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Given a double complex $C_{\ast,\ast}$ concentrated in the first quadrant. Then I understand how one can associate two spectral sequences to this complex. One that first takes horizontal and then vertical differentials and the other one vice-versa. Both converge to the homology of the total complex. In signs:

$$E^2_{p,q}=H_p(H_q(C_{\ast,\ast},d_2),d_1)))\Longrightarrow H_\ast(Tot(C_{\ast,\ast}))$$

and

$$E^2_{p,q}=H_q(H_p(C_{\ast,\ast},d_1),d_2)))\Longrightarrow H_\ast(Tot(C_{\ast,\ast})).$$

My question is how this fact (i. e. that they both converge to the same object) can be used in calculating the spectral sequences. In other words how can these spectral sequences be played off against each other? I often read this, e.g. in calculation for the second term of the Grothendieck spectral sequence, but I do not see the point.

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