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From these materials, it is required to construct a proposition <math>p : X \to \underline\mathbb{B}</math> such that <math>p(x) = F(f_1 (x), \ldots, f_k (x)),</math> for all <math>x \in X.</math>  The desired construction is determined as follows:

From these materials, it is required to construct a proposition <math>p : X \to \underline\mathbb{B}</math> such that <math>p(x) = F(f_1 (x), \ldots, f_k (x)),</math> for all <math>x \in X.</math>  The desired construction is determined as follows:

The cartesian power <math>\underline\mathbb{B}^k,</math> as a cartesian product, is characterized by the possession of a projective imagination <math>\pi = (\pi_1, \ldots, \pi_k)</math> of degree <math>k\!</math> on <math>\underline\mathbb{B}^k,</math> along with the property that any imagination <math>\underline{f} = (f_1, \ldots, f_k)</math> of degree <math>k\!</math> on an arbitrary set <math>W\!</math> determines a unique map <math>f! : W \to \underline\mathbb{B}^k,</math> the play of whose projective images <math>(\pi_1 (f!(w), \ldots, \pi_k (f!(w))</math> on the functional image <math>f!(w)</math> matches the play of images <math>(f_1 (w), \ldots, f_k (w))</math> under <math>\underline{f},</math> term for term and at every element <math>w\!</math> in <math>W.\!</math>

The cartesian power <math>\underline\mathbb{B}^k,</math> as a cartesian product, is characterized by the possession of a projective imagination <math>\pi = (\pi_1, \ldots, \pi_k)</math> of degree <math>k\!</math> on <math>\underline\mathbb{B}^k,</math> along with the property that any imagination <math>\underline{f} = (f_1, \ldots, f_k)</math> of degree <math>k\!</math> on an arbitrary set <math>W\!</math> determines a unique map <math>f! : W \to \underline\mathbb{B}^k,</math> the play of whose projective images <math>(\pi_1 (f!(w), \ldots, \pi_k (f!(w))</math> on the functional image <math>f!(w)\!</math> matches the play of images <math>(f_1 (w), \ldots, f_k (w))</math> under <math>\underline{f},</math> term for term and at every element <math>w\!</math> in <math>W.\!</math>

 

Just to be on the safe side, I state this again in more standard terms.  The cartesian power <math>\underline\mathbb{B}^k,</math> as a cartesian product, is characterized by the possession of <math>k\!</math> projection maps <math>\pi_j : \underline\mathbb{B}^k \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> along with the property that any <math>k\!</math> maps <math>f_j : W \to \underline\mathbb{B},</math> from an arbitrary set <math>W\!</math> to <math>\underline\mathbb{B},</math> determine a unique map <math>f! : W \to \underline\mathbb{B}^k</math> such that <math>\pi_j (f!(w)) = f_j (w),\!</math> for all <math>j = 1 ~\text{to}~ k,</math> and for all <math>w \in W.</math>

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Now suppose that the arbitrary set W in this construction is just the relevant universe U.  Given that the function f! : U -> Bk is uniquely determined by the imagination f : (U -> B)k, that is, by the k-tuple of propositions f = <f1, ..., fk>, it is safe to identify f! and f as being a single function, and this makes it convenient on many occasions to refer to the identified function by means of its explicitly descriptive name "<f1, ..., fk>".  This facility of address is especially appropriate whenever a concrete term or a constructive precision is demanded by the context of discussion.

Now suppose that the arbitrary set W in this construction is just the relevant universe U.  Given that the function f! : U -> Bk is uniquely determined by the imagination f : (U -> B)k, that is, by the k-tuple of propositions f = <f1, ..., fk>, it is safe to identify f! and f as being a single function, and this makes it convenient on many occasions to refer to the identified function by means of its explicitly descriptive name "<f1, ..., fk>".  This facility of address is especially appropriate whenever a concrete term or a constructive precision is demanded by the context of discussion.

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