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

<b>reconsider </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = 0 as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = 0;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:23_1_1"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> = <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:23_1_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> = <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:23_1_1"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>
<a NAME="E9:23_1_1"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>

<a NAME="E1:23_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>;<br/>
<a NAME="E2:23_1_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:23_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>) &amp; <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>) )
 
<b>by </b><i><a class="txt" href="vectmetr.html#E6:23_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="vectmetr.html#E2:23_1_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:23_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:23_1_1"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>

<a NAME="E1:23_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>;<br/>
<a NAME="E2:23_1_1_2"/><b>then </b><i><font color="Green">E16</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:23_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>),<font color="Maroon">c<sub>12</sub></font>) &amp; <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>)) )
 
<b>by </b><i><a class="txt" href="vectmetr.html#E6:23_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="vectmetr.html#E2:23_1_1_2"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:23_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:23_1_1"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)</span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)</span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)</span>)</span>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font>)
<b>by </b><i><a class="txt" href="vectmetr.html#E6:23_1_1"><i><font color="Green">E12</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E3:23_1_1_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:23_1_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)ex <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) st <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) ;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>

<br/>.= 
the <a href="struct_0.html#U2">Zero</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)
<b>by </b><i><a class="txt" href="vectmetr.html#E6:23_1_1"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E13:23_1_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>



<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E2:23_1_1_5"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font>))
<b>by </b><i><a class="txt" href="vectmetr.html#E6:23_1_1"><i><font color="Green">E12</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E4:23_1_1_5"/>
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E8:23_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="vectmetr.html#E2:23_1_1_5"><i><font color="Green">E19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:23_1_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>



<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<i><font color="Green">E20</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font>)
<b>by </b><i><a class="txt" href="vectmetr.html#E8:23_1_1"><i><font color="Green">E13</font></i></a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font>))
<b>by </b><i><a class="txt" href="vectmetr.html#E6:23_1_1"><i><font color="Green">E12</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E4:23_1_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E2:23_1_1_6"><i><font color="Green">E20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:23_1_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>



<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<i><font color="Green">E21</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font>)
<b>by </b><i><a class="txt" href="vectmetr.html#E8:23_1_1"><i><font color="Green">E13</font></i></a></i>
;<br/>
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font>))
<b>by </b><i><a class="txt" href="vectmetr.html#E8:23_1_1"><i><font color="Green">E13</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E4:23_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E2:23_1_1_7"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:23_1_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) holds 1 <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>)
<b>by </b><i><a class="txt" href="vectmetr.html#E8:23_1_1"><i><font color="Green">E13</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E4:23_1_1_8"/>
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:23_1_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) holds <br/>(  <a href="metric_1.html#K2">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; (  <a href="metric_1.html#K2">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp;  <a href="metric_1.html#K2">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="metric_1.html#K2">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>

<a NAME="E1:23_1_1_9"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><a NAME="E2:23_1_1_9"/><i><font color="Green">E24</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_9_1"><b>proof </b></a><div class="add">

<b>thus </b> <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="vectmetr.html#E4:23_1_1"><i><font color="Green">E11</font></i></a></i>
;<br/>


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

<b>thus </b><a NAME="E3:23_1_1_9"/>
(  <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0 implies <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="vectmetr.html#E1:23_1_1_9"><i><font color="Green">E23</font></i></a></i>;<br/>

<b>thus </b><a NAME="E4:23_1_1_9"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E1:23_1_1_9"><i><font color="Green">E23</font></i></a></i>;<br/>

<b>thus </b><a NAME="E5:23_1_1_9"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E1:23_1_1_9"><i><font color="Green">E23</font></i></a>, <a class="txt" href="vectmetr.html#E2:23_1_1_9"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:23_1_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) holds  <a href="metric_1.html#K2">dist</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>



<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>thus </b> <a href="metric_1.html#K2">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 0
<b>by </b><i><a class="txt" href="vectmetr.html#E4:23_1_1"><i><font color="Green">E11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:23_1_1"><i><font color="Green">E11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E19:23_1_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) holds  <a href="metric_1.html#K2">dist</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>

<b>thus </b> <a href="metric_1.html#K2">dist</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="vectmetr.html#E4:23_1_1"><i><font color="Green">E11</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:23_1_1"><i><font color="Green">E11</font></i></a></i>
<br/>.= 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<b>take </b>
 <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)
;<br/>

<b>thus </b><a NAME="E20:23_1_1"/>
 <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="vectmetr.html#V4" target="_self">strict</a>
 ;<br/>

<b>thus </b><a NAME="E21:23_1_1"/>
 <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="rlvect_1.html#V3">Abelian</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E9:23_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E22:23_1_1"/>
 <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="rlvect_1.html#V4">add-associative</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E10:23_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>

<b>thus </b><a NAME="E23:23_1_1"/>
 <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="rlvect_1.html#V5">right_zeroed</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E11:23_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>;<br/>

<b>thus </b><a NAME="E24:23_1_1"/>
 <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="rlvect_1.html#V6">right_complementable</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E12:23_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a></i>;<br/>

<b>thus </b><a NAME="E25:23_1_1"/>
 <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="rlvect_1.html#V7">RealLinearSpace-like</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E13:23_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="vectmetr.html#E14:23_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="vectmetr.html#E15:23_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="vectmetr.html#E16:23_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/>

<b>thus </b><a NAME="E26:23_1_1"/>
 <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="metric_1.html#V6">Reflexive</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E17:23_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="metric_1.html#T1">METRIC_1:1</a></i>;<br/>

<b>thus </b><a NAME="E27:23_1_1"/>
 <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="metric_1.html#V7">discerning</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E17:23_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="metric_1.html#T2">METRIC_1:2</a></i>;<br/>

<b>thus </b><a NAME="E28:23_1_1"/>
 <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="metric_1.html#V8">symmetric</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E17:23_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="metric_1.html#T3">METRIC_1:3</a></i>;<br/>

<b>thus </b><a NAME="E29:23_1_1"/>
 <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="metric_1.html#V9">triangle</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E17:23_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>

<b>thus </b><a NAME="E30:23_1_1"/>
 <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="vectmetr.html#V5" target="_self">homogeneous</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E18:23_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="vectmetr.html#D5" target="_self">Def5</a></i>;<br/>

<b>thus </b><a NAME="E31:23_1_1"/>
 <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="vectmetr.html#V6" target="_self">translatible</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E19:23_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="vectmetr.html#D6" target="_self">Def6</a></i>;<br/>


</div>
