Hmmm, I don't think I'm expressing myself clearly enough.
There are actually two kind of proofs: informal and formal proofs.
By the very definition, a proof is formal. A proof is defined as a sequence of logical algebraic steps. Each step follows from another step by applying an inference rule.
These things are the actual proofs. However, they are EXTREMELY hard to read. If all math books would consist out of formal proofs, then math books would be unreadable.
That's why we allow informal proofs. An informal proof presents people the basic ideas and reasoning of a proof. An informal proof suggests that the formal proof can be carried out if one would want to do so.
All text books contain informal proofs (and it's good that they do). Text books like to work with a lot of text and a lot of pictures to make it easier on the student. But one should always keep in mind that one should be able to formalize the proof.
And indeed, one can formalize all the proofs. An example is given by http://us.metamath.org/index.html
which contains thousands of proofs including the celebrated Hahn-Banach theorem. This site shows what actual formal proofs look like. And as one can see, it is completely unreadable. (the site also shows formal proofs in plane geometry, including constructibility).
I am NOT advocating that everybody should be using formal proofs in journals or education. Quite the contrary. But the students should be well-aware that the formal proof can be carried it if one wants to do it.