Results Have Both a Logical Role and a Job in the Argument
In Writing Definitions Precisely, the focus was on fixing the meaning of terms by stating their domains and conditions. A proved result uses those established meanings to make a claim. Within a proof, some results are most useful as intermediate steps, and others follow readily from something already proved. The labels “lemma” and “corollary” help a reader see those roles.
The labels do not change the logical requirements. A lemma still needs a definite statement and a valid proof. A corollary also needs a definite statement, and its derivation must follow from the result it cites. Calling a claim a corollary does not make it true automatically; the reader must be able to identify the earlier result and see why its hypotheses apply.
This tutorial concentrates on how to write and use these statements. In particular, a useful lemma should be precise enough to apply later without guessing what its variables mean. A useful corollary should show which established statement it depends on and how the present claim fits that statement.
Give a Lemma a Reusable Form
A lemma often packages one step that would otherwise have to be repeated inside a larger proof. To make it reusable, write it as a complete conditional or quantified statement. Identify every object under discussion, state any assumptions they must satisfy, and make the conclusion explicit. Do not rely on a nearby sentence to supply a missing domain or hypothesis if the lemma may later be read on its own.
For example, compare “Adding inequalities preserves the order” with the statement: “Let \(a,b,c,d\in\mathbb R\). If \(a\leq b\) and \(c\leq d\), then \(a+c\leq b+d\).” The second version identifies the number system, includes both assumptions, and states exactly what the reader may use. A proof can then point to the two assumptions separately.
A lemma should be sized for its mathematical purpose. It need not be more general than is useful, but it should not depend on an accidental choice of symbols or numerical values from the proof in which it first appears. If a later argument needs a fact for arbitrary real inputs, state it for arbitrary real inputs at the outset. If the fact requires a condition such as \(r>0\), include that condition rather than leaving its necessity to be inferred.
Worked Example: Stating an Order Lemma Completely
Consider the claim that adding two inequalities gives an inequality between the sums. A precise version is the following.
Lemma. Let \(a,b,c,d\in\mathbb R\). If \(a\leq b\) and \(c\leq d\), then \(a+c\leq b+d\).
Proof. Since \(a\leq b\), adding \(c\) to both sides gives \(a+c\leq b+c\). Since \(c\leq d\), adding \(b\) to both sides gives \(b+c\leq b+d\). By transitivity of the order, \(a+c\leq b+d\), as required.
Each hypothesis has a job: the first compares the first terms, and the second compares the second terms. If either inequality is omitted, the conclusion need not follow. For instance, \(a=4\), \(b=1\), \(c=0\), and \(d=2\) satisfy \(c\leq d\) but not \(a\leq b\), and their sums satisfy \(4\not\leq3\). Explicit hypotheses make the result safe to reuse.
Make the Proof's Dependency Visible
When a lemma is used later, its proof should leave a clear path from the stated assumptions to its conclusion. A short proof is welcome when every step is justified; a short proof is not complete if it skips an essential condition. A useful way to organize the writing is to move from the assumptions, through the established facts they permit, to the exact conclusion in the lemma.
Check that each variable in the statement has a stated domain and that every needed assumption is present.
Translate assumptions into inequalities, equalities, or other facts that can be used in the proof.
Use an earlier result or a justified operation, and name the reasoning that joins the steps.
The proof must establish the claimed conclusion, including its strict or weak inequality and any stated boundary case.
The proof of the order lemma illustrates this pattern. The assumptions \(a\leq b\) and \(c\leq d\) are each used once. Adding the same real number to both sides preserves each inequality, and transitivity joins the resulting comparisons. If a proof uses only one of two hypotheses, that may signal an unnecessarily strong statement, or it may mean that a step has not yet been justified.
Worked Example: Applying a Lemma as a Corollary
Use the order lemma to establish a numerical consequence.
Corollary. If \(x,y\in\mathbb R\), \(x\leq4\), and \(y\leq-1\), then \(x+y\leq3\).
Proof. Apply the lemma with \(a=x\), \(b=4\), \(c=y\), and \(d=-1\). Its assumptions hold because \(x\leq4\) and \(y\leq-1\). Its conclusion is $$ x+y\leq4+(-1)=3. $$ This proves the corollary.
The substitution is explicit: each variable in the lemma is assigned a value, and each required hypothesis is checked. The result is a corollary because it follows directly from the lemma with those choices. If one of the input inequalities had been reversed, the lemma would not apply as written, even if the numerical conclusion happened to be true for some particular values.
Write a Corollary With Its Parent Result in View
A corollary is clearest when it tells the reader which previously established result it uses. If the deduction is immediate, a brief proof may name that result, make the substitution or specialization, and state the resulting conclusion. If there is an additional step, show it. The word “corollary” is not a substitute for explaining the dependency.
One common route is to specialize a general statement to particular inputs. Another is to combine a general statement with a simple observation. In either case, check the original hypotheses before invoking the result. If the earlier result applies only to positive real numbers, for example, the corollary must establish positivity for the chosen inputs. A conclusion that resembles the earlier theorem is not enough; its assumptions must actually be met.
Sometimes a corollary is useful because it presents a consequence in the exact form needed next, even though it contains little new reasoning. Naming it can save the reader from repeatedly translating a general result into a particular setting. But excessive labeling can interrupt the argument. If a one-line substitution has no later role and adds no clarity, it may be better to include it in the surrounding proof without giving it a separate heading.
Worked Example: Translating an Open Interval
A general result about intervals can be stated as a lemma and then used to obtain a specific corollary.
Lemma. Let \(a,b,c\in\mathbb R\) with \(a<b\). For every \(x\in\mathbb R\), $$ x\in(a,b)\quad\Longleftrightarrow\quad x+c\in(a+c,b+c). $$
Proof. By the definition of the open interval, \(x\in(a,b)\) means \(a<x<b\). Adding \(c\) to all three parts of this compound inequality gives \(a+c<x+c<b+c\), which means \(x+c\in(a+c,b+c)\). Conversely, if \(x+c\in(a+c,b+c)\), then \(a+c<x+c<b+c\). Subtracting \(c\) from all three parts gives \(a<x<b\), so \(x\in(a,b)\). Both directions hold, proving the equivalence.
Corollary. If \(x\in(2,7)\), then \(x+3\in(5,10)\).
Proof. Apply the lemma with \(a=2\), \(b=7\), and \(c=3\). The condition \(a<b\) holds because \(2<7\). The lemma gives \(x\in(2,7)\) if and only if \(x+3\in(2+3,7+3)=(5,10)\). In particular, the stated membership follows.
The lemma states a reusable equivalence for every allowed interval and translation. The corollary selects one interval and one translation. The same hypotheses are checked, and the specific endpoints are calculated rather than left implicit.
Do Not Confuse the Label With the Logic
The distinction among theorem, lemma, and corollary is partly a matter of organization. A lemma is commonly positioned as support for another result; a corollary is presented as a direct consequence of an earlier result. Either way, the mathematical standard is unchanged: the statement must have a definite truth value, and the proof must justify it from definitions and established results.
This has two practical consequences. First, a statement does not become a lemma merely because it is placed before a theorem. Its proof still needs to establish all of its claims. Second, a result described as a corollary should have a visible route back to an earlier result. If that route requires a substantial new argument, the statement may still be correct, but the reader should be shown the argument rather than asked to accept the label.
It is also possible for a result to play different roles in different arguments. A statement used as an intermediate step in one proof may be a main result in another. The label reflects the role chosen for the current exposition, not an intrinsic level of mathematical importance. The proof and its hypotheses, rather than the heading alone, determine what has actually been established.
A Final Check Before Presenting a Result
Before writing a lemma or corollary into a proof, read its statement independently of the surrounding paragraph. Can a reader identify the domain of every variable? Are the hypotheses sufficient, and are they written explicitly? Does the conclusion say exactly what the proof will establish? Then inspect the proof dependency: is the result proved here, or does it follow from a named earlier result? These checks make an argument easier to verify and to reuse.
For a corollary, it is helpful to write down the substitution before composing the prose. Match each variable in the earlier result with the object in the new claim, then verify each assumption one by one. For a lemma, test the endpoint of the proof against the exact conclusion, especially when the statement involves strict inequalities, equality cases, or an equivalence with two directions.
Finally, use labels in moderation. A labeled result is helpful when it marks a reusable step, makes a dependency easy to find, or gives a consequence a convenient form for later use. When it does none of these things, a direct sentence in the proof may be clearer. Precise mathematical writing is not measured by the number of labeled statements; it is measured by whether the reader can follow exactly what is assumed and why the conclusion follows.
Check Your Understanding
For each question, focus on the statement's assumptions, the role of its label, and the justification for its conclusion.
- A proposed lemma says, “If \(a\leq b\), then \(a+c\leq b+d\).” What additional condition on \(c\) and \(d\) would make this conclusion follow from the order lemma proved above?
- In the corollary about \(x+y\leq3\), identify the values assigned to \(a,b,c,d\) and the two hypotheses that must be checked.
- Why does calling a statement a corollary not remove the need to check the hypotheses of the earlier result it uses?
- In the open-interval lemma, why must the proof establish both directions of the displayed equivalence?
- A proof uses a lemma but never uses one of its hypotheses. What are two possible issues this may indicate?
- When might a direct sentence in a proof be clearer than presenting a separate labeled corollary?