaluated with respect to the chemical potential µ . 

The approach proposed in this paper generalizes the diagrammatic theory of normal and superconducting phases of strongly correlated systems proposed in pre- vious papers [10 − $^{19]}$. 

The strong on-site repulsion U between f − electrons of opposite spins is the main term in the Hamiltonian. 

As the conduction electrons can belong not only to the s − but also to the d − atomic shell, their Coulomb re- pulsion can also be important. In this case the extended PAM must be used $^{[20]}$. For simplicity the correlations of c − electrons are not considered and one subsystem is of c − uncorrelated and the second of f − correlated electrons. Because of strong localization of the f − elec- trons they cannot hope from one lattice site to another and their delocalization is due only to the hybridization of the f − and c − states with matrix element V . It is obvious that at V ̸ = 0 in the given model with two sub- systems superconductivity arises simultaneously in both subsystems. 

In the present paper we develop the thermodynamic perturbation theory for the system in the superconduct- ing state with Hamiltonian (1) under the assumption that the term responsible for hybridization of c − and f − elec- trons is a perturbation. 0 

The Hamiltonian H c of the uncorrelated c − electrons is diagonal in band representation, where as the Hamil- tonian H 0 f is diagonalized by using Hubbard transfer op- erators $^{[21]}$. Therefore in the zeroth-order of the pertur- bation theory the statistical operator of grand canonical ensemble of the system is factorized in the momentum representation for c − and in local representation for f − electrons: 

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We use the series expansion for evolution operator: 

in the interaction representation for electron operators ( a = c, f ): 

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Here by x means ( x , σ, τ ). 

We shall denote by 〈 T AB... 〉 0 the thermodynamic av- erage with zeroth-order statistical operator (3) of the 

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chronological product of electron operators ( AB... ). Such averages are calculated independently for c − and f − op- erators with using for c − electrons the Wick Theorem of weak quantum field theory and by using for f − electrons the Generalized Wick Theorem (GWT) proposed by us in papers [10 − 13] for strongly correlated electron systems. 

In the superconducting state, unlike the normal one, nontrivial statistical averages of operator products with even total number but inequal number of creation and annihilation electron operators are possible. They re- alize the Bogoliubov quasi-averages [22] or Gor’kov [23] anomalous Green’s functions. To unify the calculation of statistical averages for normal and superconducting phases it is useful to assign an additional quantum num- ber α , called by us charge number $^{[15]}$, with the values ± 1, which can be add to electron operators according the rule ( a = c, f ): 

In this representation the interaction operator H$_{int}$ be- comes: 

Obviously, introducing a new quantum charge number leads to additional summation over it values in all dia- gram lines and to an additional factor α in the vertices of diagrams. 

Now, after such introducing, it is irrelevant whether one deals with creation or annihilation operators. First of all we shall enumerate the main results of diagram- matic theory obtained in the previous paper [15] neces- sary to our proving of stationary theorem. Such theorem for uncorrelated many-electron systems in normal state has been proved by Luttinger and Word $^{[24]}$. 

## III. PERTURBATIVE TREATMENT [15] 

We use the definition of the one-particle Matsubara Green’s functions for c − and f − electrons c − α c 

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where index 〈 〉 0 agrams which are taken into account in the right-hand part of definition (8). 

The following condition is fulfilled 

Between this new definition and traditional one [23] there is a relation 

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