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<b>let </b><font color="Maroon">PCPP</font> be  <a href="anproj_2.html#V6">Pappian</a> <a href="anproj_2.html#NM2">CollProjectivePlane</a>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">b2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">a2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">b3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">a3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">c3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">c1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">c2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/>





















<b>assume </b><a NAME="E1:20"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; not <font color="Maroon">o</font>,<font color="Maroon">a1</font>,<font color="Maroon">a2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">o</font>,<font color="Maroon">a1</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">o</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a2</font>,<font color="Maroon">a3</font>,<font color="Maroon">c1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b2</font>,<font color="Maroon">b3</font>,<font color="Maroon">c1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a1</font>,<font color="Maroon">a3</font>,<font color="Maroon">c2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b1</font>,<font color="Maroon">b3</font>,<font color="Maroon">c2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">a2</font>,<font color="Maroon">b2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">a3</font>,<font color="Maroon">b3</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<a NAME="E2:20"/><b>then </b><i><font color="Green">E35</font></i>: 
not <font color="Maroon">o</font>,<font color="Maroon">a3</font>,<font color="Maroon">a2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:20"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">a3</font>,<font color="Maroon">a2</font>,<font color="Maroon">c1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b3</font>,<font color="Maroon">b2</font>,<font color="Maroon">c1</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:20"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">c1</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><b>assume </b><a NAME="E2:20_1"/><i><font color="Green">E58</font></i>: 
not <font color="Maroon">c1</font>,<font color="Maroon">c2</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><a NAME="E3:20_1"/><b>then </b><i><font color="Green">E59</font></i>: 
not <font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">c2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/><a NAME="E4:20_1"/>
( <font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font> )
 <b>by </b><i>, , , <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/><a NAME="E5:20_1"/><b>then </b>
not <font color="Maroon">c1</font>,<font color="Maroon">o</font>,<font color="Maroon">c2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , </i>;<br/><a NAME="E6:20_1"/><b>then </b>
not <font color="Maroon">o</font>,<font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:20_1"/>
contradiction
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:20"/>
<font color="Maroon">c1</font>,<font color="Maroon">c2</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/>


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