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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/>







<a NAME="E1:13"/>
(  ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font> st <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="incproj.html#R4">on</a> <font color="Olive">P</font> iff  ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>given </b><font color="Maroon">P</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E1:13_1_1"/><i><font color="Green">E36</font></i>: 
<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/><b>take </b><font color="Maroon">P</font> = <font color="Maroon">P</font>;<br/><b>thus </b><a NAME="E2:13_1_1"/>
( <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/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>given </b><font color="Maroon">P</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E1:13_1_2"/><i><font color="Green">E37</font></i>: 
( <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> )
 ;<br/><b>take </b><font color="Maroon">P</font> = <font color="Maroon">P</font>;<br/><b>thus </b><a NAME="E2:13_1_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>
 <b>by </b><i>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:13"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="projpl_1.html#R4" target="_self">is_collinear</a>  iff  ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Olive">P</font> ) )
 <b>by </b><i/>;<br/>


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