Lab color differences between different documents

I opened a new document in Lab space and paint it with the color L: 50, a: 100, b: 100.

My RGB working space is set to ProPhoto in my color settings, therefore I read R: 182, G: 43, B: 0 on the info panel as RGB values of this color.

Then, I opened an another document in ProPhoto and I paint it with the same color, R: 182, G: 43, B: 0.

And, when I looked at the info panel for Lab values, I read those values ... L: 50, a: 100, b: 86.

Lab values for the same color for the document opened in AdobeRGB ... L: 55, a: 82, b: 71.

I can not understand why we have different Lab values for the same color between those documents.

I think that Lab color readings need to be same for all of those documents, am I wrong?