Editing Surreal number (section)

===Addition===
The definition of addition is also a recursive formula:
<math display=block>x + y = \{ X_L \mid X_R \} + \{ Y_L \mid Y_R \} = \{ X_L + y, x + Y_L \mid X_R + y, x + Y_R \},</math>
where

<math display=block>X + y = \{ x' + y: x' \in X \} , \quad x + Y = \{ x + y': y' \in Y \}</math>

This formula involves sums of one of the original operands and a surreal number drawn from the left or right set of the other. It can be proved inductively with the special cases:
<math display=block>0 + 0 = \{ {}\mid{} \} + \{ {}\mid{} \} = \{ {}\mid{} \} = 0</math>
<math display=block>x + 0 = x + \{ {}\mid{} \} = \{ X_L + 0 \mid X_R + 0 \} = \{ X_L \mid X_R \} = x</math>
<math display=block>0 + y = \{ {}\mid{} \} + y = \{ 0 + Y_L \mid 0 + Y_R \} = \{ Y_L \mid Y_R \} = y</math>
For example:
:{{math|1={{sfrac|1|2}} + {{sfrac|1|2}} = {{mset| 0 {{!}} 1 }} + {{mset| 0 {{!}} 1 }} = {{mset| {{sfrac|1|2}} {{!}} {{sfrac|3|2}} }}}},
which by the birthday property is a form of 1. This justifies the label used in the previous section.

====Subtraction====
Subtraction is defined with addition and negation:
<math display=block>x - y = \{ X_L \mid X_R \} + \{ -Y_R \mid -Y_L \} = \{ X_L - y, x - Y_R \mid X_R - y, x - Y_L \}\,.</math>