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<b>let </b><font color="Maroon">S</font> be    <a href="incsp_1.html#L2" target="_self">IncStruct</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">L</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="incsp_1.html#U1" target="_self">Points</a> of <font color="Maroon">S</font>;<br/>







<b>thus </b><a NAME="E1:21"/>
( <font color="Maroon">F</font> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L</font> &amp; <font color="Maroon">A</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font> implies <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">F</font> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L</font> &amp; <font color="Maroon">A</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font> )
 ;<br/>

<b>let </b><font color="Maroon">C</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="incsp_1.html#D4" target="_self">INCSP_1:def 4</a><br/>

<b>assume </b><a NAME="E2:21_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E3:21_1"/><b>then </b>
( <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> or <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:21_1"/>
<font color="Maroon">C</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D2" target="_self">Def2</a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>assume </b><a NAME="E2:21"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E3:21"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E4:21"/>
<font color="Maroon">F</font> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D2" target="_self">Def2</a>, </i>;<br/>

<a NAME="E5:21"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E6:21"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/>
<a NAME="E7:21"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#D2" target="_self">Def2</a>, </i>;<br/>
<b>hence </b><a NAME="E8:21"/>
<font color="Maroon">A</font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


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