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<div class="add">

<b>let </b><font color="Maroon">A</font> be  <a href="metric_1.html#V1">strict</a>  <a href="metric_1.html#L1">MetrStruct</a> , <font color="Maroon">B</font> be  <a href="metric_1.html#V1">strict</a>  <a href="metric_1.html#L1">MetrStruct</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:19_1_2"/><i><font color="Green">E49</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E2:19_1_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">A</font>  ex <font color="Olive">x1</font>, <font color="Olive">y1</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> ex <font color="Olive">x2</font>, <font color="Olive">y2</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y1</font>,<font color="Olive">y2</font></span><a href="domain_1.html#K1">]</a></span> &amp; the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">A</font> <a href="metric_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1">.</a> <font color="Olive">x1</font>,<font color="Olive">y1</font></span>)</span>,<span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">N</font> <a href="metric_1.html#K1">.</a> <font color="Olive">x2</font>,<font color="Olive">y2</font></span>)</span> )
 <b>and </b><br/><a NAME="E3:19_1_2"/><i><font color="Green">E52</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E4:19_1_2"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">B</font>  ex <font color="Olive">x1</font>, <font color="Olive">y1</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> ex <font color="Olive">x2</font>, <font color="Olive">y2</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y1</font>,<font color="Olive">y2</font></span><a href="domain_1.html#K1">]</a></span> &amp; the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">B</font> <a href="metric_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1">.</a> <font color="Olive">x1</font>,<font color="Olive">y1</font></span>)</span>,<span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">N</font> <a href="metric_1.html#K1">.</a> <font color="Olive">x2</font>,<font color="Olive">y2</font></span>)</span> )
 ;<br/>

<b>set </b><font color="Maroon">f</font> = the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">g</font> = the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">B</font>;<br/>
<a NAME="E5:19_1_2"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> holds <br/>the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">B</font> <a href="binop_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:19_1_2_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Maroon">y1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font>, <font color="Maroon">y2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:19_1_2_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span> )
 <b>and </b><br/><a NAME="E4:19_1_2_1"/><i><font color="Green">E63</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span>,<span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">N</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">m1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Maroon">n1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Maroon">m2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font>, <font color="Maroon">n2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E6:19_1_2_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m1</font>,<font color="Maroon">m2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">n1</font>,<font color="Maroon">n2</font></span><a href="domain_1.html#K1">]</a></span> )
 <b>and </b><br/><a NAME="E7:19_1_2_1"/><i><font color="Green">E65</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">B</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">m1</font>,<font color="Maroon">n1</font></span>)</span>,<span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">N</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">m2</font>,<font color="Maroon">n2</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/>
<a NAME="E8:19_1_2_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m2</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>thus </b><a NAME="E9:19_1_2_1"/>
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">B</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:19_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i>, , <a class="ref" href="binop_1.html#T1">BINOP_1:1</a></i>;<br/>


</div>
