The following proof fits the logical definition. Analyze it to find out what is really going on. Rewrite it in sensible

[answerhappygod]

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).