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In the previous paper by Alt et al (1984 Indiana Univ. Math. J. 33 367â€"91), the existence of the symmetric plane flows of two fluids with two free boundaries has been established by variational approach. However, the uniqueness of the two-phase fluids with two free boundaries is an unsolved problem. In the present paper, we will deal with the uniqueness of the two-phase fluids. In addition, if we assume that the two nozzles both converge, we shall show the monotonicity of the two free boundaries of the incompressible two-phase fluids issuing from these two nozzles. [ABSTRACT FROM AUTHOR]