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# 定理３

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• ### 定理３

1. 1. P53 3 f' g' f g pullback g mono f' mono f' g' f g
2. 2. 3 (1) f' mono f'￮k1 = f'￮k2 k1 k2 k1 f' k2 g' f g
3. 3. 3 (2) k1 f'￮k1 g'￮k2 f'￮k1 ￮￮ ￮￮ ￮ ￮ f' g'￮k1 ￮￮ g' f g
4. 4. 3 (3) pullback e f'￮k1 e g'￮e = g'￮k1 f' g'￮k1 f'￮e = f'￮k1 g' f g
5. 5. 3 (4) k1 (3) e f'￮k1 e k1 k1 f' g'￮k1 = g'￮k1 g'￮k1 g' f f'￮k1 = f'￮k1 g
6. 6. 3 (5) k2 (3) e f'￮k1 e k2 k2 f' g'￮k2 = g'￮k1 g'￮k1 g' f f'￮k2 = f'￮k1 g
7. 7. 3 (6) f'￮k2 = f'￮k1 k1 k2 k1 f' ((1) ) k2 g' f g
8. 8. 3 (7) g mono ￮ ￮ ￮ ￮ ￮ ￮ ￮ ￮ f'￮k1 ￮ ￮ ￮ ￮ g'￮k1 ￮ ￮ f' ￮ ￮ g'￮k2 g' f mono g g'￮k1 = g'￮k2 ∴
9. 9. 3 (8) k2 (3) e f'￮k1 e k1,k2 f' k1 = k2 g'￮k1 g' f g
10. 10. 3 (9) k1 k2 f'￮k1 = f'￮k2 k1 = k2 f' mono k1 f' Q.E.D. k2 g' f g