Fact.mul_left_comm: 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 19-21 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=19&end_line=21

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

# (x a) f == a (x f)
def mul_left_comm(+x: Nat, +a: Nat, +f: Nat) -> {Nat.mul(Nat.mul(x, a), f) == Nat.mul(a, Nat.mul(x, f)) : Nat}:
  Equal.trans(Nat, Nat.mul(Nat.mul(x, a), f), Nat.mul(Nat.mul(a, x), f), Nat.mul(a, Nat.mul(x, f)), Equal.cong(Nat, Nat, z => Nat.mul(z, f), Nat.mul(x, a), Nat.mul(a, x), A.mul_comm(x, a)), A.mul_assoc(a, x, f))

Uses (6):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.cong
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.trans
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  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