Looks okay to me, but if I may just say that reading the title of this post made my morning!I have to prove this theorem:
If two different lines and are perpendicular to plane then and are parallel lines.
Lines and are perpendicular on plane in points A and B.
Through points A and B there is line which is perpendicular to lines and .
Lines and form plane perpendicular to line (in point B).
Through point B there is line which intersects line and is normal to line .
Lines and form plane which is normal to line , but then there would exist two planes that contains point B and are perpendicular to line which is possible only if .
Then plane would contain also lines and and also line (because lines and intersects line ) which means that lines and are both in plane .
Lines and don't intersect because than it would exist two lines perpendicular to line in one point (intersection point) so they are parallel.
Is my proof ok?