In Ettore Casari, La Matematica della Verita' (2006), the following theorem (attributed to Tarski and Lindenbaum 1926), is stated, which Casari calls the theorem of the middle value of Tarski-Lindenbaum (MV theorem):

If $A \subseteq B \subseteq C$ and $A' \subseteq C'$ and $f : A \to A'$, $g : C \to C'$ are bijections, then there is a bijection $h : B \to B'$, for some $B'$ such that $A' \subseteq B' \subseteq C'$.

The MV theorem is stated without proof after proving the Knaster-Tarski theorem theorem (KT) on monotone functions and the Cantor-Schroeder-Bernstein theorem (CSB).

The author claims that the MV theorem follows from what has been discussed before, presumably indicating that the MV theorem follows from either the KT theorem or from the CSB theorem. I have two questions:

1. Does the MV theorem follow from the KT theorem or from the CBS theorem. If so, how is it proved? I cannot find any proofs of the MV theorem in the literature.
2. What is the significance of the MV theorem?

In Ettore Casari, La Matematica della Verita' (2006), the following theorem (attributed to Tarski and Lindenbaum 1926), is stated, which Casari calls the theorem of the middle value of Tarski-Lindenbaum (MV theorem):

If $A \subseteq B \subseteq C$ and $A' \subseteq C'$ and $f : A \to A'$, $g : C \to C'$ are bijections, then there is a bijection $h : B \to B'$, for some $B'$ such that $A' \subseteq B' \subseteq C'$.

The MV theorem is stated without proof after proving the Knaster-Tarski theorem theorem (KT) on monotone functions and the Cantor-Schroeder-Bernstein theorem (CSB).

The author claims that the MV theorem follows from what has been discussed before, presumably indicating that the MV theorem follows from either the KT theorem or from the CSB theorem. I have two questions:

1. Does the MV theorem follow from the KT theorem or from the CBS theorem?
2. What is the significance of the MV theorem?