96 Consider a different choice of decomposition $L'$ in place of $L$ above.

    96 Consider a different choice of decomposition $L'$ in place of $L$ above.

    97 This leads to a cycle consisting of filtration degree 1 stuff.

    97 This leads to a cycle consisting of filtration degree 1 stuff.

    98 We want to show that this cycle bounds a chain of filtration degree 2 stuff.

    98 We want to show that this cycle bounds a chain of filtration degree 2 stuff.

    99 Choose a decomposition $M$ which has common refinements with each of 

    99 Choose a decomposition $M$ which has common refinements with each of 

   100 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$.

   100 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$.

   101 Then we have a filtration degree 2 chain, as shown in Figure yyyy, which does the trick.

   102 Then we have a filtration degree 2 chain, as shown in Figure yyyy, which does the trick.

   102 For example, ....

   103 For example, ....

   103 

   104 

   104 

   105 

   105 \end{proof}

   106 \end{proof}