If is a weak order on X, then for every y X, 1. (y)
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1. ≿(y) ∪ ≾(y) = X and ≿(y) ∩ ≾(y) = Iy
2. ≿(y) ∪ (y) = X and ≿(y) ∩ ≺(y) = ϕ
3. ≻(y), Iy and ≺(y) together partition X
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