Proof of two way contacts, either in the form of QSL cards or via digital entry into Logbook of The World ( LoTW ), must be submitted to qualify.

The edge lettering does not occur at the same time as the minting of the coins, allowing for the natural occurrence of the lettering in either orientation, except Proof Coins where the date and lettering are all " right-side-up.

It is awarded a stamped Proof Mark if it survives without either being destroyed or suffering damage from the proof load.

Proof of first property: from 3. we obtain ( a < sup >− 1 </ sup >)< sup >− 1 </ sup >=

Proof of second property: since a * b and b * b < sup >− 1 </ sup > are defined, so is a * b * b < sup >− 1 </ sup >=

#* Proof: Then so Thus However, is prime, so or In the former case, hence ( which is a contradiction, as neither 1 nor 0 is prime ) or In the latter case, or If however, which is not prime.

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

#* Proof: If q divides 2 < sup > p </ sup > − 1 then 2 < sup > p </ sup > ≡ 1 ( mod q ).

< div class =" NavHead " style =" background :# ccf ; text-align: left ; font-size: larger ;"> Proof of ( 1 )</ div >

Fourth Session ( October 1, 787 ) -- Proof of the lawfulness of the veneration of icons was drawn from Exodus 25: 19 sqq.

Proof: Since p is a prime number the only possible values of gcd ( p < sup > k </ sup >, m ) are 1, p, p < sup > 2 </ sup >, ..., p < sup > k </ sup >, and the only way for gcd ( p < sup > k </ sup >, m ) to not equal 1 is for m to be a multiple of p. The multiples of p that are less than or equal to p < sup > k </ sup > are p, 2p, 3p, ..., p < sup > k − 1 </ sup > p = p < sup > k </ sup >, and there are p < sup > k − 1 </ sup > of them.

* Arora, Sanjeev ; Barak, Boaz, " Complexity Theory: A Modern Approach ", Cambridge University Press, 2009, ISBN 978-0-521-42426-4, section 1. 4, " Machines as strings and the universal Turing machine " and 1. 7, " Proof of theorem 1. 9 "

*: Puncture Proof: 1

Proof of Lemma 1:

* 3200th Proof Test Group, 1 July 1948-1 July 1953

* 3200th Proof Test Wing, 1 April 1951-1 July 1952

* Proof that the Sample Bivariate Correlation Coefficient has Limits ± 1

# is a polynomial ( of degree ≤ n − 1 ) in .< ref > Proof: When A is normal, use Lagrange's interpolation formula to construct a polynomial P such that, where are the eigenvalues of A .</ ref >

Proof: By symmetry, it suffices to prove that there is some constant c such that for all bitstrings s

Generally this is in a very small detail, such as the number of leaves on the ear of corn on the recent US Wisconsin state quarter: File: 2004 WI Proof. png.

From the criterion it also follows that Carmichael numbers are cyclic .< ref > Proof sketch: If is square-free but not cyclic, for two prime factors and of.

* Demonstratio evangelica ( Proof of the Gospel ) is closely connected to the Praeparatio and comprised originally twenty books of which ten have been completely preserved as well as a fragment of the fifteenth.

He famously put the point into dramatic relief with his 1939 essay " Proof of an External World ", in which he gave a common sense argument against scepticism by raising his right hand and saying " Here is one hand ," and then raising his left and saying " And here is another ," then concluding that there are at least two external objects in the world, and therefore that he knows ( by this argument ) that an external world exists.

#* Proof:, so is a square root of 2 modulo.

* In Hal Clement's short story Proof ( 1942 ), neutronium is the only form of solid matter known to Solarians, the inhabitants of the Sun's interior.

Proof: If there is no possible move, then the lemma is vacuously true ( and the first player loses the normal play game by definition ).

Proof: Let d be the position of the leftmost ( most significant ) nonzero bit in the binary representation of s, and choose k such that the dth bit of x < sub > k </ sub > is also nonzero.

Proof by contradiction is also known as indirect proof, apagogical argument, proof by assuming the opposite, and reductio ad impossibilem.

* Proof that R is uncountable

Proof: Suppose G = ⟨ H | R ⟩ is a finitely presented, residually finite group.

* Proof of Fermat's Last Theorem is discovered by Andrew Wiles.

Proof: We only give the proof in the simplified case ; the general case is similar.

* R. Schoen and S .- T. Yau, " Proof of the positive mass theorem.

* R. Schoen and S. T. Yau, " Proof that the Bondi mass is positive ", Phys.

That he Fraunces was the Person that first discovered the Conspiracy which was formed in the Year 1776 against the Life of his Excellency General Washington and that the Suspicions Which were Entertained of his agency in that Important Discovery accationed occasioned a public Enquiry after he was made a Prisoner on which the want of positive Proof alone preserved his Life.

: Proof: r − 1 ( with r as above ) is in the Jacobson radical so r is invertible.

Proof.

Proof.

Proof.

Proof.

Proof.

* Sanjeev Saxena, " A Simple Proof of Bernoulli's Inequality ", viXra: 1205. 0068, May 2012

: Proof: Let A be infinite RE.

* Gödel's Proof ( 2002 revised edition ) by Ernest Nagel and James R. Newman, edited by Hofstadter ( ISBN 0-8147-5816-9 ).

‡ Proof.

Proof.

Proof.

Proof.

Proof.

Proof: Suppose that and are two identity elements of.

Proof: Suppose that and are two inverses of an element of.

Proof.

Proof.

Proof.

Proof.

A more successful effort was the Standard Proof for Pythagoras ' Theorem, that replaced the more than 100 incompatible existing proofs.

Proof for the existence of a common Germanic goddess once known as * Fraujon does not exist, but scholars have commented that this may simply be due to lack of evidence.