I found the theorem and its proof in the attachment file.

The thing is that I don't understand why suddenly "either disjoint or equal" become "disjoint" for "Ix=Iy" condition.

Could you explain this for me?

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- October 15th 2009, 11:28 AMgyro73Don't understand the proof of a simple theorem
I found the theorem and its proof in the attachment file.

The thing is that I don't understand why suddenly "either disjoint or equal" become "disjoint" for "Ix=Iy" condition.

Could you explain this for me? - October 15th 2009, 12:55 PMPlato
It is actually easy. But you need to understand that is a connect set.

In the open connected sets are .

If two connected set have a point in common there union is connected and has one of those three forms.

Now by definition of**is a maximum connected open set**.

Thus, either .