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Vocal eliminators use the fact that only the music is recored in stereo; the vocals (of the lead singer) are recorded in mono. This means that we only have to subtract the left channel signal from the right channel signal. Unfortunately, this also eliminates nearly all low frequence signals (bass); these have to be added afterwards. The schematic below does exactly that:

Opamps U1.A and U1.B buffer the input signals.

Opamp U1.D is used as a differential amplifier. It's a combination of an inverting and a non-inverting amplifier. Since R3 = R4 = R5 = R6, U1.D's output is RIN - LIN.

The inverting amplifier around opamp U1.C adds RIN and LIN. Its output is fed to a low pass filter (R11, C1, U2.A). The result is a signal that only contains the low frequency part of both the left and the right channel.

This signal is added to RIN - LIN by opamp U2.B. Output resistors R14 and R15 make sure U2.B survives an accidental short to ground of the output terminals.