We study stability of equilibrium positions in the spatial circular restricted four-body problem formulated on the basis of Lagrange's triangular solution of the three-body problem. Using the computer algebra system Mathematica, we have constructed Birkhoff's type canonical transformation, reducing the Hamiltonian function to the normal form up to the fourth order in perturbations. Applying Arnold's and Mar-keev's theorems, we have proved stability of three equilibrium positions for the majority of initial conditions in case of mass parameters of the system belonging to the domain of the solutions linear stability, except for the points in the parameter plane for which the third and fourth order resonance conditions are fulfilled.
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