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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <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="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><a NAME="E1:32_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E2:32_1_1_1"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="vectmetr.html#D9" target="_self">Def9</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:32_1_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:32_1_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="vectmetr.html#V3" target="_self">isometric</a>
 <b>by </b><i><a class="ref" href="vectmetr.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E6:32_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E7:32_1_1_1"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="vectmetr.html#D9" target="_self">Def9</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E9:32_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E10:32_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="vectmetr.html#V3" target="_self">isometric</a>
 <b>by </b><i><a class="ref" href="vectmetr.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E11:32_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E12:32_1_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="vectmetr.html#D9" target="_self">Def9</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E14:32_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E15:32_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="vectmetr.html#V3" target="_self">isometric</a>
 <b>by </b><i><a class="ref" href="vectmetr.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E16:32_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> is <a href="vectmetr.html#V3" target="_self">isometric</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E5:32_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="vectmetr.html#E10:32_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="vectmetr.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E17:32_1_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="vectmetr.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E18:32_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font> is <a href="vectmetr.html#V3" target="_self">isometric</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E10:32_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="vectmetr.html#E15:32_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="vectmetr.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E19:32_1_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="vectmetr.html#D4" target="_self">Def4</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <span class="p2">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:32_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="vectmetr.html#E4:32_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="vectmetr.html#E7:32_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="vectmetr.html#E9:32_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="vectmetr.html#E12:32_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="vectmetr.html#E14:32_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="vectmetr.html#E17:32_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self">Def9</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:32_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="vectmetr.html#E4:32_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="vectmetr.html#E19:32_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self">Def9</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <span class="p2">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E7:32_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="vectmetr.html#E9:32_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="vectmetr.html#E12:32_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="vectmetr.html#E14:32_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self">Def9</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:32_1_1"/>
 <a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V4">associative</a>
 <b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/>

<a NAME="E3:32_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</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="32_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_1_1_2"/>
 <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="vectmetr.html#V3" target="_self">isometric</a>
 
<b>by </b><i><a class="ref" href="vectmetr.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E2:32_1_1_2"/><b>then </b><i><font color="Green">E13</font></i>: 
 <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectmetr.html#D4" target="_self">Def4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="vectmetr.html#D9" target="_self">Def9</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<a NAME="E4:32_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
;<br/>
<a NAME="E5:32_1_1_2"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectmetr.html#D9" target="_self">Def9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E7:32_1_1_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E8:32_1_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="vectmetr.html#V3" target="_self">isometric</a>
 <b>by </b><i><a class="ref" href="vectmetr.html#D4" target="_self">Def4</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:32_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="vectmetr.html#E5:32_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="vectmetr.html#E7:32_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self">Def9</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="vectmetr.html#E7:32_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:32_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="vectmetr.html#E5:32_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="vectmetr.html#E7:32_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="vectmetr.html#E7:32_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
;<br/>

<a NAME="E11:32_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="vectmetr.html#V3" target="_self">isometric</a>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E8:32_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="vectmetr.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E12:32_1_1_2"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectmetr.html#D4" target="_self">Def4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="vectmetr.html#D9" target="_self">Def9</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<i><font color="Green">E18</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<i><font color="Green">E19</font></i>:  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E8:32_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="vectmetr.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E16:32_1_1_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E8:32_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="vectmetr.html#T2" target="_self">Th2</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E5:32_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="vectmetr.html#E7:32_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="vectmetr.html#E12:32_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self">Def9</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E15:32_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="vectmetr.html#E16:32_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E14:32_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="vectmetr.html#E15:32_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="vectmetr.html#E5:32_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="vectmetr.html#E7:32_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="vectmetr.html#E12:32_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self">Def9</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E15:32_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="vectmetr.html#E16:32_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_1_1"/>
 <a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V3">Group-like</a>
 <b>by </b><i><a class="ref" href="group_1.html#D3">GROUP_1:def 3</a></i>;<br/>


</div>
