Suppose an alloy at an equilibrium temperature T consists of  mass fraction of element B.  Suppose also that at temperature T the alloy consists of two phases, α and β, for which the α consists of , and β consists of .  Let the mass of the α phase in the alloy be  so that the mass of the β phase is , where  is the total mass of the alloy.
By definition, then, the mass of element B in the α phase is , while the mass of element B in the β phase is .  Together these two quantities sum to the total mass of element B in the alloy, which is given by .  Therefore,
By rearranging, one finds that
This final fraction is the mass fraction of the α phase in the alloy.