Combination Theorem for Continuous Mappings/Topological Ring/Negation Rule

Theorem
Let $\struct {S, \tau_{_S} }$ be a topological space.

Let $\struct {R, +, *, \tau_{_R} }$ be a topological ring.

Let $g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ be a continuous mappings.

Let $-g : S \to R$ be the mapping defined by:
 * $\forall x \in S: \map {\paren {-g} } x = -\map g x$

Then
 * $-g : \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.

Proof
By definition of a topological ring:
 * $\struct {R, +, \tau_{_R} }$ is a topological group.

From Inverse Rule for Continuous Mappings to Topological Group:
 * $-g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.

Also see

 * Inverse Rule for Continuous Mappings to Topological Group