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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  =  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">a<sub>3</sub></font>,<font color="Maroon">a<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">a<sub>5</sub></font>,<font color="Maroon">a<sub>6</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font></span><a href="zfmisc_1.html#K3">:]</a></span>,<span class="p2"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font></span><a href="zfmisc_1.html#K3">:]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:9_1_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">x1</font>, <font color="Olive">y1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><br/> for <font color="Olive">x2</font>, <font color="Olive">y2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font><br/> for <font color="Olive">x3</font>, <font color="Olive">y3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font><br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font></span><a href="zfmisc_1.html#K3">:]</a></span>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font>,<font color="Olive">x3</font></span><a href="mcart_1.html#K3">]</a></span> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Olive">y1</font>,<font color="Olive">y2</font>,<font color="Olive">y3</font></span><a href="mcart_1.html#K3">]</a></span> holds <br/><font color="Maroon">F</font> <a href="metric_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x1</font>,<font color="Olive">y1</font>,<font color="Olive">x2</font>,<font color="Olive">y2</font>,<font color="Olive">x3</font>,<font color="Olive">y3</font>)
 <b>from </b><i><a class="ref" href="metric_3.html#S2">METRIC_3:sch 2</a>();<br/></i>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>let </b><font color="Maroon">x1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">x2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>, <font color="Maroon">y2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">x3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font>, <font color="Maroon">y3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font></span><a href="zfmisc_1.html#K3">:]</a></span>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font></span><a href="zfmisc_1.html#K3">:]</a></span>;<br/>







<b>assume </b><a NAME="E3:9_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="mcart_1.html#K3">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font></span><a href="mcart_1.html#K3">]</a></span> )
 ;<br/>

<b>thus </b><a NAME="E4:9_1_1"/>
<font color="Maroon">F</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/>


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