Negation of a given number "x" = { "XL" | "XR" } is defined by where the negation of a set "S" of numbers is given by the set of the negated elements of "S":  This formula involves the negation of the surreal "numbers" appearing in the left and right sets of "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 only makes sense if the result is the same, irrespective of the choice of form of the operand. This can be proven inductively using the fact that the numbers occurring in "XL" and "XR" are drawn from generations earlier than that in which the form "x" first occurs, and observing the special case:  The definition of addition is also a recursive formula: where  This formula involves sums of one of the original operands and a surreal "number" drawn from the left or right set of the other. These are to be understood as the result of choosing a form of the numeric operand, performing the sum of the two forms, and taking the equivalence class of the resulting form. This only makes sense if the result is the same, irrespective of the choice of form of the numeric operand. This can also be proven inductively with the special cases:  The recursive formula for multiplication contains arithmetic expressions involving the operands and their left and right sets, such as the expression formula_8 that appears in the left set of the product of "x" and "y". This is to be understood as the set of surreal numbers resulting from choosing one number from each set that appears in the expression and evaluating the expression on these numbers. (In each individual evaluation of the expression, only one number is chosen from each set, and is substituted in each place where that set appears in the expression.) This depends, in turn, on the ability to (a) multiply pairs of surreal "numbers" drawn from the left and right sets of "x" and "y" to get a surreal number, and negate the result; (b) multiply the surreal number "form" "x" or "y" and a surreal "number" drawn from the left or right set of the other operand to get a surreal number; and (c) add the resulting surreal numbers. This again involves special cases, this time containing 0 = { | }, the multiplicative identity 1 = { 0 | }, and its additive inverse -1 = { | 0 }.