### 4.4.2. Functional dependency invariants involving three constraints

Proposition 188 Given the constraints

$\mathrm{\pi Έ}=\mathrm{\pi Ώ}+\mathrm{\pi  }$

Proposition 189 Given the constraints

$\mathrm{\pi ½\pi  \pi °\pi »}>1\beta \mathrm{\pi »\pi ΄\pi ½}_\mathrm{\pi ΅\pi Έ\pi \pi \pi }+\mathrm{\pi »\pi ΄\pi ½}_\mathrm{\pi »\pi °\pi \pi }+\mathrm{\pi ½\pi  \pi °\pi »}-2\beta €|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|$

Proof 187 Since we have at least two distinct values the first and last sequences do not overlap. Since we have at least $\mathrm{\pi ½\pi  \pi °\pi »}$ distinct values we have at least $\mathrm{\pi ½\pi  \pi °\pi »}-2$ additional values excluding the first and last sequences, which use at most two distinct values.

Proposition 190 Given the constraints

$\mathrm{\pi Ό\pi °\pi }-\mathrm{\pi Ό\pi Έ\pi ½}\beta ₯\mathrm{\pi »}$

Proposition 191 Given the constraints

$\mathrm{\pi Ό\pi °\pi }-\mathrm{\pi Ό\pi Έ\pi ½}\beta ₯\mathrm{\pi »}$

Proposition 192 Given the constraints

$\mathrm{\pi ½\pi  \pi °\pi »}\beta €\mathrm{\pi Ό\pi °\pi }-\mathrm{\pi Ό\pi Έ\pi ½}+1$

Proof 190 Since taking all values between $\mathrm{\pi Ό\pi Έ\pi ½}$ and $\mathrm{\pi Ό\pi °\pi }$ gives the maximum number of distinct values.

Proposition 193 Given the constraints

$\mathrm{\pi Ό\pi °\pi }>\mathrm{\pi Ό\pi Έ\pi ½}\beta \mathrm{\pi ½\pi  \pi °\pi »}>1$

Proof 191 Since at least two distinct values if $\mathrm{\pi Ό\pi Έ\pi ½}$ and $\mathrm{\pi Ό\pi °\pi }$ are distinct.

Proposition 194 Given the constraints

$\mathrm{\pi \pi \pi Ό}\beta ₯\left(|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|-1\right)Β·\mathrm{\pi Ό\pi Έ\pi ½}+\mathrm{\pi Ό\pi °\pi }$

Proof 192 Since has also to use the maximum value at least once in the sum.

Proposition 195 Given the constraints

$\mathrm{\pi \pi \pi Ό}\beta €\left(|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|-1\right)Β·\mathrm{\pi Ό\pi °\pi }+\mathrm{\pi Ό\pi Έ\pi ½}$

Proof 193 Since has also to use the minimum value at least once in the sum.