Cancellation Law for Field Product

Theorem
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$.

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

Then:
 * $a \times b = a \times c \implies a = 0_F \text { or } b = c$

Proof
Let $a \times b = a \times c$.

Then:

That is:
 * $a \ne 0 \implies b = c$

Suppose $b \ne c$.

Then by Rule of Transposition:
 * $\neg \paren {a \ne 0}$

that is:
 * $a = 0_F$

and we note that in this case:
 * $a \times b = 0_F = a \times c$