Editing Lagrange's theorem (group theory) (section)

== Proof ==
The left [[coset]]s of {{mvar|H}} in {{mvar|G}} are the [[equivalence class]]es of a certain [[equivalence relation]] on {{mvar|G}}: specifically, call {{mvar|x}} and {{mvar|y}} in {{mvar|G}} equivalent if there exists {{mvar|h}} in {{mvar|H}} such that {{math|''x'' {{=}} ''yh''}}. 
Therefore, the set of left cosets forms a [[Partition of a set|partition]] of {{mvar|G}}.
Each left coset {{math|''aH''}} has the same cardinality as {{mvar|H}} because <math>x \mapsto ax</math> defines a bijection <math>H \to aH</math> (the inverse is <math>y \mapsto a^{-1}y</math>).
The number of left cosets is the [[index of a subgroup|index]] {{math|[''G'' : ''H'']}}.
By the previous three sentences, 
:<math>\left|G\right| = \left[G : H\right] \cdot \left|H\right|.</math>