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<b>let </b><font color="Maroon">IPP</font> be   <a href="incproj.html#NM2">IncProjSp</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/>





<a NAME="E1:6"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> implies  ex <font color="Olive">a</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( not <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:6_1"/>
not for <font color="Olive">a</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> holds  <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:6_1"/>
 ex <font color="Olive">a</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( not <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6"/>
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font> implies  ex <font color="Olive">a</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( not <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_2"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>, <font color="Maroon">b</font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E3:6_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">C</font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E5:6_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font>, <font color="Maroon">d</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font>, <font color="Maroon">e</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">IPP</font><b> such that </b><br/><a NAME="E7:6_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> &amp; <font color="Maroon">e</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D12">INCPROJ:def 12</a></i>;<br/>
<a NAME="E8:6_2"/>
( ( not <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> ) or ( not <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">d</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> ) or ( not <font color="Maroon">e</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">e</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> ) )
 
<b>by </b><i>, , , <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E9:6_2"/>
 ex <font color="Olive">a</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( not <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6"/>
 ex <font color="Olive">a</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( not <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/>


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