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. Let, We demonstrate there exists u ? W |?? (?) such that zR |?? (?) u and t ? |?? (?) u. Since y ? |?? (?) z and yR |?? (?) t, therefore y ? z and yRt. Let u ? W be such that zRu and t ? u. Such u exists because M is standard. Since t ? W |?? (?) , therefore M, t |= ?? (?). Hence, there exists v ? W such that tRv and M, v |= ? (?) Let w ? W be such that uRw and v ? w, Such w exists because M is standard and t ? u. Since M is standard, ? ? F OR and M, v |= ? (?), therefore M, w |= ? (?). Since uRw, therefore M, u |= ?? (?)

. Thus, Since z, t ? W |?? (?) , zRu and t ? u, therefore zR |?? (?)

. Let, be such that yR |?? (?) z and z ? |?? (?) t. We demonstrate there exists u ? W |?? (?) such that y ? |?? (?) u and uR |?? (?) t. Since yR |?? (?) z and z ? |?? (?) t, therefore yRz and z ? t. Let u ? W be such that y ? u and uRt, therefore M, y |= ?? (?). Hence, there exists v ? W such that yRv and M, v |= ? (?)

?. We-demonstrate-?, ?. Or, ?. Or, and . Or, Note that size(?) < size(?) and size(?) < size(?) Hence Thus, the formulas ? (?) ? ? (?) and ? (?) ? ? (?) are s-valid. Let us consider the following formulas: (i) ? (?)? (?), (ii) ? (?) ? [? (?)]? (?), (iii) ? (?) ? [? (?)]? (?), iv) ? (?) ? (? (?) ? [? (?)]? (?)), (v) ? (?) ? [? (?)]? (?), (vi) (? (?) ? [? (?)]? (?)), (vii) ? (?)? (?)

M. , ?. , R. , ?. W. , M. Iff et al., Let F OR be the set of all formulas ? in the IP AL's language such that for all upward closed standard models Lemma 10.3 says that for all formulas ? in the IP AL's language, ? ? F OR. We will demonstrate it by an induction on ? based on the function size(·) defined in Section 2. Let ? be a formula such that for all formulas ?, if size(?) < size(?) then ? ? F OR, We demonstrate ? ? F OR. We only consider the case ? = ??. Note that size(?) < size(?) and size(?) < size(?). Hence, ? ? F OR and ? ? F OR

M. Suppose, |. Hence, M. , and M. , Since ? ? F OR, therefore {y ? W : M, y |= ?} = {y ? W : M, y |= ? (?)} and M |? = M |? (?)

S. Moreover, ?. Or, M. , and M. , therefore M, x |= ? (?) and M |? , x |= ? (?) Since M |? = M |? (?) , therefore M |? (?) , x |= ? (?, p.x |= ? (?)? (?)

M. Suppose and |. , ?)? (?). Hence, M, x |= ? (?) and M |? (?) , x |= ? (?

?. Since-?, ?. Or, =. , and ?. W. , ?)} and M |? = M |? (?) Moreover, since ? ? F OR, M, x |= ? (?) and M |? (?) , x |= ? (?), therefore M, x |= ? and M |? (?