Thanks @greenrun for the feedback, the proof only considered simple linear equation:
Ax+Bx+C = 0, and
Ax+Bx+D = 0.
(general equation for a straight line . i.e line CD AND line AB)
when we solve this, it gives w(C-D) =0, WHERE w =0, and C-D = 0, AND C=D. This means line C = D, AND w= 0. then if C= D, THEN two lines are identical (overlapped).

Recall

August Ferdinand Möbius which stated that in order to make calculations of graphics and geometry possible in projective space, then homogeneous coordinates must be represented with (N+1) numbers.

This results in making a point in cartesian coordinates (x,y) to becomes (x, y, w)

substituting w = 0. in this equation gives (x,y) in cartesian to becomes (x, y, w) = (x/w, y/w) = (∞,∞). and this condition justifies that line C can only meet line D at (∞,∞) since w(C-D) = 0.
The proof does not consider/replace the angles α, β, γ, and δ but based on simple linear equation of a straight line.