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Example

x/y := such q: q*y = x.

• 12/4 = 3 because
  • Existence: exists q: q*4 = 12 (because 3*4 = 12);
  • Unicity: forall q: q*4 = 12 => q = 3
• We must not assume that (1/0)*0 = 1, because we cannot prove

  exists q: q*0 = 1.

Take care when reasoning about implicitly defined functions.