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In logic, equality is a primitive predicate (a statement that may have free variables) with the reflexive property (called the Law of identity), and the substitution property. From those, one can derive the rest of the properties usually needed for equality.
The corresponding angles as well as the corresponding sides are defined as appearing in the same sequence, so for example if in a polygon with the side sequence abcde and another with the corresponding side sequence vwxyz we have vertex angle a appearing between sides a and b then its corresponding vertex angle v must appear between sides v and w.
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
A substitution is called a ground substitution if it maps all variables of its domain to ground, i.e. variable-free, terms. The substitution instance tσ of a ground substitution is a ground term if all of t ' s variables are in σ ' s domain, i.e. if vars(t) ⊆ dom(σ).
A free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not a parameter of this or any container expression. The idea is related to a placeholder (a symbol that will later be replaced by some value), or a wildcard character that stands for an unspecified symbol.
The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. [5] It is known in Russia as the universal trigonometric substitution, [6] and also known by variant names such as half-tangent substitution or half-angle substitution.