Theorem 3

$((x = y) \text{ and } (y = z)) \Rightarrow (x = z)$.

Let us adjoin the hypotheses $x = y$, $y = z$ to the axioms of $\mathscr{T}_0$. By S6 the relation $(x = y) \Rightarrow ((x = z) \Leftrightarrow (y = z))$ is true. Hence

and consequently $x = z$, are true.

C44. Let $x$ be a letter and let $T$, $U$, $V\{x\}$ be terms in $\mathscr{T}_0$. Then the relation $(T = U) \Rightarrow (V\{T\} = V\{U\})$ is a theorem in $\mathscr{T}_0$.

For let $y$ and $z$ be two distinct letters which are distinct from $x$ and from the letters which appear in $T$, $U$, $V$. Adjoin the hypothesis $y = z$. Then, by S6,

that is to say $(V\{y\} = V\{y\}) \Leftrightarrow (V\{y\} = V\{z\})$, is true. Now, $V\{y\} = V\{y\}$ is true by Theorem 1; hence $V\{y\} = V\{z\}$ is true.

From all this it follows that $(y = z) \Rightarrow (V\{y\} = V\{z\})$ is a theorem in $\mathscr{T}_0$, say $A$. But $(T|y)(U|z)A$ is precisely

¶ A relation of the form $T = U$, where $T$ and $U$ are terms in $\mathscr{T}$, is called an equation; a solution (in $\mathscr{T}$) of the relation $T = U$, considered as an equation in a letter $x$, is therefore (§2, no. 2) a term $V$ in $\mathscr{T}$ such that $T\{V\} = U\{V\}$ is a theorem in $\mathscr{T}$.

¶ Let $T$ and $U$ be two terms in $\mathscr{T}$, and let $x_1$, $x_2$, ..., $x_n$ be the letters which appear in $T$ but not in $U$. If the relation

is a theorem in $\mathscr{T}$, we say that U can be put in the form $T$ (in $\mathscr{T}$). Let $R$ be a relation in $\mathscr{T}$ and let $y$ be a letter. Let $V$ be a solution (in $\mathscr{T}$) of $R$, considered as a relation in $y$. If every solution (in $\mathscr{T}$) of $R$, considered as a relation in $y$, can be put in the form $V$, then $V$ is said to be the complete solution (or general solution) of $R$ (in $\mathscr{T}$).