proposition: Lemma bin_fact2n_e n : 0<n -> ('C(n.*2, n) - 'C(n.*2, n.-1)) * n.+1 = 'C(n.*2, n).
proof: Proof. move => h. move/eqP:(f_equal2 subn (bin_fact2n_c n) (bin_fact2n_b h)). rewrite -mulnBr -mulnBl subSnn muln1 - (bin_fact2n_a n) factS (mulnCA n`!). by rewrite mulnn mulnA eqn_mul2r expn_eq0 eqn0Ngt fact_gt0 /= => /eqP. Qed.
imports: mathcomp(ssreflect ssrfun ssrbool eqtype) mathcomp(ssrnat seq path div) mathcomp(fintype tuple finfun bigop finset binomial)