<?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>, <font color="Maroon">c<sub>3</sub></font> be    <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>thus </b><a NAME="E1:78"/>
( <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> implies ex <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:78_1"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:78_1"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="group_2.html#T13" target="_self">Th13</a></i>;<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:78_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:78_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</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="E4:78_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="group_2.html#E1:78_1_1"><i><font color="Green">E57</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</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="E6:78_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="group_2.html#E2:78_1_1"><i><font color="Green">E58</font></i></a></i>;<br/><i><font color="Green">E61</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="group_2.html#E4:78_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="group_2.html#E6:78_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/><a NAME="E8:78_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_2.html#E1:78_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="group_2.html#E4:78_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="group_2.html#E6:78_1_1"><i><font color="Green">E60</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</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="E10:78_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="group_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> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><i><font color="Green">E63</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="group_2.html#E7:78_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="group_2.html#E10:78_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/><a NAME="E12:78_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="group_2.html#E4:78_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="group_2.html#E6:78_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="group_2.html#E10:78_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="group_2.html#T89" target="_self">Th89</a></i>;<br/><b>hence </b><a NAME="E13:78_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_2.html#E11:78_1_1"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:78_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</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="E3:78_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>4</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>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</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="E5:78_1_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="group_2.html#E1:78_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="group_2.html#E1:78_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="group_2.html#E3:78_1_2"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E6:78_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="group_2.html#E3:78_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="group_2.html#E5:78_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="group_1.html#T25">GROUP_1:25</a></i>;<br/><a NAME="E7:78_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="group_2.html#E5:78_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="group_2.html#T90" target="_self">Th90</a></i>;<br/><b>hence </b><a NAME="E8:78_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_2.html#E6:78_1_2"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:78_1"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_2.html#E2:78_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="group_2.html#E3:78_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="group_2.html#T61" target="_self">Th61</a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>4</sub></font> being    <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E2:78"/><i><font color="Green">E54</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E3:78"/>
<span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="E1:78_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</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 <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:78_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="group_2.html#E2:78"><i><font color="Green">E54</font></i></a>, <a class="txt" href="group_2.html#E1:78_2"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E4:78_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#T90" target="_self">Th90</a></i>;<br/>
<a NAME="E5:78_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_2.html#E2:78"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</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="E7:78_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<i><font color="Green">E57</font></i>: <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="group_2.html#E7:78_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="group_1.html#T25">GROUP_1:25</a></i>
;<br/>
<a NAME="E9:78_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="group_2.html#E7:78_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="group_2.html#T90" target="_self">Th90</a></i>;<br/>
<b>hence </b><a NAME="E10:78_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_2.html#E8:78_2"><i><font color="Green">E57</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>5</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="E4:78"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</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 <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</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="E7:78"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="group_2.html#E4:78"><i><font color="Green">E55</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:78_3"/>
not <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:78_3"/><b>then </b><i><font color="Green">E57</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E3:78_3"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="group_2.html#E7:78"><i><font color="Green">E56</font></i></a>, <a class="ref" href="group_2.html#T90" target="_self">Th90</a>, <a class="ref" href="group_1.html#T25">GROUP_1:25</a></i>;<br/><b>hence </b><a NAME="E4:78_3"/>
contradiction
 <b>by </b><i><a class="txt" href="group_2.html#E2:78"><i><font color="Green">E54</font></i></a>, <a class="txt" href="group_2.html#E2:78_3"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:78"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#T90" target="_self">Th90</a></i>;<br/>
<a NAME="E10:78"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>;<br/>
<b>hence </b><a NAME="E11:78"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_2.html#E2:78"><i><font color="Green">E54</font></i></a></i>;<br/>


</div>
