Witt's cancellation theorem in characteristic 2 Witt's cancellation theorem states that if $(V_1,q_1)$, $(V_2,q_2)$, $(V_3,q_3)$ are quadratic forms over a field such that $$(V_1,q_1) \perp (V_3,q_3) \cong (V_2,q_2) \perp (V_3,q_3),$$ then $(V_1,q_1) \cong (V_2,q_2)$. The argument when the characteristic of the ground field is not $2$ is easy to find.
It is a widely cited but very rarely proved fact that this theorem also holds over any field of characteristic two. For example, this is claimed in the appendix of Milnor and Husemöller's Symmetric Bilinear Forms, but it is not proved there, nor does it appear to be proved in the references to Bourbaki or Chevalley (as far as I can tell).
I would be grateful for a reference to a proof of this.

Here is a short, recent article that gives examples of failure for Witt cancellation over some rings for which 2 is special in some way. Might take some work to figure out an example over a field of characteristic two... http://users.math.yale.edu/~auel/papers/docs/wittlocal.pdf
For more detail, probably worth writing to Prof. Auel at Yale.
Corollary 12.12 on page 118 of Classical Groups and Geometric Algebra by Larry C. Grove  http://bookstore.ams.org/gsm-39 
second occurrence at GROVE 

