**Proof.**

$$\begin{array}{llll} \text{1.} & \vdash\_{\text{IK}\_{\mathsf{t}}} \mathsf{g} \rightarrow \left(\psi \rightarrow \left(\varphi \wedge \psi\right)\right) & & \text{A1} \\ \text{2.} & \vdash\_{\text{IK}\_{\mathsf{t}}} \mathsf{G} \mathsf{g} \rightarrow \mathsf{G} \left(\psi \rightarrow \left(\varphi \wedge \psi\right)\right) & & \text{1, RRG} \\ \text{3.} & \vdash\_{\text{IK}\_{\mathsf{t}}} \mathsf{G} \left(\psi \rightarrow \left(\varphi \wedge \psi\right)\right) \rightarrow \left(F\psi \rightarrow F \left(\varphi \wedge \psi\right)\right) & & \text{A4} \\ \text{4.} & \vdash\_{\text{IK}\_{\mathsf{t}}} \mathsf{G} \varphi \rightarrow \left(F\psi \rightarrow F \left(\varphi \wedge \psi\right)\right) & & \text{2,3, SYL} \\ \text{5.} & \vdash\_{\text{IK}\_{\mathsf{t}}} \left(G\varphi \rightarrow \left(F\psi \rightarrow F \left(\varphi \wedge \psi\right)\right)\right) \rightarrow \left(\left(G\varphi \wedge F\psi\right) \rightarrow F \left(\varphi \wedge \psi\right)\right) & & \text{A1} \\ \text{6.} & \vdash\_{\text{IK}\_{\mathsf{t}}} \left(G\varphi \wedge F\psi\right) \rightarrow F \left(\rho \wedge \psi\right) & & \text{4,5, MP} \\ \end{array}$$

$$\supset$$

**Lemma 7. IKt** (*<sup>H</sup>ϕ* ∧ *Pψ*) → *P* (*ϕ* ∧ *ψ*)

Proof analogous to the proof of the previous lemma.

**Lemma 8. IKt** *G*¬*ϕ* → *F*¬*ϕ*
