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. Proof, We denote in this proof L (m, i) by L and T (m, i) by T . We propose an induction on e

?. If-e-=-x-then and ?. , L(x) We distinguish two cases. ? If L (x) = Raw(super(C)), as Raw(C) Raw(super, Raw(C) L(x) holds ? otherwise L (x) L(x) holds by definition of L

@. If and ?. , Raw(C) and class(f ) C, then ?, L e .f : MayBeNull and

@. If and ?. , Raw(C) and C class(f ), then ?, L e .f : ? [f ] and

C. , H. , and T. Raw, The analysis gives us

. Proof, We do a case analysis on P m [i]

?. If-e-=-e, m,i),L (m,i) holds. We distinguish two cases. ? If e = MayBeNull, then it contradicts ?, L e : ? . So e = MayBeNull, so safe expr (e), L (m,i) holds

. Proof, ?. R. Is-lf-typable-w, L. , F. Pm, H. et al., ? means that there exists L and F such that

?. If and P. , we must show that safe expr (e) C,H ,T (m,i),L (m,i) holds. By hypothesis, ?, L e : ? , so from Lemma 26 the property holds

?. If and P. , we must show that L(m, i)(x) = MayBeNull and safe expr (e) C,H ,T (m,i),L (m,i) hold. By hypothesis, L(m, i)(x) = MayBeNull and ?, L e : ? , so, pp.55-81

?. If and P. , e j ), we must show that ?j, safe expr (e j ) class(m),H ,T (m,i),L (m,i) By hypothesis, ?i, ?? i , ?, L e i : ? i ), so, from Lemma 26, the property holds

?. If and P. , we must show that L (m, i)(x) = MayBeNull and f orallj, safe expr (e j ) class(m),H ,T (m,i),L (m,i) By hypothesis, (?i, ?, L e i : ? i ) and if ? [lookup(ms)] = [? ]? ? void then ?, L x : if ? ? Val then ? else Raw(super(C )) and ? = MayBeNull

M. , H. , T. , M. , H. et al., If P is LF-typable w.r.t. ?, L, F, then there exists, Theorem 5

. Proof, H. Let-(-m, T. , L. , L. et al., Forall m and m , let ?

?. By, class(m ) M (m )[this] holds