<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">FdSp</font> be   <a href="parsp_2.html#NM1">FanodesSp</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FdSp</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FdSp</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FdSp</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FdSp</font>;<br/>









<b>assume </b><a NAME="E1:30"/><i><font color="Green">E17</font></i>: 
( not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="parsp_2.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> )
 ;<br/>

<div><i><font color="Green">E18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 ;<br/><a NAME="E2:30_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="parsp_1.html#T40">PARSP_1:40</a></i>;<br/><b>hence </b><a NAME="E3:30_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2"/>
( <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> )
 ;<br/><a NAME="E2:30_2"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> &amp; not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="parsp_2.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">c</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , <a class="ref" href="parsp_1.html#T42">PARSP_1:42</a></i>;<br/><a NAME="E3:30_2"/><b>then </b>
( not <font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">a</font> <a href="parsp_2.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">d</font>,<font color="Maroon">c</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">d</font>,<font color="Maroon">a</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> )
 <b>by </b><i>, <a class="ref" href="parsp_1.html#T40">PARSP_1:40</a>, <a class="ref" href="parsp_1.html#T42">PARSP_1:42</a></i>;<br/><a NAME="E4:30_2"/><b>then </b>
( not <font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="parsp_2.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">d</font>,<font color="Maroon">c</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">d</font>,<font color="Maroon">a</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:30_2"/>
not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="parsp_2.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:30"/>
not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="parsp_2.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i>, </i>;<br/>


</div>
