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We propose a simple alternative proof of the Wronskian representation formulaobtained by Gaillard and Matveev for the Darboux-P\"oschl-Teller (DPT)potentials. It rests on the use of singular Darboux-B\"acklund transformationsapplied to the free particle system combined to the shape invariance propertiesof the DPT.

A short proof of the Gaillard-Matveev theorem based on shape invariance arguments