cast_card_sdiff — Mathlib

English

The cardinality of the union equals the sum of cardinalities minus the cardinality of the intersection, in a group with one.

Русский

Кардинал объединения равен сумме кардиналов минус кардинал пересечения.

LaTeX

$$$\\operatorname{card}(s \\cup t)^{R} = (#s^{R} + #t^{R}) - #(s \\cap t)^{R}$$$

Lean4

theorem cast_card_sdiff (h : s ⊆ t) : (#(t \ s) : R) = #t - #s := by
  rw [card_sdiff_of_subset h, Nat.cast_sub (card_mono h)]

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