Let $a,b \in F^\times$. Show that for quadratic forms holds: $$D(\lt 1,a \gt) \cap D(\lt 1,b \gt) \subseteq D(\lt 1,-ab \gt).$$
Here these sets represent the set of elements in $F^\times$ which are represented by quadratic forms.
Proof should go in standard why by taking element in the set on left hand side and then proving that is in the set on right hand side, but i get lost later in the proof.
Suppose $c\in D(\langle 1,a \rangle )\cap D(\langle 1,b \rangle )$. We know that $\langle 1,a\rangle \cong \langle c,ac \rangle $ and $\langle 1,b\rangle \cong \langle c,bc \rangle $. It follows that $$\langle 1,a,- c,-ac \rangle = \langle 1,b, -c,-bc \rangle. $$ By Witt Cancellation theorem $\langle a,-ac \rangle = \langle b,-bc \rangle $. Therefore $\langle a,-b\rangle=c\langle a,-b \rangle $. Multiply by $a$, then $\langle 1,-ab\rangle= \langle c,-abc\rangle $, so $c\in D(\langle1,-ab\rangle) $.