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We exhibit a finite basis M for a certain variety $\mathbf{V}$ of medialgroupoids. The set M consists of the medial law (xy)(zt)=(xz)(yt) and fiveother identities involving four variables. The variety $\mathbf{V}$ isgenerated by the four groupoids $\pm x\pm y$ on the integers. Since$\mathbf{V}$ is a very natural variety, proving it to be finitely based shouldbe of interest.  In an earlier paper, we made a conjecture which implies that $\mathbf{V}$ isfinitely based. In this paper, we show that $\mathbf{V}$ is finitely based byproving that M is a basis. Based on our proof, we think that our conjecturewill be difficult to prove.  We used four medial groupoids to define $\mathbf{V}$. We also present afinite basis for the variety generated by any proper subset of these fourgroupoids. In an earlier paper with R. Padmanabhan, we gave the correspondingfinite bases when the constant zero is allowed.

Groupoid identities common to four abelian group operations