The “spacetime” (length-time) interval is invariant over the Lorentz transformation (LT). The following is a proof of this for the inverse LT with length space axes *x*, *y*, and, *z*; temporal axis *t* (time line), velocity *v*, and maximum velocity *c*, along with *β* = *v*/*c* and *γ* = 1/√(1 − *β*²):

The invariant interval is

Expand the squares and cancel the middle terms to get:

The duration-base interval is invariant over the *dual* Lorentz transformation (DLT). The following is a proof of this for the inverse DLT that follows with temporal axes *x*, *y*, and, *z*; basal axis *r* (baseline), legerity *u*, and maximum legerity *κ*, along with *ζ* = *u*/*κ* and *λ* = 1/√(1 − *ζ*²):

The invariant interval is

Expand the squares and cancel the middle terms to get:

Let Σ (Δ*r _{i}*)² = (Δ

*r*)² and Σ (Δ

*t*)² = (Δ

_{i}*t*)². Then the two invariant intervals are

These two invariant intervals are proportional if *κ* = 1/*c*:

In that case, the invariant interval for proper time is the same for both 3+1 and 1+3 dimensions. However, 3D proper time is limited to space-like intervals, while 3D proper distance is limited to time-like intervals.

If *c* = *κ* = 1 (without regard to units), then these invariant intervals are nominally the same: