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<b>let </b><font color="Maroon" title="c10">CLSP</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"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">CLSP</font><br/> for <font color="Olive" title="b4">P</font> being   <a href="collsp.html#M2" title="COLLSP:mode.2">LINE</a> of <font color="Maroon" title="c10">CLSP</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">P</font> holds <br/><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><b>let </b><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">r</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">CLSP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being   <a href="collsp.html#M2" title="COLLSP:mode.2">LINE</a> of <font color="Maroon" title="c10">CLSP</font>  st <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c13">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> holds <br/><font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><b>let </b><font color="Maroon" title="c14">P</font> be    <a href="collsp.html#M2" title="COLLSP:mode.2">LINE</a> of <font color="Maroon" title="c10">CLSP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">P</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">P</font> &amp; <font color="Maroon" title="c13">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">P</font> implies <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/>









<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">P</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">P</font> &amp; <font color="Maroon" title="c13">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/>

<b>consider </b><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">CLSP</font><b> such that </b><br/><a NAME="E3:33"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">b</font>
 <b>and </b><br/><a NAME="E4:33"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c14">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>
 <b>by </b><i><a class="ref" href="collsp.html#D7" target="_self" title="COLLSP:def.7">Def7</a></i>;<br/>
<a NAME="E5:33"/><i><font color="Green" title="E29">A4</font></i>: 
 ex <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">CLSP</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>,<font color="Olive" title="b1">x</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E4:33"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>
<a NAME="E6:33"/><i><font color="Green" title="E30">A5</font></i>: 
 ex <font color="Olive" title="b1">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">CLSP</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q</font> &amp; <font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>,<font color="Olive" title="b1">y</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E4:33"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>
<a NAME="E7:33"/>
 ex <font color="Olive" title="b1">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">CLSP</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r</font> &amp; <font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>,<font color="Olive" title="b1">z</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E4:33"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:33"/>
<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="#E3:33"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="#E5:33"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="#E6:33"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="collsp.html#T8" target="_self" title="COLLSP:th.8">Th8</a></i>; <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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