thinking aloud here - so it seems like 2 things are taken as intuitive here:

a) energy is conserved in any frame of reference. b) energy can vary in 2 frame of references.

but then what it feels like is that when you reference the energy as mE(v), the v is actually not the only variable, and it will be more like mE(v, v_moving_reference)?

so we also must take intuitive that c) E(v, v_moving_reference) == E(v - v_moving_reference)