This aphorism speaks of "playing the game" - in all its forms - "The Game Of Life."  Lines 1-2 - refers to "playing the side" - which may appear unusual - since it is the game - that is being played.  However, it is a direct reference to the team - who play the game.

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Therefore, the aphorism places the emphasis on "teamwork" and that "success" in - "playing the game" - often require the individual, successfully playing his or her own side - coalitions within the team - making alliances within the team - understanding the team environment, in as much as the player must understand the "game" - "environment" - "the field" - "the board" - "the ring" (one thinks of tag wrestling!) 

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Therefore, the "complete player" is not only proficient in "playing" - "The Game" - but also proficient in "playing" - "his or her side."