Fact.choose_mul_facts: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend as Fact
Lines 212-217 of bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend&start_line=212&end_line=217

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as A
import bend-collections-laws-math@1.0.0.0/spec/math/natural.bend as S

# C(n, k) (k! (n - k)!) == n!   (Mathlib Nat.choose_mul_factorial_mul_factorial)
def choose_mul_facts(+n: Nat, +k: Nat, +h: {Nat.is_le(k, n) == True{} : Bool}) -> {Nat.mul(S.choose(n, k), Nat.mul(S.factorial(k), S.factorial(Nat.sub(n, k)))) == S.factorial(n) : Nat}:
  %Equal.sym(Nat, Nat.mul(S.choose(n, k), Nat.mul(S.factorial(k), S.factorial(Nat.sub(n, k)))), Nat.mul(Nat.mul(S.choose(n, k), S.factorial(k)), S.factorial(Nat.sub(n, k))), Equal.sym(Nat, Nat.mul(Nat.mul(S.choose(n, k), S.factorial(k)), S.factorial(Nat.sub(n, k))), Nat.mul(S.choose(n, k), Nat.mul(S.factorial(k), S.factorial(Nat.sub(n, k)))), A.mul_assoc(S.choose(n, k), S.factorial(k), S.factorial(Nat.sub(n, k))))) : {_ == S.factorial(n) : Nat}
  %Equal.sym(Nat, Nat.mul(S.choose(n, k), S.factorial(k)), S.desc(n, k), choose_mul_factorial(n, k)) : {Nat.mul(_, S.factorial(Nat.sub(n, k))) == S.factorial(n) : Nat}
  %Equal.sym(Nat, Nat.mul(S.desc(n, k), S.factorial(Nat.sub(n, k))), Nat.mul(S.factorial(Nat.sub(n, k)), S.desc(n, k)), A.mul_comm(S.desc(n, k), S.factorial(Nat.sub(n, k)))) : {_ == S.factorial(n) : Nat}
  factorial_mul_desc(n, k, h)

Uses (13):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_le
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.sub
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_assoc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_comm
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend&name=choose_mul_factorial
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend&name=factorial_mul_desc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/natural.bend&name=choose
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/natural.bend&name=desc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/natural.bend&name=factorial