A short proof of Fisher's inequality (Q686507)

We prove the following result: Theorem. Let \(E=\{a_ 1,\dots,a_ m\}\) be a finite set of cardinality \(m\) and let \(E_ 1,\dots,E_ n\) be \(n\) subsets of \(E\) such that \[ | E_ i\cap E_ j|= \lambda\qquad\text{for } i\neq j\qquad\text{and } | E_ i|>\lambda\qquad\text{for all } i. \] Then \(m\geq n\).