Question about derivation of lorentz transformation

[physics user1]

Why does it use t' in that equation and not t? Isn't the equation relative to what an observer in the external frame of reference see? So if it is why not using the time he register?

(The equation is uploades in the photo)

 

[Nugatory]

Why does it use t' in that equation and not t? Isn't the equation relative to what an observer in the external frame of reference see? So if it is why not using the time he register?

If the origin of the primed coordinate system is moving at speed $v$ relative to the origin of the unprimed coordinate system, then the origin of the unprimed coordinate system is moving at speed $-v$ relative to the origin of the primed coordinate system. Because neither frame is prefered, the transformation from the primed coordinates to the unprimed coordinates must be the same as the transformation from the unprimed coordinates to the primed coordinates, except with $v$ replaced by $-v$. That's where the circled equation comes from.