Extended Transitivity

Theorem
Let $S$ be a set.

Let $\mathcal R$ be a transitive relation on $S$.

Let $\mathcal R^=$ be the reflexive closure of $\mathcal R$.

Let $a, b, c \in S$.

Then:

Proof
$(1)$ follows from the definition of a transitive relation.

$(4)$ follows from Reflexive Closure of Transitive Relation is Transitive.

Suppose that:
 * $\paren { a \mathrel{\RR} b } \land \paren { b \mathrel{\RR^=} c }$

By the definition of reflexive closure:
 * $b \mathrel{\RR} c$ or $b = c$

If $b = c$, then since $a \mathrel{\mathcal R} b$:
 * $a \mathrel{\RR} c$

If $b \mathrel{\RR} c$ then by transitivity of $\RR$:
 * $a \mathrel{\RR} c$

Thus $(2)$ holds.

A similar argument proves that $(3)$ holds as well:

Suppose that:
 * $\paren { a \mathrel{\RR^=} b } \land \paren { b \mathrel{\RR} c }$

By the definition of reflexive closure:
 * $a \mathrel{\RR} b$ or $a = b$

If $a = b$, then since $b \mathrel{\RR} c$:
 * $a \mathrel{\RR} c$

If $a \mathrel{\RR} b$ then by transitivity of $\RR$:
 * $a \mathrel{\RR} c$

Thus $(3)$ holds.