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To aid this development, let us define a function, <math>\rho</math>, that takes as its argument a (possibly weakly) labelled object <math>\alpha</math> and relabels its atoms in an order-consistent way so that <math>\rho(\alpha)</math> is well labelled. We then define the labelled product for two objects <math>\alpha</math> and <math>\beta</math> as
:<math>\alpha \star \beta = \{(\alpha',\beta'): (\alpha',\beta') \mbox{ is well-labelled, }
Finally, the labelled product of two classes <math>\mathcal{A}</math> and <math>\mathcal{B}</math> is
:<math>\mathcal{A} \star \mathcal{B} = \bigcup_{\alpha \in \mathcal{A}, \beta \in \mathcal{B}} (\alpha \star \beta)</math>
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