Fixpoint Theorem for Continuous Functions on Chain-Complete Posets

Let P be a non empty poset. Observe that there exists a chain of P which is non empty. Let I1 be a relational structure. We say that I1 is chain-complete if and only if: (Def. 1) I1 is lower-bounded and for every chain L of I1 such that L is non empty holds sup L exists in I1. One can prove the following proposition (1) Let P1, P2 be non empty posets, K be a non empty chain of P1, and h be a monotone function from P1 into P2. Then h◦K is a non empty chain of P2. Let us note that there exists a poset which is strict, chain-complete, and non empty. Let us mention that every relational structure which is chain-complete is also lower-bounded.