  Type I - heterotic duality in D=10 predicts various relations and constraints
on higher order F^n couplings at different string loop levels on both sides. We
prove the vanishing of two-loop corrections to the heterotic F^4 terms, which
is one of the basic predictions from this duality. Furthermore, we show that
the heterotic F^5 and (CP even) F^6 couplings are not renormalized at one loop.
These results strengthen the conjecture that in D=10 any Tr F^(2n) coupling
appears only at the disc tree-level on type I side and at (n-1)-loop level on
the heterotic side. Our non-renormalization theorems are valid in any heterotic
string vacuum with sixteen supercharges.
