**Proof.**


$$\begin{array}{llll} 4. & \vdash \mathsf{IK}\_{\mathsf{t}} \, F\psi \to F\left(\boldsymbol{\varrho} \vee \psi\right) & & & & 2, \mathsf{RF} \\ 5. & \vdash \mathsf{IK}\_{\mathsf{t}} \, \left(F\boldsymbol{\varrho} \to F\left(\boldsymbol{\varrho} \vee \psi\right)\right) & \to \left(\left(F\boldsymbol{\varphi} \to F\left(\boldsymbol{\varrho} \vee \psi\right)\right) \to \left(\left(F\boldsymbol{\varrho} \vee F\psi\right) \to F\left(\boldsymbol{\varrho} \vee \psi\right)\right)\right) & & \mathbf{A}1 \\ 6. & \vdash \mathsf{IK}\_{\mathsf{t}} \, \left(F\boldsymbol{\psi} \to F\left(\boldsymbol{\varrho} \vee \psi\right)\right) \to \left(\left(F\boldsymbol{\varrho} \vee F\psi\right) \to F\left(\boldsymbol{\varrho} \vee \psi\right)\right) & & \mathbf{3.5, \mathsf{MP} \\ 7. & \vdash \mathsf{IK}\_{\mathsf{t}} \, \left(F\boldsymbol{\varrho} \vee F\psi\right) \to F\left(\boldsymbol{\varrho} \vee \psi\right) & & & 4, 6, \mathsf{MP} \\ \sqcap \blacksquare & & & & & & \end{array}$$

**Lemma 5. IKt** (*<sup>P</sup>ϕ* ∨ *Pψ*) → *P* (*ϕ* ∨ *ψ*)

> Proof analogous to the proof of the previous lemma.

**Lemma 6. IKt** (*<sup>G</sup>ϕ* ∧ *Fψ*) → *<sup>F</sup>*(*ϕ* ∧ *ψ*)
