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This fill edge preserves the proper coloring because intersecting subtrees from the same character would contradict conditions 3 and 4 of the perfect phylogeny definition.
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This contradicts condition (17).
which contradicts condition (1.13).
which contradicts condition ( H 9 ).
However, this contradicts condition (C5).
So,, which contradicts condition (ii).
But this contradicts condition (2.15).
end{aligned} But this contradicts condition (3.13).
So, we have which contradicts condition (h3).
By condition (ii), we have F ⊄ C, which contradicts condition (i).
It follows from being an -subspace of that which contradicts condition for each.
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CEO of Professional Science Editing for Scientists @ prosciediting.com