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r rs ttrs strt st qt P

❚❤❡ ❙❲❋▲ ❛♣♣r♦❛❝❤ ❲❡❧❢❛r✐s♠ ❯t✐❧✐t❛r✐❛♥✐s♠ ▲❡①✐♠✐♥ ❉✐str✐❜✉t✐✈❡ ❏✉st✐❝❡ ❛♥❞ ❊❝♦♥♦♠✐❝ ■♥❡q✉❛❧✐t② P✳ ●✳ P✐❛❝q✉❛❞✐♦ ❯✐❖✱ ❏❛♥✉❛r② ✶✼t❤✱ ✷✵✶✼ P✳ ●✳ P✐❛❝q✉❛❞✐♦ ❉✐str✐❜✉t✐✈❡ ❏✉st✐❝❡ ❛♥❞ ❊❝♦♥♦♠✐❝ ■♥❡q✉❛❧✐t②
❚❤❡ ❙❲❋▲ ❛♣♣r♦❛❝❤ ❲❡❧❢❛r✐s♠ ❯t✐❧✐t❛r✐❛♥✐s♠ ▲❡①✐♠✐♥ ❋r❛♠❡✇♦r❦ ❆ ♣❛✐r ( ① , ✐ ) ∈ ❳ × ◆ ✐s ❝❛❧❧❡❞ ❛ st❛t✐♦♥ ✳ ❆♥ ❡✈❛❧✉❛t✐♦♥ ♣r♦✜❧❡ ✐s ❞❡♥♦t❡❞ ❯ ✳ ■t ❝♦❧❧❡❝ts ❛❧❧ ✐♥❞✐✈✐❞✉❛❧ ❡✈❛❧✉❛t✐♦♥s ♦♥ ❳ × ◆ ✳ ■❢ ❳ ✐s ✜♥✐t❡✱ ❯ ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❜② ❛ ♠❛tr✐① | ❳ |×| ◆ | ✇✐t❤ ❣❡♥❡r✐❝ ❡❧❡♠❡♥t ❯ ( ① , ✐ ) ✳ ❯ ① ≡ ❯ ( ① , · ) ✐s t❤❡ r♦✇ ✈❡❝t♦r ❛♥❞ ❯ ✐ ≡ ❯ ( · , ✐ ) ✐s t❤❡ ❝♦❧✉♠♥ ✈❡❝t♦r✳ ▲❡t U ≡ { ❯ | ❯ : ❳ × ◆ → R } ✳ ❆ ❞♦♠❛✐♥ ✐s ❛ s✉❜s❡t D ⊆ U ✳ ❆ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥❛❧ ✐s ❛ ♠❛♣ ❋ : D → R ✳ ■t ❛ss✐❣♥s s♦❝✐❛❧ ♣r❡❢❡r❡♥❝❡s ❘ ❯ t♦ ❡❛❝❤ ❉ ∈ D ✱ ✐✳❡✳ ❘ ❯ = ❋ ( ❯ ) ✳ P✳ ●✳ P✐❛❝q✉❛❞✐♦ ❉✐str✐❜✉t✐✈❡ ❏✉st✐❝❡ ❛♥❞ ❊❝♦♥♦♠✐❝ ■♥❡q✉❛❧✐t②
❚❤❡ ❙❲❋▲ ❛♣♣r♦❛❝❤ ❲❡❧❢❛r✐s♠ ❯t✐❧✐t❛r✐❛♥✐s♠ ▲❡①✐♠✐♥ ❋r❛♠❡✇♦r❦ ❆ ♣❛✐r ( ① , ✐ ) ∈ ❳ × ◆ ✐s ❝❛❧❧❡❞ ❛ st❛t✐♦♥ ✳ ❆♥ ❡✈❛❧✉❛t✐♦♥ ♣r♦✜❧❡ ✐s ❞❡♥♦t❡❞ ❯ ✳ ■t ❝♦❧❧❡❝ts ❛❧❧ ✐♥❞✐✈✐❞✉❛❧ ❡✈❛❧✉❛t✐♦♥s ♦♥ ❳ × ◆ ✳ ■❢ ❳ ✐s ✜♥✐t❡✱ ❯ ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❜② ❛ ♠❛tr✐① | ❳ |×| ◆ | ✇✐t❤ ❣❡♥❡r✐❝ ❡❧❡♠❡♥t ❯ ( ① , ✐ ) ✳ ❯ ① ≡ ❯ ( ① , · ) ✐s t❤❡ r♦✇ ✈❡❝t♦r ❛♥❞ ❯ ✐ ≡ ❯ ( · , ✐ ) ✐s t❤❡ ❝♦❧✉♠♥ ✈❡❝t♦r✳ ▲❡t U ≡ { ❯ | ❯ : ❳ × ◆ → R } ✳ ❆ ❞♦♠❛✐♥ ✐s ❛ s✉❜s❡t D ⊆ U ✳ ❆ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥❛❧ ✐s ❛ ♠❛♣ ❋ : D → R ✳ ■t ❛ss✐❣♥s s♦❝✐❛❧ ♣r❡❢❡r❡♥❝❡s ❘ ❯ t♦ ❡❛❝❤ ❉ ∈ D ✱ ✐✳❡✳ ❘ ❯ = ❋ ( ❯ ) ✳ P✳ ●✳ P✐❛❝q✉❛❞✐♦ ❉✐str✐❜✉t✐✈❡ ❏✉st✐❝❡ ❛♥❞ ❊❝♦♥♦♠✐❝ ■♥❡q✉❛❧✐t②
❚❤❡ ❙❲❋▲ ❛♣♣r♦❛❝❤ ❲❡❧❢❛r✐s♠ ❯t✐❧✐t❛r✐❛♥✐s♠ ▲❡①✐♠✐♥ ❋r❛♠❡✇♦r❦ ❆ ♣❛✐r ( ① , ✐ ) ∈ ❳ × ◆ ✐s ❝❛❧❧❡❞ ❛ st❛t✐♦♥ ✳ ❆♥ ❡✈❛❧✉❛t✐♦♥ ♣r♦✜❧❡ ✐s ❞❡♥♦t❡❞ ❯ ✳ ■t ❝♦❧❧❡❝ts ❛❧❧ ✐♥❞✐✈✐❞✉❛❧ ❡✈❛❧✉❛t✐♦♥s ♦♥ ❳ × ◆ ✳ ■❢ ❳ ✐s ✜♥✐t❡✱ ❯ ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❜② ❛ ♠❛tr✐① | ❳ |×| ◆ | ✇✐t❤ ❣❡♥❡r✐❝ ❡❧❡♠❡♥t ❯ ( ① , ✐ ) ✳ ❯ ① ≡ ❯ ( ① , · ) ✐s t❤❡ r♦✇ ✈❡❝t♦r ❛♥❞ ❯ ✐ ≡ ❯ ( · , ✐ ) ✐s t❤❡ ❝♦❧✉♠♥ ✈❡❝t♦r✳ ▲❡t U ≡ { ❯ | ❯ : ❳ × ◆ → R } ✳ ❆ ❞♦♠❛✐♥ ✐s ❛ s✉❜s❡t D ⊆ U ✳ ❆ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥❛❧ ✐s ❛ ♠❛♣ ❋ : D → R ✳ ■t ❛ss✐❣♥s s♦❝✐❛❧ ♣r❡❢❡r❡♥❝❡s ❘ ❯ t♦ ❡❛❝❤ ❉ ∈ D ✱ ✐✳❡✳ ❘ ❯ = ❋ ( ❯ ) ✳ P✳ ●✳ P✐❛❝q✉❛❞✐♦ ❉✐str✐❜✉t✐✈❡ ❏✉st✐❝❡ ❛♥❞ ❊❝♦♥♦♠✐❝ ■♥❡q✉❛❧✐t②
❚❤❡ ❙❲❋▲ ❛♣♣r♦❛❝❤ ❲❡❧❢❛r✐s♠ ❯t✐❧✐t❛r✐❛♥✐s♠ ▲❡①✐♠✐♥ ❋r❛♠❡✇♦r❦ ❆ ♣❛✐r ( ① , ✐ ) ∈ ❳ × ◆ ✐s ❝❛❧❧❡❞ ❛ st❛t✐♦♥ ✳ ❆♥ ❡✈❛❧✉❛t✐♦♥ ♣r♦✜❧❡ ✐s ❞❡♥♦t❡❞ ❯ ✳ ■t ❝♦❧❧❡❝ts ❛❧❧ ✐♥❞✐✈✐❞✉❛❧ ❡✈❛❧✉❛t✐♦♥s ♦♥ ❳ × ◆ ✳ ■❢ ❳ ✐s ✜♥✐t❡✱ ❯ ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❜② ❛ ♠❛tr✐① | ❳ |×| ◆ | ✇✐t❤ ❣❡♥❡r✐❝ ❡❧❡♠❡♥t ❯ ( ① , ✐ ) ✳ ❯ ① ≡ ❯ ( ① , · ) ✐s t❤❡ r♦✇ ✈❡❝t♦r ❛♥❞ ❯ ✐ ≡ ❯ ( · , ✐ ) ✐s t❤❡ ❝♦❧✉♠♥ ✈❡❝t♦r✳ ▲❡t U ≡ { ❯ | ❯ : ❳ × ◆ → R } ✳ ❆ ❞♦♠❛✐♥ ✐s ❛ s✉❜s❡t D ⊆ U ✳ ❆ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥❛❧ ✐s ❛ ♠❛♣ ❋ : D → R ✳ ■t ❛ss✐❣♥s s♦❝✐❛❧ ♣r❡❢❡r❡♥❝❡s ❘ ❯ t♦ ❡❛❝❤ ❉ ∈ D ✱ ✐✳❡✳ ❘ ❯ = ❋ ( ❯ ) ✳ P✳ ●✳ P✐❛❝q✉❛❞✐♦ ❉✐str✐❜✉t✐✈❡ ❏✉st✐❝❡ ❛♥❞ ❊❝♦♥♦♠✐❝ ■♥❡q✉❛❧✐t②
❚❤❡ ❙❲❋▲ ❛♣♣r♦❛❝❤ ❲❡❧❢❛r✐s♠ ❯t✐❧✐t❛r✐❛♥✐s♠ ▲❡①✐♠✐♥ ❋r❛♠❡✇♦r❦ ❆ ♣❛✐r ( ① , ✐ ) ∈ ❳ × ◆ ✐s ❝❛❧❧❡❞ ❛ st❛t✐♦♥ ✳ ❆♥ ❡✈❛❧✉❛t✐♦♥ ♣r♦✜❧❡ ✐s ❞❡♥♦t❡❞ ❯ ✳ ■t ❝♦❧❧❡❝ts ❛❧❧ ✐♥❞✐✈✐❞✉❛❧ ❡✈❛❧✉❛t✐♦♥s ♦♥ ❳ × ◆ ✳ ■❢ ❳ ✐s ✜♥✐t❡✱ ❯ ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❜② ❛ ♠❛tr✐① | ❳ |×| ◆ | ✇✐t❤ ❣❡♥❡r✐❝ ❡❧❡♠❡♥t ❯ ( ① , ✐ ) ✳ ❯ ① ≡ ❯ ( ① , · ) ✐s t❤❡ r♦✇ ✈❡❝t♦r ❛♥❞ ❯ ✐ ≡ ❯ ( · , ✐ ) ✐s t❤❡ ❝♦❧✉♠♥ ✈❡❝t♦r✳ ▲❡t U ≡ { ❯ | ❯ : ❳ × ◆ → R } ✳ ❆ ❞♦♠❛✐♥ ✐s ❛ s✉❜s❡t D ⊆ U ✳ ❆ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥❛❧ ✐s ❛ ♠❛♣ ❋ : D → R ✳ ■t ❛ss✐❣♥s s♦❝✐❛❧ ♣r❡❢❡r❡♥❝❡s ❘ ❯ t♦ ❡❛❝❤ ❉ ∈ D ✱ ✐✳❡✳ ❘ ❯ = ❋ ( ❯ ) ✳ P✳ ●✳ P✐❛❝q✉❛❞✐♦ ❉✐str✐❜✉t✐✈❡ ❏✉st✐❝❡ ❛♥❞ ❊❝♦♥♦♠✐❝ ■♥❡q✉❛❧✐t②
❚❤❡ ❙❲❋▲ ❛♣♣r♦❛❝❤ ❲❡❧❢❛r✐s♠ ❯t✐❧✐t❛r✐❛♥✐s♠ ▲❡①✐♠✐♥ ❋r❛♠❡✇♦r❦ ❆ ♣❛✐r ( ① , ✐ ) ∈ ❳ × ◆ ✐s ❝❛❧❧❡❞ ❛ st❛t✐♦♥ ✳ ❆♥ ❡✈❛❧✉❛t✐♦♥ ♣r♦✜❧❡ ✐s ❞❡♥♦t❡❞ ❯ ✳ ■t ❝♦❧❧❡❝ts ❛❧❧ ✐♥❞✐✈✐❞✉❛❧ ❡✈❛❧✉❛t✐♦♥s ♦♥ ❳ × ◆ ✳ ■❢ ❳ ✐s ✜♥✐t❡✱ ❯ ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❜② ❛ ♠❛tr✐① | ❳ |×| ◆ | ✇✐t❤ ❣❡♥❡r✐❝ ❡❧❡♠❡♥t ❯ ( ① , ✐ ) ✳ ❯ ① ≡ ❯ ( ① , · ) ✐s t❤❡ r♦✇ ✈❡❝t♦r ❛♥❞ ❯ ✐ ≡ ❯ ( · , ✐ ) ✐s t❤❡ ❝♦❧✉♠♥ ✈❡❝t♦r✳ ▲❡t U ≡ { ❯ | ❯ : ❳ × ◆ → R } ✳ ❆ ❞♦♠❛✐♥ ✐s ❛ s✉❜s❡t D ⊆ U ✳ ❆ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥❛❧ ✐s ❛ ♠❛♣ ❋ : D → R ✳ ■t ❛ss✐❣♥s s♦❝✐❛❧ ♣r❡❢❡r❡♥❝❡s ❘ ❯ t♦ ❡❛❝❤ ❉ ∈ D ✱ ✐✳❡✳ ❘ ❯ = ❋ ( ❯ ) ✳ P✳ ●✳ P✐❛❝q✉❛❞✐♦ ❉✐str✐❜✉t✐✈❡ ❏✉st✐❝❡ ❛♥❞ ❊❝♦♥♦♠✐❝ ■♥❡q✉❛❧✐t②
❚❤❡ ❙❲❋▲ ❛♣♣r♦❛❝❤ ❲❡❧❢❛r✐s♠ ❯t✐❧✐t❛r✐❛♥✐s♠ ▲❡①✐♠✐♥ ❉✐s❝✉ss✐♦♥ ❲❡ ❛❧r❡❛❞② ❞✐s❝✉ss❡❞ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ❇❙✲❙❲❋ ❛♥❞ ❆rr♦✇✐❛♥ ❙❲❋✿ t❤❡ ❆rr♦✇✐❛♥ ❙❲❋ ❞❡✜♥❡s ❛ ❇❙✲❙❲❋ ❢♦r ❡❛❝❤ ♣♦ss✐❜❧❡ s♦❝✐❡t②✳ ❚❤❡ ❙❲❋▲ ❛♣♣r♦❛❝❤ ✐s ❡✈❡♥ ♠♦r❡ ❣❡♥❡r❛❧✿ t❤❡ ❙❲❋▲ ❢r❛♠❡✇♦r❦ ✐♥❝❧✉❞❡s ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❛ s♣❡❝✐✜❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♣r❡❢❡r❡♥❝❡s✱ ✐✳❡✳ ❯ ❀ ✐❢ t❤✐s ✐♥❢♦r♠❛t✐♦♥ ✐s ❞✐sr❡❣❛r❞❡❞ ❛♥❞ ♦♥❧② ♣r❡❢❡r❡♥❝❡s ❛r❡ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t✱ t❤❡♥ t❤❡ ❙❲❋▲ ✐s ✏❡q✉✐✈❛❧❡♥t✑ t♦ ❛♥ ❆rr♦✇✐❛♥ ❙❲❋❀ ✐❢ t❤✐s ✐♥❢♦r♠❛t✐♦♥ ✐s ♥♦t ❞✐sr❡❣❛r❞❡❞✱ ♠♦r❡ ✇❡❧❢❛r❡ ❝r✐t❡r✐❛ ❜❡❝♦♠❡ ❛✈❛✐❧❛❜❧❡✳ P✳ ●✳ P✐❛❝q✉❛❞✐♦ ❉✐str✐❜✉t✐✈❡ ❏✉st✐❝❡ ❛♥❞ ❊❝♦♥♦♠✐❝ ■♥❡q✉❛❧✐t②

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