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<b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#V9">triangle</a>  <a href="metric_1.html#L1">MetrStruct</a> , <font color="Maroon">N</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#V9">triangle</a>  <a href="metric_1.html#L1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D6">METRIC_1:def 6</a>,<a class="ref" href="metric_1.html#D10">METRIC_1:def 10</a><br/>



<b>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="E2:30"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="topreal7.html#K2" target="_self">]</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="topreal7.html#K2" target="_self">]</a></span> )
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green">E47</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span> <a href="metric_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="E5:30"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">m1</font>,<font color="Maroon">m2</font></span><a href="topreal7.html#K2" target="_self">]</a></span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">n1</font>,<font color="Maroon">n2</font></span><a href="topreal7.html#K2" target="_self">]</a></span> )
 <b>and </b><br/><a NAME="E6:30"/><i><font color="Green">E49</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</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/>;<br/>
<b>consider </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Maroon">q1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font>, <font color="Maroon">q2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E8:30"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="topreal7.html#K2" target="_self">]</a></span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="topreal7.html#K2" target="_self">]</a></span> )
 <b>and </b><br/><a NAME="E9:30"/><i><font color="Green">E52</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</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">p1</font>,<font color="Maroon">q1</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">p2</font>,<font color="Maroon">q2</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E10:30"/>
( the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> is <a href="metric_1.html#V5">triangle</a> &amp; the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">N</font> is <a href="metric_1.html#V5">triangle</a> )
 
<b>by </b><i><a class="ref" href="metric_1.html#D10">METRIC_1:def 10</a></i>;<br/>
<a NAME="E11:30"/><b>then </b><i><font color="Green">E60</font></i>: 
( 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">p1</font>,<font color="Maroon">q1</font> <a href="xxreal_0.html#R1">&lt;=</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">p1</font>,<font color="Maroon">y1</font></span>)</span> <a href="real_1.html#K3">+</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">y1</font>,<font color="Maroon">q1</font></span>)</span> &amp; 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">p2</font>,<font color="Maroon">q2</font> <a href="xxreal_0.html#R1">&lt;=</a> <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">p2</font>,<font color="Maroon">y2</font></span>)</span> <a href="real_1.html#K3">+</a> <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">y2</font>,<font color="Maroon">q2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="metric_1.html#D6">METRIC_1:def 6</a></i>;<br/>
<a NAME="E12:30"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m2</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">q2</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>hence </b><a NAME="E13:30"/>
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <span class="p2">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <span class="p2">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, , , , </i>;<br/>


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