More precisely, let be a first quadrant spectral sequence, meaning that
Therefore the (1,0) term of the spectral sequence has converged, meaning that it is isomorphic to the degree one graded piece of the abutment H 1(A).
Because the spectral sequence lies in the first quadrant, the degree one graded piece is equal to the first subgroup in the filtration defining the graded pieces.
The inclusion of this subgroup yields the injection E21,0 → H 1(A) which begins the five-term exact sequence.
At the third page, the (0, 1) term of the spectral sequence has converged, because all the differentials into and out of Er0,1 either begin or end outside the first quadrant when r ≥ 3.
This graded piece is the quotient of H 1(A) by the first subgroup in the filtration, and hence it is the cokernel of the edge map from E21,0.
This yields a short exact sequence Because E30,1 is the kernel of the differential E20,1 → E22,0, the last term in the short exact sequence can be replaced with the differential.
The composite E22,0 → E32,0 → H2(A), which is another edge map, therefore has kernel equal to the differential landing at E22,0.
The five-term exact sequence can be extended at the cost of making one of the terms less explicit.
While there is an edge map E23,0 → H3(A), its kernel is not the previous term in the seven-term exact sequence.