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Let f be a path in X , and let a

_{0}, ... , a

_{n}be numbers such that 0= a

_{0}< a

_{1}< ... < 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] = [f

_{1}] * ... * [f

_{n}]

any idea to start prove this theorems?