Composition of Galois Connections is Galois Connection

Theorem
Let $L_1 = \left({S_1, \preceq_1}\right), L_2 = \left({S_2, \preceq_2}\right), L_3 = \left({S_3, \preceq_3}\right)$ be ordered sets.

Let $g_1:S_1 \to S_2, g_2:S_2 \to S_3, d_1:S_2 \to S_1, d_2:S_3 \to S_2$ be mappings such that
 * $\left({g_1, d_1}\right)$ and $\left({g_2, d_2}\right)$ are Galois connections.

Then $\left({g_2 \circ g_1, d_1 \circ d_2}\right)$ is also Galois connection.

Proof
By definition of Galois connection:
 * $g_1$, $g_2$, $d_2$, and $d_1$ are increasing mappings.

Thus by Composition of Increasing Mappings is Increasing:
 * $g_2 \circ g_1$ and $d_1 \circ d_2$ are increasing mappings.

Let $s \in S_3, t \in S_1$.

We will prove that
 * $s \preceq_3 \left({g_2 \circ g_1}\right)\left({t}\right) \implies \left({d_1 \circ d_2}\right)\left({s}\right) \preceq_1 t$

Assume that
 * $s \preceq_3 \left({g_2 \circ g_1}\right)\left({t}\right)$

By definition of composition of mappings:
 * $s \preceq_3 g_2\left({g_1\left({t}\right)}\right)$

By definition of Galois connection:
 * $d_2\left({s}\right) \preceq_2 g_1\left({t}\right)$

By definition of increasing mapping:
 * $d_1\left({d_2\left({s}\right)}\right) \preceq_1 d_1\left({g_1\left({t}\right)}\right)$

By Galois Connection Implies Order on Mappings
 * $d_1 \circ g_1 \preceq_1 I_{S_1}$

By definitions of ordering on mappings and composition of mappings:
 * $g_1\left({d_1\left({t}\right)}\right) \preceq_1 I_{S_1}\left({t}\right)$

By definition of identity mapping:
 * $g_1\left({d_1\left({t}\right)}\right) \preceq_1 t$

By definition of transitivity:
 * $d_1\left({d_2\left({s}\right)}\right) \preceq_1 t$

Thus by definition of composition of mappings:
 * $\left({d_1 \circ d_2}\right)\left({s}\right) preceq_1 t$

Assume that
 * $\left({d_1 \circ d_2}\right)\left({s}\right) preceq_1 t$

By definition of composition of mappings:
 * $d_1\left({d_2\left({s}\right)}\right) preceq_1 t$

By definition of Galois connection:
 * $d_2\left({s}\right) \preceq_2 g_1\left({t}\right)$

By definition of increasing mapping:
 * $g_2\left({d_2\left({s}\right)}\right) \preceq_1 g_2\left({g_1\left({t}\right)}\right)$

By Galois Connection Implies Order on Mappings
 * $I_{S_3} \preceq_3 g_2 \circ d_2$

By definitions of ordering on mappings and composition of mappings:
 * $I_{S_3}\left({s}\right) \preceq_3 g_2\left({d_2\left({s}\right)}\right)$

By definition of identity mapping:
 * $s \preceq_3 g_2\left({d_2\left({s}\right)}\right)$

By definition:Transitivity|transitivity]]:
 * $s \preceq_3 g_2\left({g_1\left({t}\right)}\right)$

Thus by definition of composition of mappings:
 * $s \preceq_3 \left({g_2 \circ g_1}\right)\left({t}\right)$