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Re: "...what are its uses?" eqn = Sqrt[x + 4] == x - 2; soln = Solve[eqn, x, VerifySolutions -> False] (* {{x -> 0}, {x -> 5}} *) Results from Solve (also NSolve, DSolve, NDSolve, and others) are returned as Rules rather than setting the value of any variables. To verify solutions: eqn /. soln (* {False, True} *) The first solution was ...


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Your constructions were very instructive. Are you, to put your first question differently, asking why it is mod gets the downvalues in all three case; especially when examples 2 and 3 clearly show that Pattern is the head? This may not help but the documentation of the symbol ":" says that s_h is equivalent to s: _h--the name s is given to the pattern with ...



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