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The amount of oxygen is same. The partial pressure of oxygen is lower at higher altitude.

The reason why lungs must work harder is because oxygen is required in the body to supply hemoglobins in the blood. The affinity of hemoglobin to oxygen becomes lower as the partial pressure of oxygen decreases, creating a higher breathing rate in order to acquire more oxygen. Take an example of llama that lives up high in mountains. Their body is well adapted to it, such that their hemoglobin's affinity to oxygen is much higher than that of humans, and does not require extra work in order to acquire more oxygen.

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