. Preuve, En juxtaposant les propositions 4

. Argmin, Argmax(R v(t) ) et Argmax(R s ) = Argmin(W v+(s) )

. Par, si v + est continue en v(t) alors v + (v(t)) = t. Donc si t ? ? [v = v(t)] alors v + est continue en v(t ? ) et on a t ? = v + (v(t ? )) = v + (v(t))

. De, 4 que si v est continue en v + (s) alors v(v + (s)) = s. Donc si s ? ? [v + = v + (s)] alors v est continue en v + (s ? ) et on a s ? = v(v + (s ? )) = v(v + (s)) = s. D'où

. Preuve, Soit x ? X, d'après (4.12), on a ?(?, ?) ? I × J, wc(x, ?) ? ? ?? ? ? r(x, ?)

. Preuve, résulte d'une application directe de la proposition 4.3.6. Par ailleurs, comme (H) est vérifiée par la fonction wc, il résulte de (4.28) que (4.30) ?(?, ?) ? I

. De, = ?, alors en prenant ? = v r (?) dans (4.30), on obtient [wc(., v r (?)) ? v wc (v r (?))] = [v r (?) ? r(

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