Explanation: 1.3: Subtracting Equations Preserves Equality
(related to Subsection: Common Notions (all Books))
(Common Notion 3 from Book 1 of Euclid's “Elements”)
And if equal things are subtracted from equal things then the remainders are equal.
Modern Formulation
Since $xa=xb$ is equivalent to $x+(a)=x+(b)$, this is only a special case of the previous lemma. In Ancient mathematics, subtraction was considered as a separate operation rather than a special case of addition.
Mentioned in:
Proofs: 1 2 3 4
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"