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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D4">METRIC_1:def 4</a>,<a class="ref" href="metric_1.html#D8">METRIC_1:def 8</a><br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green">E18</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:27"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="topreal7.html#K2" target="_self">]</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="topreal7.html#K2" target="_self">]</a></span> )
 <b>and </b><br/><a NAME="E4:27"/><i><font color="Green">E20</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal7.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:27"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>
<a NAME="E6:27"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 
;<br/>
<a NAME="E7:27"/><b>then </b><i><font color="Green">E21</font></i>: 
( the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="topreal7.html#E1:27"><i><font color="Green">E18</font></i></a>, <a class="txt" href="topreal7.html#E4:27"><i><font color="Green">E20</font></i></a>, <a class="ref" href="topreal7.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E8:27"/>
( the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V3">discerning</a> &amp; the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V3">discerning</a> )
 
<b>by </b><i><a class="ref" href="metric_1.html#D8">METRIC_1:def 8</a></i>;<br/>
<a NAME="E9:27"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="topreal7.html#E7:27"><i><font color="Green">E21</font></i></a>, <a class="ref" href="metric_1.html#D4">METRIC_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E10:27"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="topreal7.html#E3:27"><i><font color="Green">E19</font></i></a></i>;<br/>


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