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The real projective plane can be viewed as a a square [0,1]x[0,1] with the edges identified as in the diagram above. Let b be our base point. The green loop going across the square from the 'b' in the lower left to the 'b' in the top right (which are identified) cannot be homotoped relative to its endpoints to the trivial loop. However if we compose it with the blue loop then the end of the green line no longer has to stick to 'b' during a homotopy relative to endpoints so we can 'detach' it as in the right figure and then this clearly homotopes to the trivial loop.

To clarify, the blue loop (although it doesn't look like it in my picture) is the same loop as the green one, if you can imagine drawing the green one, then rotating yourself to the opposite side of the square, then drawing the same loop.