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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/><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#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:10"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V4">associative</a>
 ;<br/>

<b>thus </b><a NAME="E2:10"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">c<sub>4</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="10_1"><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="E1:10_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then 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="E3:10_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>5</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>
 <b>and </b><br/><a NAME="E4:10_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:10_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<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="E7:10_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>6</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>
 <b>and </b><br/><a NAME="E8:10_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E9:10_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="group_2.html#E4:10_1"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E10:10_1"/>
( <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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="group_2.html#E1:10"><i><font color="Green">E7</font></i></a>, <a class="txt" href="group_2.html#E3:10_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="group_2.html#E5:10_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="group_2.html#E7:10_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="group_2.html#E9:10_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E11:10_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_2.html#E8:10_1"><i><font color="Green">E12</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="E3:10"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then 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="E5:10"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>5</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>
 <b>and </b><br/><a NAME="E6:10"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E7:10"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>
 ;<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="E9:10"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>7</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>
 <b>and </b><br/><a NAME="E10:10"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E11:10"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="group_2.html#E7:10"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E12:10"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <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>8</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="group_2.html#E1:10"><i><font color="Green">E7</font></i></a>, <a class="txt" href="group_2.html#E5:10"><i><font color="Green">E8</font></i></a>, <a class="txt" href="group_2.html#E6:10"><i><font color="Green">E9</font></i></a>, <a class="txt" href="group_2.html#E9:10"><i><font color="Green">E11</font></i></a>, <a class="txt" href="group_2.html#E10:10"><i><font color="Green">E12</font></i></a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E13:10"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="group_2.html#E11:10"><i><font color="Green">E13</font></i></a></i>;<br/>


</div>
