Originally Posted by

**Plato** “Lines p and q determine one plane beta.

Then planes alpha and beta have one common point A (A belongs to line q).

Then there is common line q' for alpha and beta.

Line p is parallel to line q' (otherwise it would intersect plane alpha) but then is q'=q because only one line is parallel to line p which goes through point A, so q belongs to plane alpha.”

It seems to me as if there is problem with “Line p is parallel to line q' (otherwise it would intersect plane alpha) but then is q'=q”. Just because two lines do not intersect does not make them parallel.

Skew lines have that property.

You can say that both p & q’ are in plane beta. Because q’ is a subset of alpha it cannot intersect p. Therefore in plane beta we have two non-intersecting lines so they must be parallel.

Now the rest of your proof works.