apply funext
case cons
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
a b : R
h : R × X
t : List (R × X)
ih : scmul a t ++ scmul b t ≈ scmul (a + b) t
⊢ scmul a (h :: t) ++ scmul b (h :: t) ≈ scmul (a + b) (h :: t)
case cons.h
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
a b : R
h : R × X
t : List (R × X)
ih : scmul a t ++ scmul b t ≈ scmul (a + b) t
⊢ ∀ (x : X), coords (scmul a (h :: t) ++ scmul b (h :: t)) x = coords (scmul (a + b) (h :: t)) x