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<div class="add">

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:53_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:53_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:53"><i><font color="Green">E36</font></i></a>, <a class="ref" href="dickson.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E4:53_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E2:53_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E5:53_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E2:53_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:53_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:53_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E5:53_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E6:53_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:53_1_1_1"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:53_1_1"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="dickson.html#E5:53_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:53_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:53_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:53_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>thus </b><a NAME="E5:53_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:53_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="dickson.html#E4:53_1_1_2"><i><font color="Green">E40</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E10:53_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E11:53_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="dickson.html#D9" target="_self">Def9</a></i>;<br/>
<b>hence </b><a NAME="E12:53_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E10:53_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>


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