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<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b3">r</font> <a href="tarski.html#R2" title="TARSKI: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">are_collinear</a> </span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c12">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c13">r</font> <a href="tarski.html#R2" title="TARSKI: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">are_collinear</a> </span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">P</font> &amp; <font color="Maroon" title="c12">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">P</font> &amp; <font color="Maroon" title="c13">r</font> <a href="tarski.html#R2" title="TARSKI: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">are_collinear</a>  )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
( <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">P</font> &amp; <font color="Maroon" title="c12">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">P</font> )
 <span class="kw">and </span><br/><a NAME="E2:31"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c13">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">P</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">are_collinear</a> </span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:31"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:31"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="collsp.html#D7" target="_self" onmouseover="rs('collsp/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E6:31"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
 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">are_collinear</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">A4</font></span></a></span>;<br/>
<a NAME="E7:31"/>
(  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">are_collinear</a>  ) &amp;  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">are_collinear</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:31"/>
<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">are_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="collsp.html#T3" target="_self" onmouseover="rs('collsp/T3')" onmouseout="rh()">Th3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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