A proof was the evidence or argument which established a fact or the truth of a statement. The proof was the most logical conclusion arrived at after analyzing all evidence. It was conclusive and concrete.
As one of the conditions for staying with Vina, Captain Christopher Pike requested from the Talosians that they give him proof that the USS Enterprise was all right. (TOS: "The Cage")
In 2266, after learning about the corbomite device, Balok of the Fesarius delayed the destruction of the Enterprise and would relent in its destruction if the crew of the starship provided proof of this device. (TOS: "The Corbomite Maneuver")
In 2370, proof was discovered that Cardassians supplied The Circle and were behind Minister Jaro Essa's attempt to take control of the Bajoran Provisional Government. (DS9: "The Siege")