What does the symbol ⊃ mean in equations 3.1 and 3.2?

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Looks very odd to me. From the surrounding text they appear to only be discussing a "subset" of the total Lagrangian, ##\mathcal{L}##. That's my best guess. The usual option would be to define a new variable, but I guess this is meant to avoid doing that.
 
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It's the interaction Lagrangian and not the total Lagrangian.
 
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