If σ an > π an, it is assumed that σ an is p times π an.

σ (pair I) π (pair j)f(i, j) = π (pair j) σ (pair i)f(i, j)

Remove f(i, j) from both sides.

σ (for I) π (for J) = π (for J) σ (for I)

σ (for I) = π (for J)

∑ stands for summation symbol in mathematics, which is mainly used to find the sum of multiple numbers.

The small print ∑ below, i= 1 indicates the sum from 1, and the small print above indicates the number that requires the sum.

For example, n is the sum of all the numbers added to n from 1.

The first number n representing A represents the N power of A, the second represents the sum, and the Theory of Three Represents multiplication such as ∏i= 1 (below the symbol) n (above the symbol) ai (to the right of the symbol) represents A 1 * A2...* an, which means the same as sum. The following symbol represents the lower limit (or initial value) of the right variable parameter.

Extended data:

Mathematical proofs are expressed as formal mathematical objects, thus simplifying their analysis through mathematical techniques. Proof is usually represented by inductive data structures, such as linked list, box linked list or tree, which are constructed according to axioms and reasoning rules of logical systems. Therefore, the theory of proof is essentially grammatical logic, which is contrary to the model theory of semantics. Together with model theory, axiomatic set theory and recursion theory, proof theory is called one of the four pillars of mathematical foundation.

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