Prove the highlighted "If and Only If" using structural induction on t

THEOREM (SEQUENCING IS A DERIVED FORM): Write A (E for external lan- guage) for the simply typed lambda-calculus with the Unit type, the se- quencing construct, and the rules E-SEQ, E-SEQNEXT, and T-SEO, and A' ("I" for internal language) for the simply typed lambda-calculus with Unit only. Let e ele - ll be the elaboration function that translates from the ex- ternal to the internal language by replacing every occurrence of t;t2 with (Ax:Unit.tz) t, where x is chosen fresh in each case. Now, for each term t of AF, we have .t-Et' iff e(t) - e(t') .I Et: Tiffre(t): T where the evaluation and typing relations of AF and I are annotated with E and I, respectively, to show which is which. THEOREM (SEQUENCING IS A DERIVED FORM): Write A (E for external lan- guage) for the simply typed lambda-calculus with the Unit type, the se- quencing construct, and the rules E-SEQ, E-SEQNEXT, and T-SEO, and A' ("I" for internal language) for the simply typed lambda-calculus with Unit only. Let e ele - ll be the elaboration function that translates from the ex- ternal to the internal language by replacing every occurrence of t;t2 with (Ax:Unit.tz) t, where x is chosen fresh in each case. Now, for each term t of AF, we have .t-Et' iff e(t) - e(t') .I Et: Tiffre(t): T where the evaluation and typing relations of AF and I are annotated with E and I, respectively, to show which is which