(see Definition of proportion for the story behind this proof)
A:B = C:D iff AD:BD = BC:BD iff AD = BC.
Q.E.D.
All the steps in this proof are propositions from Euclid. One of them is V.9. Its proof implicitly depends on Archimedes' Axiom which I see no reason to believe. If Archimedes' axiom is not true, and E is an infinitely small length, then A:B = A+E:B, but A != E. This leads to a huge discussion -- Rewriting Euclid book V (about proportion) with a non-Archimedean definition of proportion.