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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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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="b3">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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">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="b3">P</font> holds <br/> <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</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:()"><span class="comment"><font color="Red">::  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="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q</font> &amp; <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> holds <br/> <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c13">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">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="c13">P</font> implies  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">P</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q</font>
 <span class="kw">and </span><br/><a NAME="E2:45"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">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="c13">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">P</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">x</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="c11">p</font>,<font color="Maroon" title="c12">q</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font> )</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="E4:45"/>
<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="E5:45"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c13">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" title="COLLSP:def.7">Def7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:45"/>
<font color="Maroon" title="c14">x</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="c11">p</font>,<font color="Maroon" title="c12">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">r</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="E8:45"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">and </span><br/><a NAME="E9:45"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c17">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ;<br/>
<a NAME="E10:45"/>
( <font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c11">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c12">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="collsp.html#T17" target="_self" title="COLLSP:th.17">Th17</a></span>;<br/>
<a NAME="E11:45"/><span class="kw">then </span>
<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c17">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E9:45"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="collsp.html#T14" target="_self" title="COLLSP:th.14">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:45"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E32">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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