Definition:Subfield/Proper Subfield

Definition
Let $\struct {K, +, \circ}$ be a subfield of $\struct {F, +, \circ}$.

Then $\struct {K, +, \circ}$ is a proper subfield of $\struct {F, +, \circ}$ $K \ne F$.

That is, $\struct {K, +, \circ}$ is a proper subfield of $\struct {F, +, \circ}$ :
 * $(1): \quad \struct {K, +, \circ}$ is a subfield of $\struct {F, +, \circ}$
 * $(2): \quad K$ is a proper subset of $F$.