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This is not only a logically equivalent DNF but exactly the same DNF expression that we obtained before, so we have established the given equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))}.\!</math> Incidentally, one may wish to note that this DNF expression quickly folds into the following form:

This is not only a logically equivalent DNF but exactly the same DNF expression that we obtained before, so we have established the given equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))}.\!</math>&nbsp; Incidentally, one may wish to note that this DNF expression quickly folds into the following form:

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In other words, <math>{}^{\backprime\backprime} p ~\mathrm{is~equivalent~to}~ p ~\mathrm{and}~ q ~\mathrm{and}~ r {}^{\prime\prime}.\!</math>

In other words, <math>{}^{\backprime\backprime} p ~\mathrm{is~equivalent~to}~ p ~\mathrm{and}~ q ~\mathrm{and}~ r {}^{\prime\prime}.\!</math>

One lemma that suggests itself at this point is a principle that may be canonized as the ''Emptiness Rule''.  It says that a bare lobe expression like <math>\texttt{( \_, \_, \ldots )},\!</math> with any number of places for arguments but nothing but blanks as filler, is logically tantamount to the proto-typical expression of its type, namely, the constant expression <math>\texttt{(~)}\!</math> that <math>\mathrm{Ex}\!</math> interprets as denoting the logical value <math>\mathrm{false}.\!</math>  To depict the rule in graphical form, we have the continuing sequence of equations:

One lemma that suggests itself at this point is a principle that may be canonized as the ''Emptiness Rule''.  It says that a bare lobe expression like <math>\texttt{(} \_ \texttt{,} \_ \texttt{,} \ldots \texttt{)},\!</math> with any number of places for arguments but nothing but blanks as filler, is logically tantamount to the proto-typical expression of its type, namely, the constant expression <math>\texttt{(} ~ \texttt{)}\!</math> that <math>\mathrm{Ex}~\!</math> interprets as denoting the logical value <math>\mathrm{false}.~\!</math>  To depict the rule in graphical form, we have the continuing sequence of equations:

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