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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> 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#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Q</font> holds <br/><font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Q</font></span><br/><b>let </b><font color="Maroon" title="c11">P</font>, <font color="Maroon" title="c12">Q</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q</font> implies <font color="Maroon" title="c11">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Q</font> )</span><br/>





<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c11">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q</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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Q</font></span><br/>

<a NAME="E2:35"/>
<font color="Maroon" title="c12">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">r</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font> or <font color="Maroon" title="c13">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">P</font> )</span><br/>

<b>assume </b><a NAME="E1:35_1"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c13">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">P</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">r</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">CLSP</font> <b>by </b><i><a class="txt" href="collsp.html#E47"><i><font color="Green" title="E26">Lm9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">p</font>, <font color="Maroon" title="c16">q</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="E4:35_1"/>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">q</font>
 <b>and </b><br/><a NAME="E5:35_1"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c11">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">p</font>,<font color="Maroon" title="c16">q</font>
 <b>by </b><i><a class="ref" href="collsp.html#D7" target="_self" title="COLLSP:def.7">Def7</a></i>;<br/>
<a NAME="E6:35_1"/>
( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">P</font> &amp; <font color="Maroon" title="c16">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">P</font> )
 
<b>by </b><i><a class="txt" href="collsp.html#E5:35_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="collsp.html#T16" target="_self" title="COLLSP:th.16">Th16</a></i>;<br/>
<a NAME="E7:35_1"/><b>then </b>
<font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c16">q</font>,<font color="Maroon" title="c14">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="collsp.html#E1:35"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="collsp.html#E1:35_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="collsp.html#T25" target="_self" title="COLLSP:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E8:35_1"/>
<font color="Maroon" title="c13">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">P</font>
 <b>by </b><i><a class="txt" href="collsp.html#E5:35_1"><i><font color="Green" title="E30">A3</font></i></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="E3:35"/>
<font color="Maroon" title="c11">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="txt" href="collsp.html#E1:35"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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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