### 4.4.2. Functional dependency invariants involving three constraints

Proposition 188 Given the constraints

$𝙸=𝙿+𝚅$

Proof 186

Proposition 189 Given the constraints

$\mathrm{𝙽𝚅𝙰𝙻}>1⇒\mathrm{𝙻𝙴𝙽}_\mathrm{𝙵𝙸𝚁𝚂𝚃}+\mathrm{𝙻𝙴𝙽}_\mathrm{𝙻𝙰𝚂𝚃}+\mathrm{𝙽𝚅𝙰𝙻}-2\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$

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

Proposition 190 Given the constraints

$\mathrm{𝙼𝙰𝚇}-\mathrm{𝙼𝙸𝙽}\ge 𝙻$

Proposition 191 Given the constraints

$\mathrm{𝙼𝙰𝚇}-\mathrm{𝙼𝙸𝙽}\ge 𝙻$

Proposition 192 Given the constraints

$\mathrm{𝙽𝚅𝙰𝙻}\le \mathrm{𝙼𝙰𝚇}-\mathrm{𝙼𝙸𝙽}+1$

Proof 190 Since taking all values between $\mathrm{𝙼𝙸𝙽}$ and $\mathrm{𝙼𝙰𝚇}$ gives the maximum number of distinct values.

Proposition 193 Given the constraints

$\mathrm{𝙼𝙰𝚇}>\mathrm{𝙼𝙸𝙽}⇒\mathrm{𝙽𝚅𝙰𝙻}>1$

Proof 191 Since at least two distinct values if $\mathrm{𝙼𝙸𝙽}$ and $\mathrm{𝙼𝙰𝚇}$ are distinct.

Proposition 194 Given the constraints

$\mathrm{𝚂𝚄𝙼}\ge \left(|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|-1\right)·\mathrm{𝙼𝙸𝙽}+\mathrm{𝙼𝙰𝚇}$

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

Proposition 195 Given the constraints

$\mathrm{𝚂𝚄𝙼}\le \left(|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|-1\right)·\mathrm{𝙼𝙰𝚇}+\mathrm{𝙼𝙸𝙽}$

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