** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

* If two small categories are weakly equivalent, then they are equivalent.

If two equivalent amplifiers are being compared, the amplifier with higher gain settings would be more sensitive as it would take less input signal to produce a given amount of power.

If, in England, the wine sold for 70 francs ( or the pound equivalent ), which he then used to buy coal, which he imported into France, and was found to be worth 90 francs in France, he would have made a profit of 40 francs.

If a suitably sized quantum computer capable of running Grover's algorithm reliably becomes available, it would reduce a 128-bit key down to 64-bit security, roughly a DES equivalent.

If the axiom of choice is assumed, as will be the case in the rest of this article, then the two definitions are equivalent.

If ƒ is differentiable, this is equivalent to:

If it is the fraction of impinging atoms getting through the foil which is measured, the result can still be expressed in terms of the equivalent stopping cross section of the atoms.

To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m. If ( x, y ) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates

If the franchisor requires purchase from his stores, it may come under anti-trust legislation or equivalent laws of other countries.

If some specific deductive system of first-order logic is sound and complete, then is it " perfect " ( a formula is provable iff it is a semantic consequence of the axioms ), thus equivalent to any other deductive system with the same quality ( any proof in one system can be converted into the other ).

If is refutable, then so is which is equivalent to it ; thus is provable.

If adults could effectively be baptised through a desire for the sacrament when prevented from actually receiving it, some speculated that perhaps sacramentally unbaptised infants too might be saved by some waterless equivalent of ordinary baptism when prevented.

If an object ’ s weight was equivalent to 1728 carob seeds, then the object was said to weigh one Roman pound.

If, on the other hand, the object ’ s weight was equivalent to 144 carob seeds then the object was said to weigh one Roman ounce ( uncia ).

If the first-countability axiom were imposed on the topological spaces in question, the two above conditions would be equivalent.

The rules are equivalent ( If you divide both sides of inequality TR > TVC by Q gives P > AVC ).

If we allow a finite automaton access to two stacks instead of just one, we obtain a more powerful device, equivalent in power to a Turing machine.

If the object is moving either toward or away from the transmitter, there is a slight equivalent change in the frequency of the radio waves, caused by the Doppler effect.

If dark decided to put a piece in the topmost location ( all choices are strategically equivalent at this time ), one piece gets turned over, so that the board appears thus:

If converted to the metal equivalent, they were about 19, 000 per tonne in 2009.

If an argument form is valid, the conjunction of the premises will be logically equivalent to the conclusion and this can be clearly seen in a truth table ; it is displayed.

If the axiom of choice holds, the following conditions on a cardinal are equivalent:

If the Euclidean space is equipped with a choice of origin, then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin.

If it is " observed " ( measured with a photon ) not at a particular slit but rather at the screen, then there is no " which path " information as part of the interaction, so the electron's " observed " position on the screen is determined strictly by its probability function.

If the photon energy is too low, the electron is unable to escape the material.

If the photon energy is absorbed, some of the energy liberates the electron from the atom, and the rest contributes to the electron's kinetic energy as a free particle.

If an atom is already in the excited state, it may be perturbed by the passage of a photon that has a frequency ν < sub > 21 </ sub > corresponding to the energy gap ΔE of the excited state to ground state transition.

If a photon moves from one place and time – in shorthand, A – to another place and time – shorthand, B – the associated quantity is written in Feynman's shorthand as P ( A to B ).

If a light source (' the atom ') is in the excited state with energy, it may spontaneously decay to a lower lying level ( e. g., the ground state ) with energy, releasing the difference in energy between the two states as a photon.

If the atom is in the excited state, it may decay into the lower state by the process of spontaneous emission, releasing the difference in energies between the two states as a photon.

If we were to set 80 % of the mass of the photon rocket

If a photon rocket begins its journey in low earth orbit, then one year of thrusting may be required to achieve an earth escape velocity of 11. 2 km / s if the vehicle is already in orbit at a velocity of 9, 100 m / s, and 400 m / s additional velocity is obtained from the east to west rotation of the earth.

If the signal data are not constant, then can be calculated from data where the signal is zero or relatively constant .</ ref > Notice that such an alternative definition is only useful for variables that are always non-negative ( such as photon counts and luminance ).

If the scattered photon still has enough energy left, the process may be repeated.

If the photon is of lower energy, but still has sufficient energy ( in general a few eV to a few keV, corresponding to visible light through soft X-rays ), it can eject an electron from its host atom entirely ( a process known as the photoelectric effect ), instead of undergoing Compton scattering.

If the final vibrational state of the molecule is more energetic than the initial state, then the emitted photon will be shifted to a lower frequency in order for the total energy of the system to remain balanced.

If the final vibrational state is less energetic than the initial state, then the emitted photon will be shifted to a higher frequency, and this is designated as an Anti-Stokes shift.

Prior to any measurements being made, the polarizations of the photons are indeterminate ; If a measurement is made on one of the photons using a two-channel polarizer aligned with the axes of the coordinate system, the photon assumes either horizontal or vertical polarization, with 50 % probability for each orientation, and the other two photons immediately assume the identical polarization.

If the atom, which is now in the excited state, emits a photon spontaneously, it will be kicked by the same amount of momentum but in a random direction.

If the photon is absorbed, no new direction is given, and tracing for that photon ends.

If the photon reflects, the surface's bidirectional reflectance distribution function is used to determine the ratio of reflected radiance.

If an excited atom then emits a photon spontaneously, it will be accelerated.

If a wave containing exactly one photon passes over some atoms, and one of them absorbs it, that atom needs to tell the others that they can't absorb the photon anymore.

If an observer runs away from a photon in the direction it travels from a source, having it catch up with the observer, then when the photon catches up it will be seen as having less energy than it had at the source.