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

<b>set </b><font color="Maroon">c<sub>3</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>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="vectmetr.html#K6" target="_self">SubIsomGroupRel</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1_1_1"/>
<font color="Maroon">c<sub>5</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#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</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> ;<br/><a NAME="E3:36_1_1_1"/>
 <a href="group_1.html#K2">1.</a> <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="hidden.html#R1">=</a>  <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>
 <b>by </b><i><a class="ref" href="vectmetr.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E4:36_1_1_1"/><b>then </b>
 <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="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T55">GROUP_2:55</a></i>;<br/><a NAME="E5:36_1_1_1"/><b>then </b><i><font color="Green">E16</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_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="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<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="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</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>6</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/><b>hence </b><a NAME="E7:36_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K6" target="_self">SubIsomGroupRel</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E5:36_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="vectmetr.html#D10" target="_self">Def10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:36_1_1"/><b>then </b>
 <a href="vectmetr.html#K6" target="_self">SubIsomGroupRel</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_2.html#R1">is_reflexive_in</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="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/>
<a NAME="E3:36_1_1"/><b>then </b><i><font color="Green">E16</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K6" target="_self">SubIsomGroupRel</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</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> &amp;  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K6" target="_self">SubIsomGroupRel</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</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="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<b>hence </b><a NAME="E4:36_1_1"/>
 <a href="vectmetr.html#K6" target="_self">SubIsomGroupRel</a> <font color="Maroon">c<sub>2</sub></font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:36_1_1_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</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#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:36_1_1_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</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#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:36_1_1_2"/><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K6" target="_self">SubIsomGroupRel</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</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#E1:36_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="vectmetr.html#E2:36_1_1_2"><i><font color="Green">E18</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:36_1_1_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E7:36_1_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <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#E3:36_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="vectmetr.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E8:36_1_1_2"/><i><font color="Green">E22</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</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>
 <b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/><a NAME="E9:36_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</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>
 <b>by </b><i><a class="txt" href="vectmetr.html#E6:36_1_1_2"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E10:36_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</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>10</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="E12:36_1_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E13:36_1_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>10</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>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="vectmetr.html#E6:36_1_1_2"><i><font color="Green">E20</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> 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="txt" href="vectmetr.html#E8:36_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><i><font color="Green">E25</font></i>:  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</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#E13:36_1_1_2"><i><font color="Green">E24</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="E17:36_1_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E13:36_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="vectmetr.html#T2" target="_self">Th2</a></i>;<br/><i><font color="Green">E27</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K3">"</a>  = 
<font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="ref" href="group_2.html#T57">GROUP_2:57</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E12:36_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="vectmetr.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K2">"</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E12:36_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="vectmetr.html#E16:36_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="vectmetr.html#E17:36_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
;<br/><a NAME="E19:36_1_1_2"/>
<font color="Maroon">c<sub>7</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#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E20:36_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E12:36_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E21:36_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E7:36_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="vectmetr.html#E12:36_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="vectmetr.html#E17:36_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/><b>hence </b><a NAME="E22:36_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K6" target="_self">SubIsomGroupRel</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E18:36_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="vectmetr.html#D10" target="_self">Def10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:36_1_1"/><b>then </b>
 <a href="vectmetr.html#K6" target="_self">SubIsomGroupRel</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_2.html#R3">is_symmetric_in</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="relat_2.html#D3">RELAT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:36_1_1"/>
 <a href="vectmetr.html#K6" target="_self">SubIsomGroupRel</a> <font color="Maroon">c<sub>2</sub></font> is <a href="relat_2.html#V3">symmetric</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E3:36_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:36_1_1_3"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</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#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:36_1_1_3"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</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#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:36_1_1_3"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>7</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#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:36_1_1_3"/><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K6" target="_self">SubIsomGroupRel</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:36_1_1_3"/><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K6" target="_self">SubIsomGroupRel</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</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#E1:36_1_1_3"><i><font color="Green">E17</font></i></a>, <a class="txt" href="vectmetr.html#E2:36_1_1_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="vectmetr.html#E3:36_1_1_3"><i><font color="Green">E19</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:36_1_1_3"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E9:36_1_1_3"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E4:36_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="vectmetr.html#D10" target="_self">Def10</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:36_1_1_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E12:36_1_1_3"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E5:36_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="vectmetr.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E13:36_1_1_3"/><i><font color="Green">E26</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</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>
 <b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/><a NAME="E14:36_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>11</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>
 <b>by </b><i><a class="txt" href="vectmetr.html#E8:36_1_1_3"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E15:36_1_1_3"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>11</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>13</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="E17:36_1_1_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E18:36_1_1_3"/>
<font color="Maroon">c<sub>13</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>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="vectmetr.html#E8:36_1_1_3"><i><font color="Green">E22</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> 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="txt" href="vectmetr.html#E13:36_1_1_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E21:36_1_1_3"/>
<font color="Maroon">c<sub>12</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>
 <b>by </b><i><a class="txt" href="vectmetr.html#E11:36_1_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="vectmetr.html#E13:36_1_1_3"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E22:36_1_1_3"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>12</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>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="vectmetr.html#E11:36_1_1_3"><i><font color="Green">E24</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> 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="txt" href="vectmetr.html#E13:36_1_1_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><i><font color="Green">E30</font></i>: <font color="Maroon">c<sub>17</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>15</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#K1">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</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>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="vectmetr.html#E15:36_1_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="vectmetr.html#E22:36_1_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self">Def9</a></i>
;<br/><a NAME="E26:36_1_1_3"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E27:36_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/><a NAME="E28:36_1_1_3"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E25:36_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E29:36_1_1_3"/>
<font color="Maroon">c<sub>8</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#K4" target="_self">RLMSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E30:36_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E17:36_1_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E31:36_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E9:36_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="vectmetr.html#E12:36_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E32:36_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K6" target="_self">SubIsomGroupRel</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E28:36_1_1_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="vectmetr.html#D10" target="_self">Def10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:36_1_1"/><b>then </b>
 <a href="vectmetr.html#K6" target="_self">SubIsomGroupRel</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_2.html#R8">is_transitive_in</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="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E10:36_1_1"/>
 <a href="vectmetr.html#K6" target="_self">SubIsomGroupRel</a> <font color="Maroon">c<sub>2</sub></font> is <a href="relat_2.html#V8">transitive</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E3:36_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>


</div>
