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text: 'The real numbers, when [50g constructed as Dedekind cuts] over the [4zq rationals], form a [481 field].\n\nWe shall often write the one-sided [dedekind_cut Dedekind cut] $(A, B)$ %%note:Recall: "one-sided" means that $A$ has no greatest element.%% as simply $\\mathbf{A}$ (using bold face for Dedekind cuts); we can do this because if we already know $A$ then $B$ is completely determined.\nThis will make our notation less messy.\n\nThe field structure, together with the [total_order total ordering] on it, is as follows (where we write $\\mathbf{0}$ for the Dedekind cut $(\\{ r \\in \\mathbb{Q} \\mid r < 0\\}, \\{ r \\in \\mathbb{Q} \\mid r \\geq 0 \\})$): \n\n- $(A, B) + (C, D) = (A+C, B+D)$\n- $\\mathbf{A} \\leq \\mathbf{C}$ if and only if everything in $A$ lies in $C$.\n- Multiplication is somewhat complicated.\n - If $\\mathbf{0} \\leq \\mathbf{A}$, then $\\mathbf{A} \\times \\mathbf{C} = \\{ a c \\mid a \\in A, a > 0, c \\in C \\}$. [todo: we've missed out the complement in this notation, and can't put set-builder sets in boldface]\n - If $\\mathbf{A} < \\mathbf{0}$ and $\\mathbf{0} \\leq \\mathbf{C}$, then $\\mathbf{A} \\times \\mathbf{C} = \\{ a c \\mid a \\in A, c \\in C, c > 0 \\}$.\n - If $\\mathbf{A} < \\mathbf{0}$ and $\\mathbf{C} < \\mathbf{0}$, then $\\mathbf{A} \\times \\mathbf{C} = \\{\\} $ [todo: write down the form of the set]\n\nwhere $(A, B)$ is a one-sided [dedekind_cut Dedekind cut] (so that $A$ has no greatest element).\n\n(Here, the "set sum" $A+C$ is defined as "everything that can be made by adding one thing from $A$ to one thing from $C$": namely, $\\{ a+c \\mid a \\in A, c \\in C \\}$ in [-3lj]; and $A \\times C$ is similarly $\\{ a \\times c \\mid a \\in A, c \\in C \\}$.)\n\n# Proof\n\n## Well-definedness\n\nWe need to show firstly that these operations do in fact produce [dedekind_cut Dedekind cuts].\n\n### Addition\nFirstly, we need everything in $A+C$ to be less than everything in $B+D$.\nThis is true: if $a+c \\in A+C$, and $b+d \\in B+D$, then since $a < b$ and $c < d$, we have $a+c < b+d$.\n\nNext, we need $A+C$ and $B+D$ together to contain all the rationals.\nThis is true: [todo: this, it's quite boring]\n\nFinally, we need $(A+C, B+D)$ to be one-sided: that is, $A+C$ needs to have no top element, or equivalently, if $a+c \\in A+C$ then we can find a bigger $a' + c'$ in $A+C$.\nThis is also true: if $a+c$ is an element of $A+C$, then we can find an element $a'$ of $A$ which is bigger than $a$, and an element $c'$ of $C$ which is bigger than $C$ (since both $A$ and $C$ have no top elements, because the respective Dedekind cuts are one-sided); then $a' + c'$ is in $A+C$ and is bigger than $a+c$.\n\n### Multiplication\n[todo: this section]\n\n### Ordering\n[todo: this section]\n\n## Additive [3jb commutative] [3gd group structure]\n\n[todo: identity, associativity, inverse, commutativity]\n\n## [3gq Ring structure]\n\n[todo: multiplicative identity, associativity, distributivity]\n\n## [481 Field structure]\n\n[todo: inverses]\n\n## Ordering on the field\n\n[todo: a <= b implies a+c <= b+c, and 0 <= a, 0 <= b implies 0 <= ab]',
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