from the last declaration in \Delta (which is p:'). (oe-E) In fact the ([\Theta]) is not exactly the ([\Theta]) that is found by induction. Possibly some of the free variables in ([\Theta]) are renamed. What happens is the following: 1. Consider the proof-context \Delta 1 ] \Delta 2 and especially the renaming of the declared variables in \Delta 2 that has been caused by the operation ]. 2. Rename the free proof-variables in ([\Theta]) accordingly, obtaining say, ([\Theta 0 ]). 3. Apply ([\Sigma]) to ([\Theta 0 ]). (There will in practice be no confusion if we just write ([\Theta]) instead.) Of course the intended meaning is that the judgement below the double lines is derivable if the judgement above the lines is. This will be proved later in Theorem 3.2.8. It should be clear at this point however that there is a one-to-one correspondence between the occurrences of ' as a (non-discharged) premise in the deduction and declarations p:' in \Delta. Notation. If for \Sigma a deducti...