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t Prs rt rt r tr t r

  1. ▲✐❢t✐♥❣ Pr♦❜❧❡♠s ✐♥ ❛ ●r♦t❤❡♥❞✐❡❝❦ ❋✐❜r❛t✐♦♥ ❆♥❞r❡✇ ❙✇❛♥ ❙❡♣t❡♠❜❡r ✶✷✱ ✷✵✶✼

  2. ❆ ❧✐❢t✐♥❣ ♣r♦❜❧❡♠ ♦❢ ❛❣❛✐♥st ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ sq✉❛r❡ ✇✐t❤ ♦♥ t❤❡ ❧❡❢t ❛♥❞ ♦♥ t❤❡ r✐❣❤t✳ ❆ ☞❧❧❡r ♦❢ ❛ ❧✐❢t✐♥❣ ♣r♦❜❧❡♠ ✐s ❛ ❞✐❛❣♦♥❛❧ ♠❛♣ ❢r♦♠ t♦ ♠❛❦✐♥❣ t✇♦ ❝♦♠♠✉t❛t✐✈❡ tr✐❛♥❣❧❡s✳ ❲❡ s❛② ❤❛s t❤❡ ❧❡❢t ❧✐❢t✐♥❣ ♣r♦♣❡rt② ❛❣❛✐♥st ❛♥❞ ❤❛s t❤❡ r✐❣❤t ❧✐❢t✐♥❣ ♣r♦♣❡rt② ❛❣❛✐♥st ❛♥❞ ✇r✐t❡ t ✐❢ ❡✈❡r② ❧✐❢t✐♥❣ ♣r♦❜❧❡♠ ❤❛s ❛ ☞❧❧❡r✳ ❲❡ s❛② ❛ ❝❧❛ss ♦❢ ♠❛♣s ❘ ✐s ❝♦☞❜r❛♥t❧② ❣❡♥❡r❛t❡❞ ✐❢ ❢♦r s♦♠❡ ☞①❡❞ s❡t ✱ ❜❡❧♦♥❣s t♦ ❘ ♣r❡❝✐s❡❧② ✇❤❡♥ t ❢♦r ❛❧❧ ✷ ✳ ✎ ✎ ❉❡☞♥✐t✐♦♥ ❙✉♣♣♦s❡ ✇❡ ❛r❡ ❣✐✈❡♥ ♠❛♣s f ❛♥❞ g ✐♥ ❛ ❝❛t❡❣♦r② ❈ ✱ ❛s ❜❡❧♦✇✿ A X g f B Y

  3. ❆ ☞❧❧❡r ♦❢ ❛ ❧✐❢t✐♥❣ ♣r♦❜❧❡♠ ✐s ❛ ❞✐❛❣♦♥❛❧ ♠❛♣ ❢r♦♠ t♦ ♠❛❦✐♥❣ t✇♦ ❝♦♠♠✉t❛t✐✈❡ tr✐❛♥❣❧❡s✳ ❲❡ s❛② ❤❛s t❤❡ ❧❡❢t ❧✐❢t✐♥❣ ♣r♦♣❡rt② ❛❣❛✐♥st ❛♥❞ ❤❛s t❤❡ r✐❣❤t ❧✐❢t✐♥❣ ♣r♦♣❡rt② ❛❣❛✐♥st ❛♥❞ ✇r✐t❡ t ✐❢ ❡✈❡r② ❧✐❢t✐♥❣ ♣r♦❜❧❡♠ ❤❛s ❛ ☞❧❧❡r✳ ❲❡ s❛② ❛ ❝❧❛ss ♦❢ ♠❛♣s ❘ ✐s ❝♦☞❜r❛♥t❧② ❣❡♥❡r❛t❡❞ ✐❢ ❢♦r s♦♠❡ ☞①❡❞ s❡t ✱ ❜❡❧♦♥❣s t♦ ❘ ♣r❡❝✐s❡❧② ✇❤❡♥ t ❢♦r ❛❧❧ ✷ ✳ ✎ ✎ ❉❡☞♥✐t✐♦♥ ❙✉♣♣♦s❡ ✇❡ ❛r❡ ❣✐✈❡♥ ♠❛♣s f ❛♥❞ g ✐♥ ❛ ❝❛t❡❣♦r② ❈ ✱ ❛s ❜❡❧♦✇✿ ✴ X A g f ✴ Y B ❆ ❧✐❢t✐♥❣ ♣r♦❜❧❡♠ ♦❢ f ❛❣❛✐♥st g ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ sq✉❛r❡ ✇✐t❤ f ♦♥ t❤❡ ❧❡❢t ❛♥❞ g ♦♥ t❤❡ r✐❣❤t✳

  4. ❲❡ s❛② ❤❛s t❤❡ ❧❡❢t ❧✐❢t✐♥❣ ♣r♦♣❡rt② ❛❣❛✐♥st ❛♥❞ ❤❛s t❤❡ r✐❣❤t ❧✐❢t✐♥❣ ♣r♦♣❡rt② ❛❣❛✐♥st ❛♥❞ ✇r✐t❡ t ✐❢ ❡✈❡r② ❧✐❢t✐♥❣ ♣r♦❜❧❡♠ ❤❛s ❛ ☞❧❧❡r✳ ❲❡ s❛② ❛ ❝❧❛ss ♦❢ ♠❛♣s ❘ ✐s ❝♦☞❜r❛♥t❧② ❣❡♥❡r❛t❡❞ ✐❢ ❢♦r s♦♠❡ ☞①❡❞ s❡t ✱ ❜❡❧♦♥❣s t♦ ❘ ♣r❡❝✐s❡❧② ✇❤❡♥ t ❢♦r ❛❧❧ ✷ ✳ ❄ ✎ ✎ ✴ ❉❡☞♥✐t✐♦♥ ❙✉♣♣♦s❡ ✇❡ ❛r❡ ❣✐✈❡♥ ♠❛♣s f ❛♥❞ g ✐♥ ❛ ❝❛t❡❣♦r② ❈ ✱ ❛s ❜❡❧♦✇✿ ✴ X A g f B Y ❆ ❧✐❢t✐♥❣ ♣r♦❜❧❡♠ ♦❢ f ❛❣❛✐♥st g ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ sq✉❛r❡ ✇✐t❤ f ♦♥ t❤❡ ❧❡❢t ❛♥❞ g ♦♥ t❤❡ r✐❣❤t✳ ❆ ☞❧❧❡r ♦❢ ❛ ❧✐❢t✐♥❣ ♣r♦❜❧❡♠ ✐s ❛ ❞✐❛❣♦♥❛❧ ♠❛♣ ❢r♦♠ B t♦ X ♠❛❦✐♥❣ t✇♦ ❝♦♠♠✉t❛t✐✈❡ tr✐❛♥❣❧❡s✳

  5. ❲❡ s❛② ❛ ❝❧❛ss ♦❢ ♠❛♣s ❘ ✐s ❝♦☞❜r❛♥t❧② ❣❡♥❡r❛t❡❞ ✐❢ ❢♦r s♦♠❡ ☞①❡❞ s❡t ✱ ❜❡❧♦♥❣s t♦ ❘ ♣r❡❝✐s❡❧② ✇❤❡♥ t ❢♦r ❛❧❧ ✷ ✳ ❄ ✎ ✎ ✴ ❉❡☞♥✐t✐♦♥ ❙✉♣♣♦s❡ ✇❡ ❛r❡ ❣✐✈❡♥ ♠❛♣s f ❛♥❞ g ✐♥ ❛ ❝❛t❡❣♦r② ❈ ✱ ❛s ❜❡❧♦✇✿ ✴ X A g f B Y ❆ ❧✐❢t✐♥❣ ♣r♦❜❧❡♠ ♦❢ f ❛❣❛✐♥st g ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ sq✉❛r❡ ✇✐t❤ f ♦♥ t❤❡ ❧❡❢t ❛♥❞ g ♦♥ t❤❡ r✐❣❤t✳ ❆ ☞❧❧❡r ♦❢ ❛ ❧✐❢t✐♥❣ ♣r♦❜❧❡♠ ✐s ❛ ❞✐❛❣♦♥❛❧ ♠❛♣ ❢r♦♠ B t♦ X ♠❛❦✐♥❣ t✇♦ ❝♦♠♠✉t❛t✐✈❡ tr✐❛♥❣❧❡s✳ ❲❡ s❛② f ❤❛s t❤❡ ❧❡❢t ❧✐❢t✐♥❣ ♣r♦♣❡rt② ❛❣❛✐♥st g ❛♥❞ g ❤❛s t❤❡ r✐❣❤t ❧✐❢t✐♥❣ ♣r♦♣❡rt② ❛❣❛✐♥st f ❛♥❞ ✇r✐t❡ f t g ✐❢ ❡✈❡r② ❧✐❢t✐♥❣ ♣r♦❜❧❡♠ ❤❛s ❛ ☞❧❧❡r✳

  6. ❄ ✎ ✴ ✎ ❉❡☞♥✐t✐♦♥ ❙✉♣♣♦s❡ ✇❡ ❛r❡ ❣✐✈❡♥ ♠❛♣s f ❛♥❞ g ✐♥ ❛ ❝❛t❡❣♦r② ❈ ✱ ❛s ❜❡❧♦✇✿ ✴ X A g f B Y ❆ ❧✐❢t✐♥❣ ♣r♦❜❧❡♠ ♦❢ f ❛❣❛✐♥st g ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ sq✉❛r❡ ✇✐t❤ f ♦♥ t❤❡ ❧❡❢t ❛♥❞ g ♦♥ t❤❡ r✐❣❤t✳ ❆ ☞❧❧❡r ♦❢ ❛ ❧✐❢t✐♥❣ ♣r♦❜❧❡♠ ✐s ❛ ❞✐❛❣♦♥❛❧ ♠❛♣ ❢r♦♠ B t♦ X ♠❛❦✐♥❣ t✇♦ ❝♦♠♠✉t❛t✐✈❡ tr✐❛♥❣❧❡s✳ ❲❡ s❛② f ❤❛s t❤❡ ❧❡❢t ❧✐❢t✐♥❣ ♣r♦♣❡rt② ❛❣❛✐♥st g ❛♥❞ g ❤❛s t❤❡ r✐❣❤t ❧✐❢t✐♥❣ ♣r♦♣❡rt② ❛❣❛✐♥st f ❛♥❞ ✇r✐t❡ f t g ✐❢ ❡✈❡r② ❧✐❢t✐♥❣ ♣r♦❜❧❡♠ ❤❛s ❛ ☞❧❧❡r✳ ❲❡ s❛② ❛ ❝❧❛ss ♦❢ ♠❛♣s ❘ ✐s ❝♦☞❜r❛♥t❧② ❣❡♥❡r❛t❡❞ ✐❢ ❢♦r s♦♠❡ ☞①❡❞ s❡t I ✱ g ❜❡❧♦♥❣s t♦ ❘ ♣r❡❝✐s❡❧② ✇❤❡♥ f t g ❢♦r ❛❧❧ f ✷ I ✳

  7. ❆♥♦t❤❡r ❡①❛♠♣❧❡ ♦❢ ❛ ❝♦☞❜r❛♥t❧② ❣❡♥❡r❛t❡❞ ❝❧❛ss ✐s ❑❛♥ ☞❜r❛t✐♦♥s ✐♥ s✐♠♣❧✐❝✐❛❧ s❡ts✱ ❛s ✉s❡❞ ✐♥ t❤❡ s✐♠♣❧✐❝✐❛❧ s❡t ♠♦❞❡❧ ♦❢ t②♣❡ t❤❡♦r②✳ ❋♦r ❡①❛♠♣❧❡✱ ❧❡t ❈ ✿❂ ❃ t❤❡ ❝❛t❡❣♦r② ♦❢ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s✱ ❛♥❞ ❧❡t f ✿❂ e ✵ ✿ ✶ ✱ ✦ ❬✵ ❀ ✶❪ ❜❡ ❛♥ ❡♥❞♣♦✐♥t ✐♥❝❧✉s✐♦♥ ✐♥t♦ t❤❡ ✉♥✐t ✐♥t❡r✈❛❧✳ ❆ ♠❛♣ g ✿ X ✦ Y ❤❛s t❤❡ ♣❛t❤ ❧✐❢t✐♥❣ ♣r♦♣❡rt② ✐❢ ✐t ❤❛s t❤❡ r✐❣❤t ❧✐❢t✐♥❣ ♣r♦♣❡rt② ❛❣❛✐♥st e ✵ ✳ ❊①❛♠♣❧❡s ✐♥❝❧✉❞❡ t❤❡ ♠❛♣ ❘ ✦ ❙ ✶ ❣✐✈❡♥ ❜② t ✼✦ ✭❝♦s✭ t ✮ ❀ s✐♥✭ t ✮✮✱ ❛♥❞ ✭tr✐✈✐❛❧❧②✮ t❤❡ ✉♥✐q✉❡ ♠❛♣ X ✦ ✶ ❢♦r ❛♥② t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✳ ❚❤❡ ❝❧❛ss ♦❢ ♠❛♣s ✇✐t❤ ♣❛t❤ ❧✐❢t✐♥❣ ♣r♦♣❡rt② ✐s ❝♦☞❜r❛♥t❧② ❣❡♥❡r❛t❡❞ ❜② ❢ e ✵ ❣ ❜② ❞❡☞♥✐t✐♦♥✳

  8. ❋♦r ❡①❛♠♣❧❡✱ ❧❡t ❈ ✿❂ ❃ t❤❡ ❝❛t❡❣♦r② ♦❢ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s✱ ❛♥❞ ❧❡t f ✿❂ e ✵ ✿ ✶ ✱ ✦ ❬✵ ❀ ✶❪ ❜❡ ❛♥ ❡♥❞♣♦✐♥t ✐♥❝❧✉s✐♦♥ ✐♥t♦ t❤❡ ✉♥✐t ✐♥t❡r✈❛❧✳ ❆ ♠❛♣ g ✿ X ✦ Y ❤❛s t❤❡ ♣❛t❤ ❧✐❢t✐♥❣ ♣r♦♣❡rt② ✐❢ ✐t ❤❛s t❤❡ r✐❣❤t ❧✐❢t✐♥❣ ♣r♦♣❡rt② ❛❣❛✐♥st e ✵ ✳ ❊①❛♠♣❧❡s ✐♥❝❧✉❞❡ t❤❡ ♠❛♣ ❘ ✦ ❙ ✶ ❣✐✈❡♥ ❜② t ✼✦ ✭❝♦s✭ t ✮ ❀ s✐♥✭ t ✮✮✱ ❛♥❞ ✭tr✐✈✐❛❧❧②✮ t❤❡ ✉♥✐q✉❡ ♠❛♣ X ✦ ✶ ❢♦r ❛♥② t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✳ ❚❤❡ ❝❧❛ss ♦❢ ♠❛♣s ✇✐t❤ ♣❛t❤ ❧✐❢t✐♥❣ ♣r♦♣❡rt② ✐s ❝♦☞❜r❛♥t❧② ❣❡♥❡r❛t❡❞ ❜② ❢ e ✵ ❣ ❜② ❞❡☞♥✐t✐♦♥✳ ❆♥♦t❤❡r ❡①❛♠♣❧❡ ♦❢ ❛ ❝♦☞❜r❛♥t❧② ❣❡♥❡r❛t❡❞ ❝❧❛ss ✐s ❑❛♥ ☞❜r❛t✐♦♥s ✐♥ s✐♠♣❧✐❝✐❛❧ s❡ts✱ ❛s ✉s❡❞ ✐♥ t❤❡ s✐♠♣❧✐❝✐❛❧ s❡t ♠♦❞❡❧ ♦❢ t②♣❡ t❤❡♦r②✳

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