cast_card_erase_of_mem — Mathlib

English

If a ∈ s, then (#(s.erase a) : R) = #s - 1 in any additive group with one.

Русский

Если a ∈ s, то (#(s.erase a) : R) = #s - 1 в любом/additive group with one.

LaTeX

$$$\\mathrm{cast}\\,(#(s.erase a)) = #s - 1$$$

Lean4

/-- $\#(s \setminus \{a\}) = \#s - 1$ if $a \in s$.
  This result is casted to any additive group with 1,
  so that we don't have to work with `ℕ`-subtraction. -/
-- @[simp] -- removed because LHS is not in simp normal form

theorem cast_card_erase_of_mem (hs : a ∈ s) : (#(s.erase a) : R) = #s - 1 := by
  rw [← card_erase_add_one hs, cast_add, cast_one, eq_sub_iff_add_eq]

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