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Curious about skips above and below them and the indentation.

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opinions differ on this matter.

amsthm treats each theorem-class object as a separate unit, beginning and ending with a paragraph break. depending on the \theoremstyle (three are predefined), there may or may not be extra space above or below. the text that starts immediately after a theorem-class object is always assumed to start a new paragraph, and unless that is overridden, it will be indented; this is true whether or not a blank line is present (but for ease of reading the input file, a blank line is recommended.)

the same paragraphing convention is true for the proof environment of amsthm; however, here there is also the extra visual cue of the "tombstone" at the end, as well as extra space above and below.

for details, have a look at the amsthm user's guide -- texdoc amsthm.

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Have a look at the amsthm package. It provides commands and environments for typesetting all kinds of theorems. Also the spaces above and below a paragraph will be set.

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