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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="group_3.html#V1">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>thus </b><a NAME="E1:12"/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> is  <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> as    <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="group_2.html#T106">GROUP_2:106</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>thus </b><a NAME="E1:12_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:12_1_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:12_1_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T125">GROUP_2:125</a></i>;<br/>
<a NAME="E5:12_1_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="group_6.html#E4:12_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="group_2.html#T99">GROUP_2:99</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/>
<a NAME="E7:12_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="group_6.html#E3:12_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/>
<a NAME="E8:12_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="group_6.html#E5:12_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="group_2.html#T125">GROUP_2:125</a>, <a class="ref" href="group_3.html#T141">GROUP_3:141</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:12_1_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E11:12_1_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T126">GROUP_2:126</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</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="group_6.html#E3:12_1_1_1"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="group_6.html#E10:12_1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E14:12_1_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a>  )
 
<b>by </b><i><a class="txt" href="group_6.html#E10:12_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="group_1.html#T22">GROUP_1:22</a>, <a class="ref" href="group_2.html#T57">GROUP_2:57</a></i>;<br/>
<a NAME="E15:12_1_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/>
<a NAME="E16:12_1_1_1"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>10</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/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</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="group_6.html#E15:12_1_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E18:12_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="group_6.html#E10:12_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="group_6.html#E11:12_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="group_6.html#E16:12_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="group_2.html#T52">GROUP_2:52</a>, <a class="ref" href="group_2.html#T99">GROUP_2:99</a></i>;<br/>
<b>hence </b><a NAME="E19:12_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T126">GROUP_2:126</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:12_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> is  <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="group_3.html#T141">GROUP_3:141</a></i>;<br/>


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

<b>hence </b><a NAME="E2:12"/>
<font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> is  <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div>
