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<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" target="_self">HGrStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">b<sub>3</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 <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/>

<b>thus </b><a NAME="E3:10"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_1.html#E1:10"><i><font color="Green">E5</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" target="_self">GROUP_1:def 4</a><br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <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>3</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="E6:10"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</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="E8:10"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</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="E10:10"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E11:10"/><i><font color="Green">E10</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green">E6</font></i></a></i>;<br/>
<i><font color="Green">E11</font></i>: <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E1:10"><i><font color="Green">E5</font></i></a>, <a class="txt" href="group_1.html#E8:10"><i><font color="Green">E8</font></i></a>, <a class="txt" href="group_1.html#E10:10"><i><font color="Green">E9</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="group_1.html#E1:10"><i><font color="Green">E5</font></i></a>, <a class="txt" href="group_1.html#E6:10"><i><font color="Green">E7</font></i></a>, <a class="txt" href="group_1.html#E10:10"><i><font color="Green">E9</font></i></a>, <a class="txt" href="group_1.html#E11:10"><i><font color="Green">E10</font></i></a></i>
;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3" target="_self">GROUP_1:def 3</a><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E13:10"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:10"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="group_1.html#E1:10"><i><font color="Green">E5</font></i></a>, <a class="txt" href="group_1.html#E6:10"><i><font color="Green">E7</font></i></a></i>;<br/>

<a NAME="E15:10"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:10"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="group_1.html#E1:10"><i><font color="Green">E5</font></i></a>, <a class="txt" href="group_1.html#E8:10"><i><font color="Green">E8</font></i></a>, <a class="txt" href="group_1.html#E12:10"><i><font color="Green">E11</font></i></a></i>;<br/>

<b>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>1</sub></font><b> such that </b><br/><a NAME="E18:10"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>consider </b><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="E20:10"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>consider </b><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="E22:10"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E23:10"/><i><font color="Green">E17</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="group_1.html#E2:10"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E1:10"><i><font color="Green">E5</font></i></a>, <a class="txt" href="group_1.html#E16:10"><i><font color="Green">E13</font></i></a>, <a class="txt" href="group_1.html#E22:10"><i><font color="Green">E16</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E1:10"><i><font color="Green">E5</font></i></a>, <a class="txt" href="group_1.html#E20:10"><i><font color="Green">E15</font></i></a>, <a class="txt" href="group_1.html#E22:10"><i><font color="Green">E16</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="group_1.html#E1:10"><i><font color="Green">E5</font></i></a>, <a class="txt" href="group_1.html#E14:10"><i><font color="Green">E12</font></i></a>, <a class="txt" href="group_1.html#E18:10"><i><font color="Green">E14</font></i></a>, <a class="txt" href="group_1.html#E23:10"><i><font color="Green">E17</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E25:10"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="group_1.html#E18:10"><i><font color="Green">E14</font></i></a>, <a class="txt" href="group_1.html#E20:10"><i><font color="Green">E15</font></i></a></i>;<br/>


</div>
