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Idea

Also converse is true:

• For each a*b, left hand side of only one equation "matches":
  • Either b = 0 or b = y' for some y.
  • Consequence of first Peano axiom.
• Equality b = y' determines unique y.
  • (b = y₀' /\  b = y₁') => y₀ = y₁ for all y₀ and y₁.
  • Consequence of second Peano axiom.

Pair of equations uniquely determines a function.