A map of exact couples induces a map of spectral sequences

Since $ui=\tilde{i}u$, the component of $u$ at $(p,q)$ sends $D_{p,q}^{\mathrm{\prime }}=\mathrm{im}{i}_{p-1,q+1}$ into ${\tilde{D}}_{p,q}^{\mathrm{\prime }}$, giving a restriction $u^{\mathrm{\prime }}$. Also $vjk=\tilde{j}uk=\tilde{j}\tilde{k}v$, so $v$ commutes with the page differential, sends cycles to cycles and boundaries to boundaries, and induces $v^{\mathrm{\prime }}:E_{p,q}^{\mathrm{\prime }}\to {\tilde{E}}_{p,q}^{\mathrm{\prime }}$. Both maps preserve the bidegree.

For $a\in D_{p,q}^{\mathrm{\prime }}$, the equality $u^{\mathrm{\prime }}{i}^{\mathrm{\prime }}a=u(ia)=\tilde{i}(ua)={\tilde{i}}^{\mathrm{\prime }}{u}^{\mathrm{\prime }}a$ proves the derived $i$ square. Locally write $a=ix$, where $x$ has bidegree $(p-1,q+1)$. Then $v^{\mathrm{\prime }}{j}^{\mathrm{\prime }}a=[vjx]=[\tilde{j}ux]={\tilde{j}}^{\mathrm{\prime }}(\tilde{i}ux)={\tilde{j}}^{\mathrm{\prime }}{u}^{\mathrm{\prime }}a$, at bidegree $(p-r,q+r)$. The formula is independent of the local lift by the already defined derived maps, and equality descends by epic cancellation. For a cycle $e\in E_{p,q}$, $u^{\mathrm{\prime }}{k}^{\mathrm{\prime }}[e]=uke=\tilde{k}ve={\tilde{k}}^{\mathrm{\prime }}{v}^{\mathrm{\prime }}[e]$, with target ${\tilde{D}}_{p-1,q}^{\mathrm{\prime }}$. Quotient descent proves this last equality on all of $E^{\mathrm{\prime }}$. These are every derived-couple commutation square with its required degrees.

Repeat steps 1.1–2.1 at each derived couple. On its $E$ terms the next map is precisely the map induced on homology by the current $v$ map. Thus the maps commute with every differential and with each transition in [F2], as required by [F3]. Image restrictions of identity maps are identities; quotient maps induced by identities are identities. Restrictions and quotient descents of a composite agree with composites of the restrictions and descents by their uniqueness. This proves identity and composition compatibility at every finite stage. Zero images, zero homology quotients and the initial $r=1$ case all use the same formulas; the latter sends the derived $j$ to degree $(-1,1)$ as required. No global lifts or AC are used.