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<b>let </b><font color="Maroon">CPS</font> be  <a href="collsp.html#V4">proper</a> <a href="collsp.html#NM2">CollSp</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font>;<br/><b>let </b><font color="Maroon">b'</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font>;<br/><b>let </b><font color="Maroon">c'</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CPS</font>;<br/>













<b>assume </b><a NAME="E1:13"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font>
 ;<br/><b>set </b><font color="Maroon">X</font> =  <a href="collsp.html#K1">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>;<br/><a NAME="E2:13_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="collsp.html#K1">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="collsp.html#K1">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a>  <a href="collsp.html#K1">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> )
 <b>by </b><i>, <a class="ref" href="collsp.html#T16">COLLSP:16</a>, <a class="ref" href="collsp.html#T17">COLLSP:17</a></i>;<br/><b>reconsider </b><font color="Maroon">P'</font> =  <a href="collsp.html#K1">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> as    <a href="incproj.html#M1" target="_self">LINE</a> of <font color="Maroon">CPS</font> <b>by </b><i>, <a class="ref" href="collsp.html#D7">COLLSP:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P'</font> as   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E5:13_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>, , </i>;<br/><b>hence </b><a NAME="E6:13_1_1"/>
 ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span> 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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 ;<br/><a NAME="E2:13_1_2"/>
 ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span> st <br/>( <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/><b>hence </b><a NAME="E3:13_1_2"/>
 ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span> 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>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:13_1"/>
 ex <font color="Olive">P</font> being  <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span> 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/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">P</font> being   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">CPS</font></span>)</span><b> such that </b><a NAME="E3:13"/><i><font color="Green">E60</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>reconsider </b><font color="Maroon">P'</font> = <font color="Maroon">P</font> as    <a href="incproj.html#M1" target="_self">LINE</a> of <font color="Maroon">CPS</font> <b>by </b><i/>;<br/>
<a NAME="E5:13"/>
( <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P'</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P'</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P'</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:13"/>
<font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="collsp.html#T25">COLLSP:25</a></i>;<br/>


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