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<b>let </b><font color="Maroon" title="c1">CPS</font> be  <a href="collsp.html#V4" title="COLLSP:attr.4">proper</a> <a href="collsp.html#NM2" title="COLLSP:NM.2">CollSp</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not for <font color="Olive" title="b1">p</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span><br/> for <font color="Olive" title="b2">P</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span> holds  <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">P</font></span><br/>

<b>consider </b><font color="Maroon" title="c2">p'</font>, <font color="Maroon" title="c3">q'</font>, <font color="Maroon" title="c4">r'</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CPS</font><b> such that </b><br/><a NAME="E2:14"/><i><font color="Green" title="E10">A1</font></i>: 
not <font color="Maroon" title="c2">p'</font>,<font color="Maroon" title="c3">q'</font>,<font color="Maroon" title="c4">r'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="collsp.html#D6" title="COLLSP:def.6">COLLSP:def 6</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">X</font> =  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c2">p'</font>,<font color="Maroon" title="c3">q'</font>;<br/>
<a NAME="E3:14"/>
<font color="Maroon" title="c2">p'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q'</font>
 
<b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="collsp.html#T7" title="COLLSP:th.7">COLLSP:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">P'</font> =  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c2">p'</font>,<font color="Maroon" title="c3">q'</font> as    <a href="incproj.html#M1" title="INCPROJ:mode.1">LINE</a> of <font color="Maroon" title="c1">CPS</font> <b>by </b><i><a class="ref" href="collsp.html#D7" title="COLLSP:def.7">COLLSP:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">P</font> = <font color="Maroon" title="c6">P'</font> as   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span> <b>by </b><i><a class="ref" href="incproj.html#T2" target="_self" title="INCPROJ:th.2">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c4">r'</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span> ;<br/>
<a NAME="E7:14"/>
not <font color="Maroon" title="c8">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon" title="c8">r</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_1"/><b>then </b>
<font color="Maroon" title="c4">r'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c2">p'</font>,<font color="Maroon" title="c3">q'</font>
 
<b>by </b><i><a class="ref" href="incproj.html#T9" target="_self" title="INCPROJ:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E3:14_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="collsp.html#T17" title="COLLSP:th.17">COLLSP:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:14"/>
not for <font color="Olive" title="b1">p</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span><br/> for <font color="Olive" title="b2">P</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span> holds  <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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