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<b>let </b><font color="Maroon">G</font> be   <a href="incproj.html#NM2">IncProjSp</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">G</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:37"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="projpl_1.html#R5" target="_self">is_a_quadrangle</a> 
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green">E37</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="incproj.html#R2">are_mutually_different</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> or <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 </span><b>by </b><i>, <a class="ref" href="incproj.html#D6">INCPROJ:def 6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><b>case </b></a><a NAME="E1:37_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:37_1_1_1"/><b>then </b>
not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="ref" href="incproj.html#D5">INCPROJ:def 5</a></i>;<br/><a NAME="E3:37_1_1_1"/><b>then </b>
not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="projpl_1.html#NR6" target="_self">is_a_triangle</a> 
 <b>by </b><i><a class="ref" href="projpl_1.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E4:37_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="projpl_1.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2"><b>case </b></a><a NAME="E1:37_1_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/><div class="add"><a NAME="E2:37_1_1_2"/><b>then </b>
not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="ref" href="incproj.html#D5">INCPROJ:def 5</a></i>;<br/><a NAME="E3:37_1_1_2"/><b>then </b>
not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="projpl_1.html#NR6" target="_self">is_a_triangle</a> 
 <b>by </b><i><a class="ref" href="projpl_1.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E4:37_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="projpl_1.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_3"><b>case </b></a><a NAME="E1:37_1_1_3"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:37_1_1_3"/><b>then </b>
not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="ref" href="incproj.html#D5">INCPROJ:def 5</a></i>;<br/><a NAME="E3:37_1_1_3"/><b>then </b>
not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="projpl_1.html#NR6" target="_self">is_a_triangle</a> 
 <b>by </b><i><a class="ref" href="projpl_1.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E4:37_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="projpl_1.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_4"><b>case </b></a><a NAME="E1:37_1_1_4"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:37_1_1_4"/><b>then </b>
not <font color="Maroon">d</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="ref" href="incproj.html#D5">INCPROJ:def 5</a></i>;<br/><a NAME="E3:37_1_1_4"/><b>then </b>
not <font color="Maroon">d</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="projpl_1.html#NR6" target="_self">is_a_triangle</a> 
 <b>by </b><i><a class="ref" href="projpl_1.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E4:37_1_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="projpl_1.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_5"><b>case </b></a><a NAME="E1:37_1_1_5"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:37_1_1_5"/><b>then </b>
not <font color="Maroon">d</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="ref" href="incproj.html#D5">INCPROJ:def 5</a></i>;<br/><a NAME="E3:37_1_1_5"/><b>then </b>
not <font color="Maroon">d</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="projpl_1.html#NR6" target="_self">is_a_triangle</a> 
 <b>by </b><i><a class="ref" href="projpl_1.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E4:37_1_1_5"/>
contradiction
 <b>by </b><i><a class="ref" href="projpl_1.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_6"><b>case </b></a><a NAME="E1:37_1_1_6"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/><div class="add"><a NAME="E2:37_1_1_6"/><b>then </b>
not <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="ref" href="incproj.html#D5">INCPROJ:def 5</a></i>;<br/><a NAME="E3:37_1_1_6"/><b>then </b>
not <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="projpl_1.html#NR6" target="_self">is_a_triangle</a> 
 <b>by </b><i><a class="ref" href="projpl_1.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E4:37_1_1_6"/>
contradiction
 <b>by </b><i><a class="ref" href="projpl_1.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:37"/>
contradiction
 ;<br/>


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