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<b>let </b><font color="Maroon">G</font> be    <a href="incsp_1.html#L1">IncProjStr</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">Q</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font>;<br/>













<b>thus </b><a NAME="E1:5"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="incproj.html#R3">on</a> <font color="Maroon">P</font> implies <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="incproj.html#R3">on</a> <font color="Maroon">P</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="incproj.html#R3">on</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:5_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="incproj.html#D7">INCPROJ:def 7</a></i>;<br/>
<b>hence </b><a NAME="E3:5_1"/>
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="incproj.html#R3">on</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="incproj.html#D7">INCPROJ:def 7</a></i>;<br/>


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

<b>thus </b><a NAME="E2:5"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font> implies ( <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_2"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:5_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<b>hence </b><a NAME="E3:5_2"/>
( <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="incproj.html#R4">on</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>


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

<b>thus </b><a NAME="E3:5"/>
( <font color="Maroon">a</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font> implies <font color="Maroon">a</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">Q</font>,<font color="Maroon">P</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_3"/>
<font color="Maroon">a</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font>
 ;<br/>

<a NAME="E2:5_3"/><b>then </b>
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:5_3"/>
<font color="Maroon">a</font> <a href="projpl_1.html#R2" target="_self">on</a> <font color="Maroon">Q</font>,<font color="Maroon">P</font>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E4:5"/>
<font color="Maroon">a</font> <a href="projpl_1.html#R3" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font>,<font color="Maroon">R</font>
 ;<br/>

<a NAME="E5:5"/><b>then </b>
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">P</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">R</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:5"/>
( <font color="Maroon">a</font> <a href="projpl_1.html#R3" target="_self">on</a> <font color="Maroon">P</font>,<font color="Maroon">R</font>,<font color="Maroon">Q</font> &amp; <font color="Maroon">a</font> <a href="projpl_1.html#R3" target="_self">on</a> <font color="Maroon">Q</font>,<font color="Maroon">P</font>,<font color="Maroon">R</font> &amp; <font color="Maroon">a</font> <a href="projpl_1.html#R3" target="_self">on</a> <font color="Maroon">Q</font>,<font color="Maroon">R</font>,<font color="Maroon">P</font> &amp; <font color="Maroon">a</font> <a href="projpl_1.html#R3" target="_self">on</a> <font color="Maroon">R</font>,<font color="Maroon">P</font>,<font color="Maroon">Q</font> &amp; <font color="Maroon">a</font> <a href="projpl_1.html#R3" target="_self">on</a> <font color="Maroon">R</font>,<font color="Maroon">Q</font>,<font color="Maroon">P</font> )
 <b>by </b><i/>;<br/>


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