It is frequently stated in the following equivalent form: Suppose that is continuous and that u is a real number satisfying or Then for some c ∈ b, f ( c ) = u.

Suppose G is an ordered abelian group, meaning an abelian group with a total ordering "<" respecting the group's addition, so that a < b if and only if a + c < b + c for all c. Let I be a well-ordered subset of G, meaning I contains no infinite descending chain.

Suppose the table has r rows and c columns ; the row sums are and the column sums are.

Suppose a vertex joins three units with spin numbers a, b, and c. Then, these requirements are stated as:

Suppose that one computes the MDCT of the subsequent, 50 % overlapped, 2N block ( c, d, e, f ).

Suppose that the Fermat equation with exponent &# 8467 ; ≥ 3 had a solution in non-zero integers a, b, c. Let us form the corresponding Frey curve E. It is an elliptic curve and one can show that its discriminant Δ is equal to 16 ( abc )< sup > 2 &# 8467 ;</ sup > and its conductor N is the radical of abc, i. e. the product of all distinct primes dividing abc.

Suppose that a proportion of the population q ( where q < q < sub > c </ sub >) is immunised at birth against an infection with R < sub > 0 </ sub >> 1.

Suppose that π is any degree n subgroup of the symmetric group on n points, u a cohomology class in H < sup > q </ sup >( X, B ), A an abelian group acted on by π, and c a cohomology class in H < sub > i </ sub >( π, A ).

Suppose a, b, c are numbers ( not necessarily positive or rational ) which satisfy the following two equations:

Suppose a, b, c are positive real numbers.

Suppose Af is defined in the sub-interval Af.

Suppose they both had ventured into realms which their colleagues thought infidel: is this the way gentlemen settle frank differences of opinion??

Suppose, says Dr. Lyttleton, the proton has a slightly greater charge than the electron ( so slight it is presently immeasurable ).

Suppose it is something right on the planet, native to it.

Suppose there is a program

Suppose there is a chain at 1A, 2A, 3A, and 4A, along with another chain at 6A and 7A.

If two players tie for minority, they will share the minority shareholder bonus. Suppose Festival is the chain being acquired.

Alex is the majority shareholder, and Betty is the minority shareholder. Suppose now that Worldwide is the chain being acquired.

Suppose that R ( x, y ) is a relation in the xy plane.

) Then X < sub > i </ sub > is the value ( or realization ) produced by a given run of the process at time i. Suppose that the process is further known to have defined values for mean μ < sub > i </ sub > and variance σ < sub > i </ sub >< sup > 2 </ sup > for all times i. Then the definition of the autocorrelation between times s and t is

Suppose that a car is driving up a tall mountain.

Suppose that the car is ascending at 2. 5 km / h.

Suppose the vector field describes the velocity field of a fluid flow ( such as a large tank of liquid or gas ) and a small ball is located within the fluid or gas ( the centre of the ball being fixed at a certain point ).

Suppose that F is a partial function that takes one argument, a finite binary string, and possibly returns a single binary string as output.

Suppose, says Searle, that this computer performs its task so convincingly that it comfortably passes the Turing test: it convinces a human Chinese speaker that the program is itself a live Chinese speaker.

; Dennett's reply from natural selection: Suppose that, by some mutation, a human being is born that does not have Searle's " causal properties " but nevertheless acts exactly like a human being.

Suppose that is a complex-valued function which is differentiable as a function.

Suppose a mass is attached to a spring which exerts an attractive force on the mass proportional to the extension / compression of the spring.

Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.

Suppose Eve has intercepted the cryptogram below, and it is known to be encrypted using a simple substitution cipher:

Suppose we want to define an extremely simple XML markup scheme for a book: a book is defined as a sequence of one or more pages ; each page contains text only.

Suppose that R is an algebra over the field C of complex numbers and M = N is a finite-dimensional simple module over R. Then Schur's lemma says that the endomorphism ring of the module M is a division ring ; this division ring contains C in its center, is finite-dimensional over C and is therefore equal to C. Thus the endomorphism ring of the module M is " as small as possible ".

Suppose further that the dimension of B over k is finite, i. e. that B is a central simple algebra.

Suppose there is a simple database that lists people and addresses.

Suppose that a simple robot has two wheels which can both move forward or reverse and that they are positioned parallel to one another, and equidistant from the center of the robot.

Suppose that we desire a meromorphic function with simple poles of residue 1 at all positive integers.

A simple explanation would be as follows: Suppose the total finance charge for a 12 month loan was $ 78. 00.

* Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all have Lipschitz constant bounded by some K. If ƒ < sub > n </ sub > converges to a mapping ƒ uniformly, then ƒ is also Lipschitz, with Lipschitz constant bounded by the same K. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the Banach space of continuous functions.

Suppose U is an open subset of the complex plane C, f: U → C is a holomorphic function and the closed disk

Suppose we are given a closed, oriented curve in the xy plane.

Suppose that is an open subset of the complex plane, that is a holomorphic function on, and that is a closed curve in.

Suppose S is an immersed submanifold of M. If the inclusion map i: S → M is closed then S is actually an embedded submanifold of M. Conversely, if S is an embedded submanifold which is also a closed subset then the inclusion map is closed.

Suppose a planar closed loop carries an electric current I and has vector area S ( x, y, and z coordinates of this vector are the areas of projections of the loop onto the yz, zx, and xy planes ).

This problem reduces to a question on the coframe bundle of M. Suppose we had such a closed coframe

If is closed, densely defined and continuous on its domain, then it is defined on B < sub > 1 </ sub >.< ref > Suppose f < sub > j </ sub > is a sequence in the domain of T that converges to.

Suppose is a plane curve with curvature which makes a convex curve when closed by the chord connecting its endpoints, and is a curve of the same length with curvature.

In mathematical logic, the conservativity theorem states the following: Suppose that a closed formula

: Suppose that a closed formula is a theorem of a first-order theory, where we denote.