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Recall that $d_H(A,B) = max{max_{a in A} min_{b in B} d(a,b),max_{b in B} min_{a in A} d(a,b)}$

Theorem: Let $A,B,C in H(X)$ (where $H(X)$ is the set of non-empty compact subsets of $X$). Then $d_H(A cup B,C) = max {d_H(A,C),d_H(B,C)}$

This claim appears in the book on fractals and splines by Peter Massopust. However, I'm not able to connect the hint below with the statement of the theorem:

Proof: Note that
begin{align}
d(Acup B, C) &= max_{alphain Acup B} d(alpha, C) = maxleft{ max_{alphain A} d(alpha,C), max_{alphain B} d(alpha,C)right} \
&= max{ d(A,C), d(B,C)}
end{align}
which implies that claim. (The reader is encouraged to verify this).

It seems that the definition of $d$ is (see the book Fractals everywhere) taken to be $d(A,B) = max{d(a,B):a in A}$. So the above hint makes sense.

So how can I use this hint to prove my theorem?

My try:

$d_H(A cup B,C) = max {d(A cup B,C) , d(C, A cup B) }$

by definition. Then, by the hint:

$d_H(A cup B,C) = max {max{d(A cup B,C), d(C,A cup B)} , d(C, A cup B) }$

then, I take the other member:

$max{d_H(A,C),d_H(B,C)} = max {max{d(A,C), d(C,A)} , {max{d(B,C), d(C,B)} }$

again by definition. Two terms are in the two sides, but I would need an equality of the sort $d(C,A cup B) = max {d(C,A),d(C,B)}$. This reduces to show:

$max_{c in C}{min_{a in A cup B} d(c,a)} = max{max_{c in C}{min_{a in A} d(c,a),max_{c in C}{min_{a in B} d(c,a)}}$