How do I solve part d only using propositional calculus rules of inf.pdf

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(1) The document provides instructions to solve part (d) of a propositional logic problem set using only propositional calculus rules of inference and defined macros, without quantifiers or proof techniques like proof by contradiction. (2) It defines macros for Contrapositive, De Morgan's laws, and absorption based on formal proofs from lecture. (3) Part (d) of the problem asks to find a formal proof or counterexample for the assertion LM(MN)(LK)P(QL)PK using the given rules and macros.

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How do I solve part d only using propositional calculus rules of influence (aka not using rules of
inference for quantifiers and not using proofs like proof by contradiction) Note: In lecture we
formally proved ContrapositiveQPPQPQQP De Morgan's laws
(PQ)(PR)P(QR)P(QR)(PQ)(PR)(PQ)(PR)P(QR)P(QR)(PQ)(PR) These are not strictly speaking
part of our formal proof system, but since we already worked out formal proofs for them, and
since we could simply cut-and-paste those proofs wherever needed, you may use these results as
"macros" in your solutions to the following problems. (1) For each of the following assertions, if
it is valid, find a formal proof, in the system discussed in lecture-to emphasize, you can only use
the rules of inference given in lecture and the macros aboveand if it is not valid, find a
counterexample (i.e. true/false assignments to the variables which make the assumptions true but
the conclusion false). (a) begin{tabular}{l} PQ (PQ)R hlineR end{tabular} (b) (c)
begin{tabular}{|l|} hlineXY XZ hlineZY hline hlineEF GF HI EH hlineGI
hline end{tabular} (d) LM(MN)(LK)P(QL)PK

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