Suppose that we have that Γ and "H" prove "C", and we wish to show that Γ proves "H"→"C". For each step "S" in the deduction which is a premise in Γ (a reiteration step) or an axiom, we can apply modus ponens to the axiom 1, "S"→("H"→"S"), to get "H"→"S". If the step is "H" itself (a hypothesis step), we apply the theorem schema to get "H"→"H". If the step is the result of applying modus ponens to "A" and "A"→"S", we first make sure that these have been converted to "H"→"A" and "H"→("A"→"S") and then we take the axiom 2, ("H"→("A"→"S"))→(("H"→"A")→("H"→"S")), and apply modus ponens to get ("H"→"A")→("H"→"S") and then again to get "H"→"S". At the end of the proof we will have "H"→"C" as required, except that now it only depends on Γ, not on "H". So the deduction step will disappear, consolidated into the previous step which was the conclusion derived from "H".