Proving Theorem 51.3 in Munkres 2e Edition on Homotopy Paths

[valtz]

i stuck when i want to prove theorem 51.3 in munkres 2en editions about homotopy paths

Let f be a path in X , and let a₀ , ... , a_n be numbers such that 0= a₀ < a₁ < ... < a_n. Let f_i : I → X be the path that equals the positive linear map of I onto [a_i-1, a_i] followed by f then

[f] = [f₁] * ... * [f_n]


any idea to start prove this theorems?

 

[mathwonk]

draw a picture.