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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</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">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">p3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font> holds <br/> ( <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Olive" title="b3">p3</font> <a href="rltopsp1.html#R1" title="RLTOPSP1:pred.1">are_collinear</a>  iff ( <font color="Olive" title="b1">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Olive" title="b2">p2</font>,<font color="Olive" title="b3">p3</font>) or <font color="Olive" title="b2">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Olive" title="b3">p3</font>,<font color="Olive" title="b1">p1</font>) or <font color="Olive" title="b3">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">p3</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</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="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font> <a href="rltopsp1.html#R1" title="RLTOPSP1:pred.1">are_collinear</a>  iff ( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) or <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) or <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) ) )</span><br/>



<span class="kw">thus </span><a NAME="E1:82"/>
(  not <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font> <a href="rltopsp1.html#R1" title="RLTOPSP1:pred.1">are_collinear</a>  or <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) or <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) or <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</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="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) or <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) or <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) ) implies <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font> <a href="rltopsp1.html#R1" title="RLTOPSP1:pred.1">are_collinear</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="82_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:82_1"/>
<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font> <a href="rltopsp1.html#R1" title="RLTOPSP1:pred.1">are_collinear</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"> ( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) or <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) or <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">L</font> being   <a href="rltopsp1.html#NM3" title="RLTOPSP1:NM.3">line</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E3:82_1"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">L</font>
 <span class="kw">and </span><br/><a NAME="E4:82_1"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">L</font>
 <span class="kw">and </span><br/><a NAME="E5:82_1"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">L</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E7:82_1"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#D15" onmouseover="rs('rltopsp1/D15')" onmouseout="rh()">RLTOPSP1:def 15</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:82_1_1"/>
( <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p3</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p3</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:82_1_1_1"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p3</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="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) or <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) or <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:82_1_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) or <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) or <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T68" onmouseover="rs('rltopsp1/T68')" onmouseout="rh()">RLTOPSP1:68</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:82_1_1_2"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p3</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="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) or <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) or <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) )</span><br/><div class="add"><a NAME="E2:82_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A5</font></span>: 
 <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E5:82_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="rltopsp1.html#T75" onmouseover="rs('rltopsp1/T75')" onmouseout="rh()">RLTOPSP1:75</a>, <a class="txt" href="#E7:82_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:82_1_1_2_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) or  not <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:82_1_1_2_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</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="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) or <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) or <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:82_1_1_2_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) or <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) or <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T66" target="_self" onmouseover="rs('topreal9/T66')" onmouseout="rh()">Th64</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:82_1_1_2_1_2"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
 not <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</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="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) or <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) or <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) )</span><br/><div class="add"><a NAME="E2:82_1_1_2_1_2"/>
<font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T65" target="_self" onmouseover="rs('topreal9/T65')" onmouseout="rh()">Th63</a>, <a class="txt" href="#E2:82_1_1_2"><span class="lab"><font color="Green" title="E43">A5</font></span></a></span>;<br/><a NAME="E3:82_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="txt" href="#E1:82_1_1_2_1_2"><span class="lab"><font color="Green" title="E44">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:82_1_1_2_1_2"/>
( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) or <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) or <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T66" target="_self" onmouseover="rs('topreal9/T66')" onmouseout="rh()">Th64</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:82"/>
( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) or <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) or <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</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="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font> <a href="rltopsp1.html#R1" title="RLTOPSP1:pred.1">are_collinear</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:82_2"/><span class="kw">then </span>
( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) or <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) or <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_1"><span class="kw">suppose </span></a><a NAME="E1:82_2_1"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</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="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font> <a href="rltopsp1.html#R1" title="RLTOPSP1:pred.1">are_collinear</a> </span><br/><div class="add"><span class="kw">take </span>
 <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>)
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rltopsp1.html#D16" onmouseover="rs('rltopsp1/D16')" onmouseout="rh()">RLTOPSP1:def 16</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"> ( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) &amp; <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) &amp; <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) )</span><br/><a NAME="E2:82_2_1"/>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T73" onmouseover="rs('rltopsp1/T73')" onmouseout="rh()">RLTOPSP1:73</a></span>;<br/><span class="kw">hence </span><a NAME="E3:82_2_1"/>
<font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_2_1"><span class="lab"><font color="Green" title="E39">A7</font></span></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"> ( <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) &amp; <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>) )</span><br/><span class="kw">thus </span><a NAME="E4:82_2_1"/>
<font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T72" onmouseover="rs('rltopsp1/T72')" onmouseout="rh()">RLTOPSP1:72</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"> <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>)</span><br/><span class="kw">thus </span><a NAME="E5:82_2_1"/>
<font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T72" onmouseover="rs('rltopsp1/T72')" onmouseout="rh()">RLTOPSP1:72</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_2"><span class="kw">suppose </span></a><a NAME="E1:82_2_2"/><span class="lab"><font color="Green" title="E39">A8</font></span>: 
<font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</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="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font> <a href="rltopsp1.html#R1" title="RLTOPSP1:pred.1">are_collinear</a> </span><br/><div class="add"><span class="kw">take </span>
 <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>)
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rltopsp1.html#D16" onmouseover="rs('rltopsp1/D16')" onmouseout="rh()">RLTOPSP1:def 16</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"> ( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) &amp; <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) &amp; <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) )</span><br/><span class="kw">thus </span><a NAME="E2:82_2_2"/>
<font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T72" onmouseover="rs('rltopsp1/T72')" onmouseout="rh()">RLTOPSP1:72</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"> ( <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) &amp; <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) )</span><br/><a NAME="E3:82_2_2"/>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T73" onmouseover="rs('rltopsp1/T73')" onmouseout="rh()">RLTOPSP1:73</a></span>;<br/><span class="kw">hence </span><a NAME="E4:82_2_2"/>
<font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_2_2"><span class="lab"><font color="Green" title="E39">A8</font></span></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"> <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>)</span><br/><span class="kw">thus </span><a NAME="E5:82_2_2"/>
<font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c2">p1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T72" onmouseover="rs('rltopsp1/T72')" onmouseout="rh()">RLTOPSP1:72</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_3"><span class="kw">suppose </span></a><a NAME="E1:82_2_3"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
<font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</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="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font> <a href="rltopsp1.html#R1" title="RLTOPSP1:pred.1">are_collinear</a> </span><br/><div class="add"><span class="kw">take </span>
 <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rltopsp1.html#D16" onmouseover="rs('rltopsp1/D16')" onmouseout="rh()">RLTOPSP1:def 16</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"> ( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) &amp; <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) &amp; <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) )</span><br/><span class="kw">thus </span><a NAME="E2:82_2_3"/>
<font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T72" onmouseover="rs('rltopsp1/T72')" onmouseout="rh()">RLTOPSP1:72</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"> ( <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) &amp; <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) )</span><br/><span class="kw">thus </span><a NAME="E3:82_2_3"/>
<font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T72" onmouseover="rs('rltopsp1/T72')" onmouseout="rh()">RLTOPSP1:72</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"> <font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)</span><br/><a NAME="E4:82_2_3"/>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T73" onmouseover="rs('rltopsp1/T73')" onmouseout="rh()">RLTOPSP1:73</a></span>;<br/><span class="kw">hence </span><a NAME="E5:82_2_3"/>
<font color="Maroon" title="c4">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_2_3"><span class="lab"><font color="Green" title="E39">A9</font></span></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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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