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Transitivity of parallel lines

I need to prove that $\displaystyle a\parallel b \wedge b\parallel c \Rightarrow a\parallel c$ if a,b,c are not in the same plane.

See attachment.

If we put point B in plane $\displaystyle \alpha $ then we get line d that is parallel to line b. Also if we put point C then e is parallel to c.

Now we must prove that $\displaystyle a\parallel d \wedge d\parallel e \Rightarrow a\parallel e$.

If e is not parallel to a then a and e must intersect but then we would have two lines that are parallel to d which is contradictory so its must be $\displaystyle a\parallel c$.

Is that proof ok?