Why in proc glm, the SE of a difference just simply takes on the larger SE of the two means?

[richardy]

In proc glm (simple one-way ANOVA), I noticed the SE of differences do not correspond to widely circulated uncertainty propagation technique (e.g. any variation of quadrature summation, pooled variance etc.). Instead, it just takes on the larger SE of the two means in comparison. What is the math and assumption behind this? Is this always justifiable?

[PaigeMiller]

Which command in PROC GLM produced the top table?

What are the sample sizes in the three groups At, D and Taglia?