To prove that there exists an such that holds, it suffices to give a specific and then prove that is true for that . Such a proof usually starts "let ", and then goes on to prove that holds for the given . is sometimes referred to as a witness for .
To prove that there exists an [math]x [/math] such that [math]P(x) [/math] holds, it suffices to give a specific [math]x [/math] and then prove that [math]P [/math] is true for that [math]x [/math]. Such a proof usually starts "let [math]x:= \cdots [/math]", and then goes on to prove that [math]P(x) [/math] holds for the given [math]x [/math]. [math]x [/math] is sometimes referred to as a witness for [math]P(x) [/math].
