Mapping on Cartesian Product of Substructures is Restriction of Operation

Theorem
Let $\struct {S, \circ}$ be an algebraic structure with $1$ operation.

Let $\struct {A, \circ {\restriction_A} }, \struct {B, \circ {\restriction_B} }$ be closed algebraic substructures of $\struct {S, \circ}$, where $\circ {\restriction_A}$ and $\circ {\restriction_B}$ are the operations induced by the restrictions of $\circ$ to $A$ and $B$ respectively.

Let the mapping $\phi: A \times B \to S$ be defined as:


 * $\forall \tuple {a, b} \in A \times B: \map \phi {a, b} = a \circ b$

where $A \times B$ denotes the Cartesian product of $A$ and $B$.

Then $\phi$ is the restriction to $A \times B$ of the operation $\circ$ on $S \times S \to S$.

Proof
Suppose the mapping $\phi: A \times B \to S$ is defined as:


 * $\forall \tuple {a, b} \in A \times B: \map \phi {a, b} = a \circ b$

where $A \times B$ denotes the Cartesian product of $A$ and $B$.

Then: