**Proof.**

(A) **IKt** *G* (*ϕ* ∧ *ψ*) → (*<sup>G</sup>ϕ* ∧ *Gψ*)


(B) **IKt** (*<sup>G</sup>ϕ* ∧ *Gψ*) → *G* (*ϕ* ∧ *ψ*)

$$\begin{array}{llll} 1. & \vdash\_{\text{IK}\_{\mathsf{i}}} \varrho \rightarrow \left(\psi \rightarrow (\varrho \wedge \psi)\right) & & \text{A1} \\ 2. & \vdash\_{\text{IK}\_{\mathsf{i}}} \mathbb{G}\varphi \rightarrow \mathbb{G}\left(\psi \rightarrow (\varrho \wedge \psi)\right) & & \text{1,RRG} \\ 3. & \vdash\_{\text{IK}\_{\mathsf{i}}} \mathbb{G}\left(\psi \rightarrow (\varrho \wedge \psi)\right) \rightarrow \left(\mathbb{G}\psi \rightarrow \mathbb{G}(\varrho \wedge \psi)\right) & & \text{A2} \\ 4. & \vdash\_{\text{IK}\_{\mathsf{i}}} \mathbb{G}\varphi \rightarrow \left(\mathbb{G}\psi \rightarrow \mathbb{G}(\varrho \wedge \psi)\right) & & \text{2,3} \\ 5. & \vdash\_{\text{IK}\_{\mathsf{i}}} \left(\mathbb{G}\varphi \rightarrow \left(\mathbb{G}\psi \rightarrow \mathbb{G}(\varrho \wedge \psi)\right)\right) & \rightarrow \left(\left(\mathbb{G}\varphi \wedge \mathbb{G}\psi\right) \rightarrow \mathbb{G}(\varrho \wedge \psi)\right) & & \text{A1} \\ \end{array}$$

$$\begin{array}{cccc} \text{5.} & \vdash\_{\mathbf{IK}\_{\mathsf{I}}} \left( \mathbf{G}\boldsymbol{\varphi} \rightarrow \left( \mathbf{G}\boldsymbol{\psi} \rightarrow \mathbf{G}(\boldsymbol{\varphi}\wedge\boldsymbol{\psi}) \right) \right) \rightarrow & \left( \left( \mathbf{G}\boldsymbol{\varphi}\wedge\mathbf{G}\boldsymbol{\psi} \right) \rightarrow \mathbf{G}(\boldsymbol{\varphi}\wedge\boldsymbol{\psi}) \right) \\\ \mathbf{6.} & \vdash\_{\mathbf{IK}\_{\mathsf{I}}} \left( \mathbf{G}\boldsymbol{\varphi}\wedge\mathbf{G}\boldsymbol{\psi} \right) \rightarrow \mathbf{G}(\boldsymbol{\varphi}\wedge\boldsymbol{\psi}) \end{array} \tag{4.5.\text{ $\mathsf{M}\prime}$ }$$

With (**A**) and (**B**) we ge<sup>t</sup> a thesis.

The next lemma is proved similarly.

**Lemma 3. IKt** *H* (*ϕ* ∧ *ψ*) ↔ (*<sup>H</sup>ϕ* ∧ *Hψ*)

**Lemma 4. IKt** (*<sup>F</sup>ϕ* ∨ *Fψ*) → *F* (*ϕ* ∨ *ψ*)
