ci+1 = xiyi + (xi xor yi)ci
= xiyi + xi not(yi) ci + not(xi) yi ci
= xi (yi + not(yi) ci) + not(xi) yi ci
= xi (yi + ci) + not(xi) yi ci
= xiyi + xici + not(xi) yi ci
= xici + xiyi + not(xi) yi ci
= xici + yi (xi + not(xi) ci)
= xici + yi (xi + ci)
= xici + yixi + yici
= xiyi + xi ci + yi ci

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