Editing Surreal number (section)

===Negation===

Negation of a given number {{math|1=''x'' = {{mset| ''X''{{sub|''L''}} {{!}} ''X''{{sub|''R''}} }}}} is defined by
<math display=block>-x = - \{ X_L \mid X_R \} = \{ -X_R \mid -X_L \},</math>
where the negation of a set {{mvar|S}} of numbers is given by the set of the negated elements of {{mvar|S}}:
<math display=block>-S = \{ -s: s \in S \}.</math>

This formula involves the negation of the surreal ''numbers'' appearing in the left and right sets of {{mvar|x}}, which is to be understood as the result of choosing a form of the number, evaluating the negation of this form, and taking the equivalence class of the resulting form.  This makes sense only if the result is the same, irrespective of the choice of form of the operand.  This can be proved inductively using the fact that the numbers occurring in {{math|''X''{{sub|''L''}}}} and {{math|''X''{{sub|''R''}}}} are drawn from generations earlier than that in which the form {{mvar|x}} first occurs, and observing the special case:
<math display=block>-0 = - \{ {}\mid{} \} = \{ {}\mid{} \} = 0.</math>