Proving the following theorem by using Armstrong's axioms

Question

Reflexivity: If Y ⊆ X then, X → Y . Such FDs are called trivial FDs(Functional dependencies).
Augmentation: If X → Y , then XZ → Y Z.
Transitivity: If X → Y and Y → Z, then X → Z.

Prove

Union: if X → Y and X → Z then X → Y Z.
Proof: Using Armstrong’s Axioms:
1. X → Y , Given
2. X → Z, Given
3. X → XZ, Augment 2 by X
4. XZ → Y Z, Augment 1 by Z
5. X → Y Z, Transitivity using 3 and 4

I just have a question about the third step. So it's saying that $XX=X$? or $X \implies XX$?. How do we go from $XX \implies XZ$ to $X \implies XZ$? because it doesn't agree with Amstrong's axioms.

Answer 1

Since X, Y and Z are sets of attributes, then XX (representing X U X) is X