Answer Happy

The following proof fits the logical definition. Analyze it tofind out what is really going on. Rewrite it in sensible style toreveal the structure of the argument.
If A, B, c are sets then (A∩B)∩ C = A∩(B∩C).
Proof:
L1: Let a ∈ (A ∩B)∩ C.
L2. a ∈ A ∩B
L3: Let b ∈ A ∩(B∩C).
L4: a ∈ C
L5: b∈ B∩C
L6: b ∈ B
L7: a ∈ B
L8: b ∈ C
L9: a, b ⊆ B
L10: b ∈ A
L11: a ∈ A
L12: b ∈ A ∩B
L13: a ∈ A ∩B
L14: {a, b) ⊆ A ∩B
L15: a∈ B∩C
L16: a ∈ A ∩(B∩C).
L17. (A ∩B) ∩ C ⊆ A ∩(B∩C)
L18: b ∈ (A ∩B) ∩ C
L19: (A ∩B) ∩ C ⊇ A ∩(B∩C)
L20: (A ∩B)∩ C = A ∩(B∩C).