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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><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="E1:58"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#R4" target="_self">//</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:58"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <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="aff_1.html#R4" target="_self">//</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</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="E1:58_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#R4" target="_self">//</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E2:58_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i><a class="ref" href="aff_1.html#T50" target="_self">Th50</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:58_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:58_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="aff_1.html#E1:58_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="aff_1.html#E2:58_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="aff_1.html#E1:58_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="aff_1.html#E37"><i><font color="Green">Lemma25</font></i></a></i>;<br/><a NAME="E3:58_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#R2" target="_self">//</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_1.html#E1:58_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="aff_1.html#T43" target="_self">Th43</a></i>;<br/><a NAME="E4:58_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#R2" target="_self">//</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="aff_1.html#E1:58_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="aff_1.html#T57" target="_self">Th57</a></i>;<br/><b>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="E6:58_1_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</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="aff_1.html#E2:58_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="aff_1.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E7:58_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="aff_1.html#E1:58_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="aff_1.html#E2:58_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="aff_1.html#E6:58_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="aff_1.html#E37"><i><font color="Green">Lemma25</font></i></a></i>;<br/><a NAME="E8:58_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="aff_1.html#E1:58_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="aff_1.html#E2:58_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="aff_1.html#E4:58_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="aff_1.html#E6:58_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="aff_1.html#T41" target="_self">Th41</a></i>;<br/><a NAME="E9:58_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E10:58_1_1_1"/><b>then </b>
 <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E11:58_1_1_1"/>
<font color="Maroon">c<sub>8</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="aff_1.html#E7:58_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="aff_1.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:58_1_1"/>
<font color="Maroon">c<sub>8</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="aff_1.html#E1:58_1"><i><font color="Green">E52</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:58_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:58"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_1.html#E1:58"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:58"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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