Then the topology τ < sub > 1 </ sub > is said to be a coarser ( weaker or smaller ) topology than τ < sub > 2 </ sub >, and τ < sub > 2 </ sub > is said to be a finer ( stronger or larger ) topology than τ < sub > 1 </ sub >.

Then edit the coarse mesh to create the basic shape, then edit the offsets for the next subdivision step, then repeat this at finer and finer levels.

Then in another colour he drew a second still finer line upon the first, and went away, bidding her show it to Apelles if he came again, and add that this was the man he was seeking.

#* Proof: suppose that p is composite, hence can be written with a and Then is prime, but and contradicting statement 1.

Many Christians support a secular state, and may acknowledge that the conception has support in Biblical teachings, particularly the statement of Jesus in the Book of Luke: " Then give to Caesar what is Caesar's, and to God what is God's .".

Then in December 2006 Omar reportedly issued a statement expressing confidence that foreign forces will be driven out of Afghanistan.

Then Prime Minister Keizō Obuchi confirmed this meaning with a statement on June 29, 1999:

Then, the presiding officer makes a formal, but simple statement to the House, acquainting each House that the Royal Assent has been granted to the acts mentioned.

Then whenever F ( X ) would appear in a statement, you can replace it with a new symbol Y of type U and include another statement P ( X, Y ).

Then universally quantify over each Y immediately after the corresponding X is introduced ( that is, after X is quantified over, or at the beginning of the statement if X is free ), and guard the quantification with P ( X, Y ).

Then the statement is that the sum of delta-functions at each point of Λ, and at each point of Λ ′, are again Fourier transforms as distributions, subject to correct normalization.

Then Prime Minister Tony Blair issued a statement, saying that the report " makes a well-argued and powerful case for the system it recommends " and that " it is very much a modification of the existing Westminster system, rather than any full blown PR system as practised in other countries.

Then U. S. Under-Secretary of State Walter Bedell Smith said, " In connection with the statement in the Declaration concerning free elections in Vietnam, my government wishes to make clear its position which it has expressed in a Declaration made in Washington on June 29, 1954, as follows: ' In the case of nations now divided against their will, we shall continue to seek unity through free elections, supervised by the United Nations to ensure they are conducted fairly.

Then, the ( graded ) PBW theorem can be reformulated as the statement that, under certain hypotheses, this final morphism is an isomorphism.

Then this vibrant embodiment of the international interfaith community will again come together to strive to, in the words of the CPWR mission statement, " cultivate harmony between the world's religious and spiritual communities and foster their engagement with the world and its other guiding institutions in order to achieve a peaceful, just, and sustainable world ".

More formally, let us use variable x to denote abusively the tuple of variables involved in statement S. Then, a given Hoare triple is provable in Hoare logic for total correctness if and only if the first-order predicate below holds:

Then, due to his apparent disregard for her statement, the man ( in a stern manner ) tells her to stop talking and get on his horse.

The band's statement revealed that all bass playing on the forthcoming album had in fact been performed by either John Mitchell or John Beck-in what the band referred to as " Genesis-style ", a reference to the 1978 Genesis album ... And Then There Were Three ...-and that It Bites had reluctantly parted company with Nolan due to his lack of involvement and commitment.

Then the statement is a reformulation of the going up theorem of Cohen-Seidenberg.

Let A = K be the ring of polynomials in n variables over a field K. Then the global dimension of A is equal to n. This statement goes back to David Hilbert's foundational work on homological properties of polynomial rings, see Hilbert's syzygy theorem.

Then, the first player says a simple statement starting with " Never have I ever ".

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