Then is a compact topological space ; this follows from the Tychonoff theorem.

Then by Arzelà – Ascoli theorem the space K is compact.

Then Goursat's theorem asserts that ƒ is analytic in an open complex domain Ω if and only if it satisfies the Cauchy – Riemann equation in the domain.

Then, once this claim ( expressed in the previous sentence ) is proved, it will suffice to prove " φ is either refutable or satisfiable " only for φ's belonging to the class C. Note also that if φ is provably equivalent to ψ ( i. e., ( φ ≡ ψ ) is provable ), then it is indeed the case that " ψ is either refutable or satisfiable " → " φ is either refutable or satisfiable " ( the soundness theorem is needed to show this ).

Then according to the second isomorphism theorem S ∩ T is normal in T and ST / S ≅ T /( S ∩ T ).

# Then used resolution to attempt to obtain a proof by contradiction by adding the clausal form of the negation of the theorem to be proved.

Let us suppose that L is a complete lattice and let f be a monotonic function from L into L. Then, any x ′ such that f ′( x ′) ≤ x ′ is an abstraction of the least fixed-point of f, which exists, according to the Knaster – Tarski theorem.

Then Cauchy's theorem can be stated as the integral of a function holomorphic in an open set taken around any cycle in the open set is zero.

Then use the SAS congruence theorem for triangles OPA ' and OPB ' to conclude that angles POA and POB are equal.

Then the group action is by classification of G-orbits ( also known as the orbit-stabilizer theorem ).

Abstractly, we can say that D is a linear transformation from some vector space V to another one, W. We know that D ( c ) = 0 for any constant function c. We can by general theory ( mean value theorem ) identify the subspace C of V, consisting of all constant functions as the whole kernel of D. Then by linear algebra we can establish that D < sup >− 1 </ sup > is a well-defined linear transformation that is bijective on Im D and takes values in V / C.

Then Noether's theorem states that the following quantity is conserved,

Then by the theorem, the equation also holds for D, E and F ′.

Then they are independent, but not necessarily identically distributed, and their joint probability distribution is given by the Bapat – Beg theorem.

Then one proves that if the theorem is true for pieces resulting from a cutting of a Haken manifold, that it is true for that Haken manifold.

Then there is a cyclic cubic field inside the cyclotomic field of pth roots of unity, and a normal integral basis of periods for the integers of this field ( an instance of the Hilbert – Speiser theorem ).

Then it is a theorem that

Then, the Riemann-Roch theorem states that

Then the integral in Mercer's theorem reduces to a simple summation

Then the Riemann – Roch theorem states: if g is a genus of X,

Then the theorem states that for analytic functions f, if

Then the Radon – Nikodym theorem provides the function g, equal to the density of μ with respect to Q.

Then v ( p ) is the displacement vector of this projected point relative to p. According to the hairy ball theorem, there is a p such that v ( p )

Then the Pythagorean theorem for two curves intersecting orthogonally at is:

Then Wilson's theorem says that

After this, the senior priest ( or bishop ) pours pure olive oil and a small amount of wine into the shrine lamp, and says the " Prayer of the Oil ", which calls upon God to "... sanctify this Oil, that it may be effectual for those who shall be anointed therewith, unto healing, and unto relief from every passion, every malady of the flesh and of the spirit, and every ill ..." Then follow seven series of epistles, gospels, long prayers, Ektenias ( litanies ) and anointings.

" Then the celebrant says:

The city will be taken, the houses looted, and the women raped … Then the Lord will go forth and fight against those nations … Thus the Lord my God will come, and all the saints with you .” 14: 2-5 In Matthew, Jesus says, " The sign of the Son of Man will appear in heaven, and then all the tribes of the earth will mourn, and they will see the Son of Man coming on the clouds of heaven with power and great glory.

Then Yoneda's lemma says that:

Then to perform, two's complement theory says to invert each bit with a NOT gate then add one.

Arion is alluded to in Plato's " Republic " at 453d, where Socrates says: “ Then we, too, must swim and try to escape out of the sea of argument in the hope that either some dolphin will take us on its back.

Then he climbs off and she says, " Birthday is over, Seymour ," and lights up a cigarette.

" Then, as he says the next prayer, he washes each of the places where he had been anointed with Chrism.

Then, as Eleanor weeps softly in the stands, Gehrig addresses the fans: " People all say that I've had a bad break ," he says.

As Superman brings Lucy back to the window of her apartment, Ricky is furious, and at one point yells, "... In all of the fifteen years we've been married ..." Then Superman says, " You mean to tell me that you've been married to her for fifteen years?

Then we covered them with real duck and turkey feathers ," says Chesney.

Then it gives Tristran a leaf and says to listen to it when he needs help the most.

Then, after the fight, he says, " My thoughts are whirl'd like a potter's wheel ./ I know not where I am nor what I do ./ A witch, by fear, not force, like Hannibal ,/ Drives back our troops and conquers as she lists " ( 1. 6. 19 – 22 ).

Then, after Henry praises peacemakers, Winchester hypocritically says, " Let me be blessèd for the peace I make ,/ Against this proud Protector with my sword " ( 2. 1. 35 – 36 ).

Then, when making his argument as to why he should be king, he says to Henry, " Thy hand is made to grasp a palmer's staff / And not to grace an aweful princely sceptre " ( 5. 1. 97 – 98 ).

Then in answering a question from a reporter in the White House press room, Ryan says he will seek election to the office of President of the United States.

Then the priest says :) Almighty and eternal God, long ago you gave your promise to Abraham and his posterity.

Then, he says that now that he is finally like them, he has shown them the way to make others see them.

" Then roller coasters and attractions of Six Flags are shown and says " Six Flags, More Flags, More Fun!

Then in 1575, Hernando de Escalante Fontaneda, a shipwreck survivor who had lived with the Native Americans of Florida for 17 years, published his memoir in which he locates the Fountain of Youth in Florida, and says that Ponce de León was supposed to have looked for them there.

Unlocking the repository, the Vorta says " Then accept this gift from the Founders.

Then one also says that x approximates y and writes

Then Lola's dad says he gave it " the old college try ".

" Then, Gehry came up with the curved wooden pipe concept, " like a logjam kind of thing ," says Rosales, " turned sideways.

" Then, following shouts of " Aye ", he says, " of the contrary, No ," and similar shouts of " No " may follow.