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Double de-clutching is a method of gear changing using a gearbox which is not fitted with the synchromesh devices which are assumed to be just 'normal' today.
If a driver didn't double de-clutch or didn't do it properly, there was a gear 'clash'. As well as being horribly noisy, this could be horribly expensive.
In simplified terms, in preparation for changing gear, place your left hand on the gear selector lever and your left foot in place upon the clutch pedal. Then, concurrently, release pressure on the throttle pedal and apply pressure to full depth onto the clutch pedal. This is the first de-clutch which disconnects the engine from the drive wheels at the back.
In this condition, with the clutch pedal fully depressed, move the gear selector lever into 'neutral' and release the clutch pedal. The engine, although still not connected to the rear wheels, is now re-connected to the input, output and lay shafts within the gearbox. In this condition bring the engine up to the estimated speed it would need to be running once the next gear is actually engaged. Next, a moment later, press the clutch pedal full depth - which is the second de-clutch - and move the gear selector lever to its next position. Finally gently release the clutch pedal to complete the gear change.
The above method was what I was taught when first learning to drive. Soon it became second nature for a gear change to take less than a second - except that I was not doing it many hundreds of times while, at the same time, distracted by the responsibilities of driving a London bus!