If the algorithm proves all values of x_{k+1} to be inconsistent with x_1=a_1,\ldots,x_k=a_k, then this evaluation was consistent, as otherwise the algorithm would not have evaluated x_{k+1} at all; as a result, the constraints violated by a value of x_{k+1} together with x_1=a_1,\ldots,x_k=a_k all contain x_{k+1}.
As a result, an inconsistent evaluation is the restriction of the truth evaluation of x_1,\ldots,x_k to variables that are in a constraint with x_{k+1}, provided that this constraint contains no unassigned variable.