1. Check my proof about perpendicular lines which are parallel

I have to prove this theorem:

If two different lines $\displaystyle a$ and $\displaystyle b$ are perpendicular to plane $\displaystyle \alpha$ then $\displaystyle a$ and $\displaystyle b$ are parallel lines.

My proof:
Lines $\displaystyle a$ and $\displaystyle b$ are perpendicular on plane $\displaystyle \alpha$ in points A and B.

Through points A and B there is line $\displaystyle c$ which is perpendicular to lines $\displaystyle a$ and $\displaystyle b$.
Lines $\displaystyle b$ and $\displaystyle c$ form plane $\displaystyle \beta$ perpendicular to line $\displaystyle d$ (in point B).
Through point B there is line $\displaystyle e$ which intersects line $\displaystyle a$ and is normal to line $\displaystyle d$.
Lines $\displaystyle e$ and $\displaystyle b$ form plane $\displaystyle \gamma$ which is normal to line $\displaystyle d$, but then there would exist two planes that contains point B and are perpendicular to line $\displaystyle d$ which is possible only if $\displaystyle \beta=\gamma$.
Then plane $\displaystyle \beta$ would contain also lines $\displaystyle c$ and $\displaystyle e$ and also line $\displaystyle a$ (because lines $\displaystyle c$ and $\displaystyle e$ intersects line $\displaystyle a$) which means that lines $\displaystyle a$ and $\displaystyle b$ are both in plane $\displaystyle \beta$.

Lines $\displaystyle a$ and $\displaystyle b$ don't intersect because than it would exist two lines perpendicular to line $\displaystyle c$ in one point (intersection point) so they are parallel.

Is my proof ok?

2. Originally Posted by OReilly
I have to prove this theorem:

If two different lines $\displaystyle a$ and $\displaystyle b$ are perpendicular to plane $\displaystyle \alpha$ then $\displaystyle a$ and $\displaystyle b$ are parallel lines.

My proof:
Lines $\displaystyle a$ and $\displaystyle b$ are perpendicular on plane $\displaystyle \alpha$ in points A and B.

Through points A and B there is line $\displaystyle c$ which is perpendicular to lines $\displaystyle a$ and $\displaystyle b$.
Lines $\displaystyle b$ and $\displaystyle c$ form plane $\displaystyle \beta$ perpendicular to line $\displaystyle d$ (in point B).
Through point B there is line $\displaystyle e$ which intersects line $\displaystyle a$ and is normal to line $\displaystyle d$.
Lines $\displaystyle e$ and $\displaystyle b$ form plane $\displaystyle \gamma$ which is normal to line $\displaystyle d$, but then there would exist two planes that contains point B and are perpendicular to line $\displaystyle d$ which is possible only if $\displaystyle \beta=\gamma$.
Then plane $\displaystyle \beta$ would contain also lines $\displaystyle c$ and $\displaystyle e$ and also line $\displaystyle a$ (because lines $\displaystyle c$ and $\displaystyle e$ intersects line $\displaystyle a$) which means that lines $\displaystyle a$ and $\displaystyle b$ are both in plane $\displaystyle \beta$.

Lines $\displaystyle a$ and $\displaystyle b$ don't intersect because than it would exist two lines perpendicular to line $\displaystyle c$ in one point (intersection point) so they are parallel.

Is my proof ok?
Looks okay to me, but if I may just say that reading the title of this post made my morning!

-Dan