Peano built all of arithmetic from almost nothing: a zero, a successor (“the next one”), and induction (true for 0, and if true for n then true for n+1, so true for all n). Slide n and watch induction stack the triangular number 0+1+…+n into the closed form n(n+1)/2 — proved, not asserted.
Addition and multiplication are defined by recursion on the successor: a+0=a, a+S(b)=S(a+b); a·0=0, a·S(b)=a·b+a. Those two clauses, plus induction, are enough. The instrument runs the recursive add and mul live and a fail-loud self-check throws unless they match ordinary arithmetic for all a,b in 0..11 AND the inductive step S(n)=S(n−1)+n reproduces n(n+1)/2 for every n up to 50 — the sum is grown one successor at a time, never plugged into the formula.
The visual triangle is a picture of the theorem, not the proof; the proof is the induction the self-check performs. Peano’s axioms (1889) also assume 0 is not a successor and successor is injective — structural facts we take as given here.