I think you may have had some trouble when you did your latex, because I can't understand a word of what you wrote. My specialty is epsilon-delta proofs so I would love to help. Let me know when you fix the glitch.
The epsilon delta rule states
EijkEpqk=(delta(ip)delta(jq)-delta(iq)delta(jp))
I am constantly using this but get stuck when it is applied.
For example
EijkEpqkA(j)B(l)C(m)=(delta(ip)delta(jq)-delta(iq)delta(jp))A(j)B(l)C(m)
This then becomes
A(j)B(i)C(j)-A(j)B(j)C(i)
Can anybody please explain this result????