S F ( x ) is an inductive subset of P for each x ∈ P ; A3.
From condition (2) of this theorem, for every fixed element x − i ∈ S − i, P i ( S i, x − i ) is an inductive subset of U.
2′ For every s ∈ S, the set { d X ( u, s ′ ) : u ∈ F ( s ) }, or F ( s ), is an inductive subset of X with finite number of maximal elements.
It immediately follows that ∨ { x α } ∈ A. Hence A is chain complete; and therefore, A is an inductive subset of P. Applying Zorn's lemma, A has a maximal element x ∗.
The ≽ i -downward set of the inverse image { z i ∈ S i : f i ( z i, x − i ) is a maximal element of f i ( S i, x − i ) } is an inductive subset of S i ; G3.
Assume, in addition to conditions A1 and A3, F also satisfies one of the following conditions: A2. { z ∈ P : z ≼ u for some u ∈ F ( x ) } is an inductive subset of P, for each x ∈ P. A2′.
Assume, in addition conditions A1 and A3, F also satisfies one of the conditions below: A2. S F ( x ) is an inductive subset of P, for each x ∈ P. A2′.
F is order-increasing upward; S F ( x ) is an inductive subset of P for each x ∈ P ; There is a y in P with y ≼ u for some u ∈ F ( y ).
The ≽ i -downward set of the inverse image { z i ∈ S i : P i ( z i, x − i ) is a maximum element of f i ( S i, x − i ) } is an inductive subset of S i ; g2′.
Assume, in addition to conditions A1 and A3, F also satisfies one of the following conditions: A2. S F ( x ) is an inductive subset of P, for each x ∈ P. A2′.
