Sum of Quotients of Real Numbers

Theorem

$\forall x, w \in \R, y, z \in \R_{\ne 0}: \dfrac x y + \dfrac w z = \dfrac {\paren {x \times z} + \paren {y \times w} } {y \times z}$

Proof

 $\displaystyle \frac x y + \frac w z$ $=$ $\displaystyle \paren {x \times \frac 1 y} + \paren {w \times \frac 1 z}$ Definition of Real Division $\displaystyle$ $=$ $\displaystyle \paren {x \times \frac 1 y \times 1} + \paren {1 \times w \times \frac 1 z}$ Real Number Axioms: $\R M 3$: Identity $\displaystyle$ $=$ $\displaystyle \paren {x \times \frac 1 y \times z \times \frac 1 z} + \paren {y \times \frac 1 y \times w \times \frac 1 z}$ Real Number Axioms: $\R M 4$: Inverses $\displaystyle$ $=$ $\displaystyle \paren {x \times z \times \frac 1 y \times \frac 1 z} + \paren {y \times w \times \frac 1 y \times \frac 1 z}$ Real Number Axioms: $\R M 2$: Commutativity $\displaystyle$ $=$ $\displaystyle \paren {\paren {x \times z} \times \paren {\frac 1 y \times \frac 1 z} } + \paren {\paren {y \times w} \times \paren {\frac 1 y \times \frac 1 z} }$ Real Number Axioms: $\R M 1$: Associativity $\displaystyle$ $=$ $\displaystyle \paren {\paren {x \times z} + \paren {y \times w} } \times \paren {\frac 1 y \times \frac 1 z}$ Real Number Axioms: $\R D$: Distributivity $\displaystyle$ $=$ $\displaystyle \paren {\paren {x \times z} + \paren {y \times w} } \times \frac 1 {y z}$ Product of Reciprocals of Real Numbers $\displaystyle$ $=$ $\displaystyle \frac {\paren {x \times z} + \paren {y \times w} } {y \times z}$ Definition of Real Division

$\blacksquare$