of the theorem The proof of the theorem can be presented in various ways. Here the proof is first given in the language of coordinates and Christoffel symbols, and then in the coordinate-free language of covariant derivatives. Regardless of the presentation, the idea is to use the metric-compatibility and torsion-freeness conditions to obtain a direct formula for any connection that is both metric-compatible and torsion-free. This establishes the uniqueness claim in the fundamental theorem. To establish the existence claim, it must be directly checked that the formula obtained does define a connection as desired. Local coordinates Here the Einstein summation convention will be used, which is to say that an index repeated as both subscript and superscript is being summed over all values. Let denote the dimension of . Recall that, relative to a local chart, a connection is given by smooth functions with for any vector fields and . Torsion-freeness of the connection refers to the condition that for arbitrary and . Written in terms of local coordinates, this is equivalent to which by arbitrariness of and is equivalent to the condition . Similarly, the condition of metric-compatibility is equivalent to the condition In this way, it is seen that the conditions of torsion-freeness and metric-compatibility can be viewed as a linear system of equations for the connection, in which the coefficients and 'right-hand side' of the system are given by the metric and its first derivative. The fundamental theorem of Riemannian geometry can be viewed as saying that this linear system has a unique solution. This is seen via the following computation: in which the metric-compatibility condition is used three times for the first equality and the torsion-free condition is used three times for the second equality. The resulting formula is sometimes known as the first Christoffel identity. It can be contracted with the inverse of the metric, , to find the second Christoffel identity: This proves the uniqueness of a torsion-free and metric-compatible condition; that is, any such connection must be given by the above formula. To prove the existence, it must be checked that the above formula defines a connection that is torsion-free and metric-compatible. This can be done directly. Invariant formulation The above proof can also be expressed in terms of vector fields. Torsion-freeness refers to the condition that and metric-compatibility refers to the condition that where , , and are arbitrary vector fields. The