<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K3">:]</a></span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="mcart_1.html#K5">`1</a> , <font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="mcart_1.html#K6">`2</a> , <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="mcart_1.html#K7">`3</a> , <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K5">`1</a> , <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K6">`2</a> , <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K7">`3</a>  as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E2:34"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="mcart_1.html#K3">]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="mcart_1.html#K3">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T48">MCART_1:48</a></i>;<br/>
<b>thus </b><a NAME="E3:34"/>
( <a href="metric_4.html#K13" target="_self">Eukl_dist3</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1"/><i><font color="Green">E29</font></i>: 
<a href="metric_4.html#K13" target="_self">Eukl_dist3</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>set </b><font color="Maroon">c<sub>9</sub></font> = <a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E2:34_1"/>
 <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#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</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#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</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#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="metric_4.html#E2:34"><i><font color="Green">E28</font></i></a>, <a class="txt" href="metric_4.html#E1:34_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="metric_4.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E3:34_1"/><b>then </b><i><font color="Green">E30</font></i>: 
 <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"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</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#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E4:34_1"/><i><font color="Green">E31</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E5:34_1"/><i><font color="Green">E32</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E6:34_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E7:34_1"/><b>then </b><i><font color="Green">E33</font></i>: 
0 <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="metric_4.html#E5:34_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E8:34_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="metric_4.html#E3:34_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="metric_4.html#E4:34_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="metric_4.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E9:34_1"/><b>then </b>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E10:34_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T9">METRIC_1:9</a></i>;<br/>
<a NAME="E11:34_1"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="metric_4.html#E3:34_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="metric_4.html#E4:34_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="metric_4.html#E7:34_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="metric_4.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E12:34_1"/><i><font color="Green">E36</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E13:34_1"/><i><font color="Green">E37</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E14:34_1"/><b>then </b><i><font color="Green">E38</font></i>: 
( <span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="metric_4.html#E11:34_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="metric_4.html#E12:34_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xreal_1.html#T29">XREAL_1:29</a></i>;<br/>
<a NAME="E15:34_1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="metric_4.html#E11:34_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="metric_4.html#E12:34_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="metric_4.html#E13:34_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xreal_1.html#T29">XREAL_1:29</a></i>;<br/>
<a NAME="E16:34_1"/><b>then </b>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E17:34_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T9">METRIC_1:9</a></i>;<br/>
<a NAME="E18:34_1"/>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>5</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="metric_4.html#E14:34_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<b>hence </b><a NAME="E19:34_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="metric_4.html#E2:34"><i><font color="Green">E28</font></i></a>, <a class="txt" href="metric_4.html#E10:34_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="metric_4.html#E17:34_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="metric_1.html#T9">METRIC_1:9</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E4:34"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E5:34"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T9">METRIC_1:9</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/>
<a NAME="E6:34"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="metric_4.html#E4:34"><i><font color="Green">E29</font></i></a>, <a class="ref" href="metric_1.html#T9">METRIC_1:9</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/>
<a href="metric_4.html#K13" target="_self">Eukl_dist3</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
 <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#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</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#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</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#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="metric_4.html#E2:34"><i><font color="Green">E28</font></i></a>, <a class="ref" href="metric_4.html#D13" target="_self">Def13</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="metric_4.html#E4:34"><i><font color="Green">E29</font></i></a>, <a class="txt" href="metric_4.html#E5:34"><i><font color="Green">E30</font></i></a>, <a class="txt" href="metric_4.html#E6:34"><i><font color="Green">E31</font></i></a>, <a class="ref" href="metric_1.html#T9">METRIC_1:9</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a>, <a class="ref" href="square_1.html#T82">SQUARE_1:82</a></i>
;<br/>
<b>hence </b><a NAME="E8:34"/>
<a href="metric_4.html#K13" target="_self">Eukl_dist3</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div>
