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Veblen & Young, in "projective geometry", vol. 1, section 105, use the following lemma:

If a collineation between two intersecting planes in 3d projective space is such that any two corresponding lines intersect on the common line of the two planes, then the collineation is a perspectivity.

Where one may find a proof?

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The common line $L$ of the two planes is preserved by the collineation $g$. Now, as it is where the intersection of any two corr. lines $\ell$ and $g(\ell)$ lies, each point on $L$ is preserved as well. But now you are still left with quite a bit of freedom ($GL_2$ acting on a 4-dimensional space).

Are you sure there are no more conditions there?

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  • $\begingroup$ Thank you Dima! I think the same, however they an authority, so I preferred to check this. I believe quite a different theorem must be used. $\endgroup$
    – peter smoczynski
    Commented Feb 10, 2015 at 22:30
  • $\begingroup$ perhaps you should spell out exactly what they mean by "corresponding lines" of a collineation. $\endgroup$
    – Dima Pasechnik
    Commented Feb 11, 2015 at 9:34
  • $\begingroup$ On what page of the book it was used? (Sect 105 is 6 pages long...) $\endgroup$
    – Dima Pasechnik
    Commented Feb 11, 2015 at 9:39

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