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<b>let </b><font color="Maroon">P</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green">E33</font></i>: 
( (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">P</font> ) or (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">d</font>,<font color="Maroon">a</font>,<font color="Maroon">P</font> ) or (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">d</font>,<font color="Maroon">a</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">P</font> ) or (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">d</font>,<font color="Maroon">a</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">P</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan17.html#D1" target="_self">JORDAN17:def 1</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1"/>
( (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">P</font> ) or (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">d</font>,<font color="Maroon">a</font>,<font color="Maroon">P</font> ) or (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">d</font>,<font color="Maroon">a</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">P</font> ) or (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">d</font>,<font color="Maroon">a</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">P</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>suppose </b></a><a NAME="E1:16_1_1"/><i><font color="Green">E34</font></i>: 
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">P</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">e</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:16_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">b</font>,<font color="Maroon">e</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">e</font>,<font color="Maroon">c</font>,<font color="Maroon">P</font> )
 <b>by </b><i>, </i>;<br/><b>take </b>
<font color="Maroon">e</font>
;<br/><b>thus </b><a NAME="E4:16_1_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</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">a</font>,<font color="Maroon">b</font>,<font color="Maroon">e</font>,<font color="Maroon">c</font> <a href="jordan17.html#R1" target="_self">are_in_this_order_on</a> <font color="Maroon">P</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:16_1_2"/><i><font color="Green">E36</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E2:16_1_2"/><i><font color="Green">E37</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E3:16_1_2"/><i><font color="Green">E38</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">d</font>,<font color="Maroon">a</font>,<font color="Maroon">P</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">e</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:16_1_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">b</font>,<font color="Maroon">e</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">e</font>,<font color="Maroon">c</font>,<font color="Maroon">P</font> )
 <b>by </b><i>, , </i>;<br/><b>take </b>
<font color="Maroon">e</font>
;<br/><a NAME="E6:16_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><b>hence </b><a NAME="E7:16_1_2"/>
( <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</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">a</font>,<font color="Maroon">b</font>,<font color="Maroon">e</font>,<font color="Maroon">c</font> <a href="jordan17.html#R1" target="_self">are_in_this_order_on</a> <font color="Maroon">P</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:16_1_3"/><i><font color="Green">E40</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E2:16_1_3"/><i><font color="Green">E43</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">d</font>,<font color="Maroon">a</font>,<font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E3:16_1_3"/><i><font color="Green">E44</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">P</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E4:16_1_3"/>
 ex <font color="Olive">e</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Olive">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">e</font>,<font color="Maroon">c</font> <a href="jordan17.html#R1" target="_self">are_in_this_order_on</a> <font color="Maroon">P</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_3_1"/><i><font color="Green">E46</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">P</font>
 
<b>by </b><i>, , <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_3_1_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> or <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_3_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><a NAME="E2:16_1_3_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T5">JORDAN7:5</a></i>;<br/><b>then consider </b><font color="Maroon">e</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:16_1_3_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E5:16_1_3_1_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">b</font>,<font color="Maroon">e</font>,<font color="Maroon">P</font>
 <b>by </b><i/>;<br/><b>take </b>
<font color="Maroon">e</font>
;<br/><b>thus </b><a NAME="E6:16_1_3_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E7:16_1_3_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 <b>by </b><i>, , , <a class="ref" href="jordan7.html#T2">JORDAN7:2</a></i>;<br/><b>thus </b><a NAME="E8:16_1_3_1_1_1"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">e</font>,<font color="Maroon">c</font> <a href="jordan17.html#R1" target="_self">are_in_this_order_on</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_3_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">e</font> =  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>;<br/><a NAME="E2:16_1_3_1_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T5">JORDAN7:5</a></i>;<br/><a NAME="E3:16_1_3_1_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">e</font>,<font color="Maroon">c</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan7.html#T3">JORDAN7:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_3_1_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:16_1_3_1_1_2_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E3:16_1_3_1_1_2_1"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><b>hence </b><a NAME="E4:16_1_3_1_1_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:16_1_3_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><b>thus </b><a NAME="E6:16_1_3_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E7:16_1_3_1_1_2"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">e</font>,<font color="Maroon">c</font> <a href="jordan17.html#R1" target="_self">are_in_this_order_on</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:16_1_4"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">d</font>,<font color="Maroon">a</font>,<font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E2:16_1_4"/><i><font color="Green">E54</font></i>: 
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">P</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">e</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:16_1_4"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">b</font>,<font color="Maroon">e</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">e</font>,<font color="Maroon">c</font>,<font color="Maroon">P</font> )
 <b>by </b><i>, , </i>;<br/><b>take </b>
<font color="Maroon">e</font>
;<br/><b>thus </b><a NAME="E5:16_1_4"/>
( <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</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">a</font>,<font color="Maroon">b</font>,<font color="Maroon">e</font>,<font color="Maroon">c</font> <a href="jordan17.html#R1" target="_self">are_in_this_order_on</a> <font color="Maroon">P</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
