Changes

|}

|}

Associated with the triadic relation <math>L\!</math> is a binary relation <math>L_{\underline{~} ~ \underline{~} ~ \operatorname{T}} \subseteq \mathbb{B} \times \mathbb{B}</math> that is called the ''fiber'' of <math>L\!</math> with <math>\operatorname{T}</math> in the third place.  This object is defined as follows:

Associated with the triadic relation <math>L\!</math> is a binary relation <math>L_{\underline{~} \, \underline{~} \, \operatorname{T}} \subseteq \mathbb{B} \times \mathbb{B}</math> that is called the ''fiber'' of <math>L\!</math> with <math>\operatorname{T}</math> in the third place.  This object is defined as follows:

{| align="center" cellspacing="10" width="90%"

{| align="center" cellspacing="10" width="90%"

| <math>L_{\underline{~} ~ \underline{~} ~ \operatorname{T}} = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : (p, q, \operatorname{T}) \in L \}.</math>

| <math>L_{\underline{~} \, \underline{~} \, \operatorname{T}} = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : (p, q, \operatorname{T}) \in L \}.</math>

|}

|}

: <math>\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\,.\!</math>

: <math>\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\,.\!</math>

Form the binary relation that is called the ''fiber'' of <math>\operatorname{Cond}</math> at <math>T\!</math>, notated as follows:

Form the binary relation that is called the ''fiber'' of <math>\operatorname{Cond}</math> at <math>\operatorname{T}</math>, notated as follows:

: <math>\operatorname{Cond}^{-1}(T) \subseteq \mathbb{B} \times \mathbb{B}\,.\!</math>

: <math>\operatorname{Cond}^{-1}(\operatorname{T}) \subseteq \mathbb{B} \times \mathbb{B}.</math>

This object is defined as follows:

This object is defined as follows:

: <math>\operatorname{Cond}^{-1}(T) = \{ (p,\ q) \in \mathbb{B} \times \mathbb{B}\ :\ \operatorname{Cond} (p,\ q) = T \}\,.\!</math>

: <math>\operatorname{Cond}^{-1}(\operatorname{T}) = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : \operatorname{Cond} (p, q) = \operatorname{T} \}.</math>

The implication sign "<math>\Rightarrow\!</math>" denotes the same formal object as the relation names "<math>L_{..T}\mbox{ }\!</math>" and "<math>\operatorname{Cond}^{-1}(T)\mbox{ }\!</math>", the only differences being purely syntactic.  Thus we have the following logical equivalence:

The implication sign "<math>\Rightarrow\!</math>" denotes the same formal object as the relation names "<math>L_{..T}\mbox{ }\!</math>" and "<math>\operatorname{Cond}^{-1}(T)\mbox{ }\!</math>", the only differences being purely syntactic.  Thus we have the following logical equivalence: