## Proof That Homotopy Mappings of Homotopic Functions Are Homotopic

Theorem

Ifare homotopic mappings taking values in the spaceandare defined on and homotopic, then the mappingsandare homotopic.

Proof

Letso thatis homotopic to

A continuous functionfor allexists such that

Letso thatis homotopic to

A continuous functionfor allexists such that

Define

Thenand

are continuous, nH is continuous sois a homotopy andandare homotopic.