Starting from the Navier-Stokes equation, we rigorously prove that a modifi
ed third-order structure function, (S) over tilde(3)(r), asymptotically equ
als -4/3 epsilon r (epsilon is the dissipation rate) in an inertial regime.
From this result, we rigorously confirm the Kolmogorov four-fifths law, wi
thout the Kolmogorov assumption on isotropy. Our definition of the structur
e function involves a solid angle averaging over all possible orientations
of the displacement vector y, besides space-time-averaging. Direct:numerica
l simulation for a highly symmetric flow for a Taylor Reynolds number of up
to 155 shows that the flow remains significantly anisotropic and that, wit
hout solid angle averaging, the resulting structure functions approximately
satisfy these scaling relations over some range of r = \y\ for some orient
ation of y, but not for another.