# [isabelle] Showing an "existence" axiom is satisfied.

```[A] I'm trying to show a certain space of functions forms a vector space;
one of the necessary conditions is the existence of an additive inverse.

lemma (in vectorspace) coeff_space_is_vs:
fixes S
assumes h1: "finite S" and h2: "S⊆carrier V"
shows "vectorspace K (coeff_space S)"
proof -
have 0: "vectorspace K V"..
from 0 h1 h2 show ?thesis
apply (auto intro!: vs_criteria simp add: coeff_space_def)
apply (auto simp add: vectorspace_def)

The first goal is now
1. (!!v∷'c ⇒ 'a.
finite S ⟹
S ⊆ carrier V ⟹
v ∈ S →⇩E carrier K ⟹
module K V ⟹
field K ⟹
∃*neg_v∷'c ⇒ 'a*∈S →⇩E carrier K.
(λva∷'c∈S. v va ⊕⇘K⇙ neg_v va) = (λv∷'c∈S. 𝟬⇘K⇙)

The important part of this is the neg_v. I want to substitute a value to
*neg_v* so I try (tutorial p. 85)

*apply (rule_tac x="restrict (λv'. ⊖⇘K⇙ (v v')) S" in exI)*
(i.e. the additive inverse of a function just has every term negated) but
this doesn't work. What's the right way to write this?

Thanks,

Holden

```

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