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A comparison of the corresponding projections in <math>\operatorname{Proj}^{(2)} L(\text{A})\!</math> and <math>\operatorname{Proj}^{(2)} L(\text{B})\!</math> shows that the distinction between the triadic relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> is preserved by <math>\operatorname{Proj}^{(2)},\!</math> and this circumstance allows one to say that this much information, at least, can be derived from the dyadic projections. However, to say that a triadic relation <math>L \in \operatorname{Pow} (O \times S \times I)\!</math> is reducible in this sense it is necessary to show that no distinct <math>L' \in \operatorname{Pow} (O \times S \times I)\!</math> exists such that <math>\operatorname{Proj}^{(2)} L = \operatorname{Proj}^{(2)} L',\!</math> and this can take a rather more exhaustive or comprehensive investigation of the space <math>\operatorname{Pow} (O \times S \times I).\!</math>
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As it happens, each of the relations R C {A, B} is uniquely determined by its projective triple Proj (R). This can be seen as follows. Consider any coordinate position <s, i> in the plane SxI. If <s, i> is not in RSI then there can be no element <o, s, i> in R, therefore we may restrict our attention to positions <s, i> in RSI, knowing that there exist at least |RSI| = 8 elements in R, and seeking only to determine what objects o exist such that <o, s, i> is an element in the objective "fiber" of <s, i>. In other words, for what o C O is <o, s, i> C PrSI 1(<s, i>)? The fact that ROS has exactly one element <o, s> for each coordinate s C S and that ROI has exactly one element <o, i> for each coordinate i C I, plus the "coincidence" of it being the same o at any one choice for <s, i>, tells us that R has just the one element <o, s, i> over each point of SxI. This proves that both A and B are reducible in an informational sense to triples of dyadic relations, that is, they are "dyadically reducible".
As it happens, each of the relations R C {A, B} is uniquely determined by its projective triple Proj (R). This can be seen as follows. Consider any coordinate position <s, i> in the plane SxI. If <s, i> is not in RSI then there can be no element <o, s, i> in R, therefore we may restrict our attention to positions <s, i> in RSI, knowing that there exist at least |RSI| = 8 elements in R, and seeking only to determine what objects o exist such that <o, s, i> is an element in the objective "fiber" of <s, i>. In other words, for what o C O is <o, s, i> C PrSI 1(<s, i>)? The fact that ROS has exactly one element <o, s> for each coordinate s C S and that ROI has exactly one element <o, i> for each coordinate i C I, plus the "coincidence" of it being the same o at any one choice for <s, i>, tells us that R has just the one element <o, s, i> over each point of SxI. This proves that both A and B are reducible in an informational sense to triples of dyadic relations, that is, they are "dyadically reducible".
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