Final answer:
The question seems to pertain to special relativity and the constancy of the speed of light in different reference frames. A precise proof cannot be provided due to the truncated question, but the principle in question is that the speed of light remains constant for any two observers in relative motion according to Einstein's theory of relativity.
Step-by-step explanation:
The student's question appears to be relating to the field of Physics, specifically to concepts in special relativity where velocities and light speed are considered in different reference frames. However, the question seems to be truncated and lacks clear variables or context to provide a precise proof. To prove that a beam of light approaches at speed c, regardless of the relative velocity v between two observers, involves Lorentz transformations and the invariance of the speed of light in all inertial reference frames according to Einstein's theory of relativity. The provided statements and equations hint at these principles, depicting relationships between velocities, time dilation, and length contraction.
However, without the full and correct context or formulation of the question, any specific proof cannot be accurately provided. Assuming the question pertains to the constancy of the speed of light in all inertial frames, relativity theory does state that light's speed c will remain constant for any two observers, irrespective of their relative motion, as long as their velocities are less than the speed of light itself.