I have to prove this theorem:

If two different lines

and

are perpendicular to plane

then

and

are parallel lines.

My proof:

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?