Let H be a group acting on a set A. Prove that the relation ~ on A defined by a~b iff $\displaystyle a=hb$ for some h in H is an equivalence relation.

I have already shown this.

Let H be a subgroup of the finite group G and let H act on G by left multiplication. Let x exist in G and let O be the orbit of x under the action of H. Prove that the map $\displaystyle H\to O$ defined by $\displaystyle h\mapsto hx$ is a bijection.

I have already shown this too.

From these two statements, deduce Lagrange's Theorem: if G is a finite group and $\displaystyle H\leq G$, then $\displaystyle |H|| |G|$.

I understand Lagrange's Theorem. I need an explanation on how I can deduce his theorem from the above.