Real Numbers form Subfield of Complex Numbers

Theorem
The Field of Real Numbers $$\left({\R, +, \times}\right)$$ forms a subfield of the Field of Complex Numbers $$\left({\C, +, \times}\right)$$.

Proof
From Additive Group of Reals Subgroup of Complex, $$\left({\R, +}\right)$$ is a subgroup of $$\left({\C, +}\right)$$.

From Multiplicative Group of Reals Subgroup of Complex, $$\left({\R, \times}\right)$$ is a subgroup of $$\left({\C, \times}\right)$$.

The result follows from the Subfield Test via the One-Step Subgroup Test and Two-Step Subgroup Test.