As promised, in this section I show how to derive Rules 4-7 from the original Rules 1-3.
X → X (“self determination”).
Proof: Immediate by reflexivity.
If X → Y and X → Z, then X → YZ (“union”).
Proof: X → Y (given), hence X → XY by augmentation; also X → Z (given), hence XY → YZ by augmentation; hence X → YZ by transitivity.
If X → Y and Z → W, then XZ → YW (“composition”).
Proof: X → Y (given), hence XZ → YZ by augmentation; likewise, Z → W (given), hence YZ → YW by augmentation; hence XZ → YW by transitivity.
If X → YZ, then X → Y and X → Z (“decomposition”).
Proof: X → YZ (given) and YZ → Y by reflexivity; hence X → Y by transitivity (and likewise for X → Z).
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