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Given any proposition <math>f : X \to \mathbb{B},</math> the ''tacit extension'' of <math>f\!</math> to <math>\operatorname{E}X</math> is notated <math>\varepsilon f : \operatorname{E}X \to \mathbb{B}</math> and defined by the equation <math>\varepsilon f = f,</math> so it's really just the same proposition living in a bigger universe.

Given any proposition f : X -> B, the "tacit extension" of f to EX

is notated !e!f : EX -> B and defined by the equation !e!f = f, so

it's really just the same proposition living in a bigger universe.

Tacit extensions formalize the intuitive idea that a new function

Tacit extensions formalize the intuitive idea that a new function is related to an old function in such a way that it obeys the same constraints on the old variables, with a "don't care" condition on the new variables.

is related to an old function in such a way that it obeys the same

constraints on the old variables, with a "don't care" condition on

the new variables.

Figure 24-2 illustrates the "tacit extension" of the proposition

Figure&nbsp;24-2 illustrates the tacit extension of the proposition or scalar field <math>f = pq : X \to \mathbb{B}</math> to give the extended proposition or differential field that we notate as <math>\varepsilon f = \varepsilon (pq) : \operatorname{E}X \to \mathbb{B}.</math>

or scalar field f = pq : X -> B to give the extended proposition

or differential field that we notate as !e!f = !e![pq] : EX -> B.

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