Lichens are a composite organism made up of a fungus and a photosynthetic organism?

The relationship appears to be mutualistic, with the fungal partner protecting the algae or cyanobacteria by retaining water, and the photosynthetic partner reducing atmospheric carbon dioxide into carbohydrates that provide energy for the fungus. How can we "prove" this, that is, what would be the best way to design an experiment that would provide support for this hypothesis?