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<div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM7">FinSequence</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">Q</font> be   <a href="struct_0.html#NM2">Subset</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">q</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">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>







<b>assume </b><a NAME="E1:36"/><i><font color="Green">E29</font></i>: 
(  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">f</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a>  &amp; <font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a> &amp;  <a href="jordan5c.html#K2" target="_self">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  <a href="jordan5c.html#K2" target="_self">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 ;<br/>

<a NAME="E2:36"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> =  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, <a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/>
<b>set </b><font color="Maroon">q1</font> =  <a href="jordan5c.html#K2" target="_self">Last_Point</a> <font color="Maroon">P</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">Q</font>;<br/>
<b>set </b><font color="Maroon">p2</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>thus </b><a NAME="E4:36"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:36_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:36_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:36_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:36_1"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="jordan5c.html#R1" target="_self">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>,<font color="Maroon">P</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:36_1"/>
 <a href="jordan5c.html#R1" target="_self">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>,<font color="Maroon">q</font>,<font color="Maroon">P</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:36_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="jordan5c.html#R1" target="_self">LE</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">q</font>,<font color="Maroon">P</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E2:36"><i><font color="Green">E31</font></i></a>, </i>;<br/>
<a NAME="E8:36_1"/>
 <a href="jordan5c.html#R1" target="_self">LE</a>  <a href="jordan5c.html#K2" target="_self">Last_Point</a> <font color="Maroon">P</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">Q</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:36_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="jordan5c.html#R1" target="_self">LE</a>  <a href="jordan5c.html#K2" target="_self">Last_Point</a> <font color="Maroon">P</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">Q</font>,<font color="Maroon">q</font>,<font color="Maroon">P</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E2:36_1"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<a NAME="E10:36_1"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="sppol_1.html#T49">SPPOL_1:49</a></i>;<br/>
<a NAME="E11:36_1"/><b>then </b><i><font color="Green">E37</font></i>: 
(  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q</font> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T31">TOPREAL1:31</a>, <a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<a NAME="E12:36_1"/><b>then </b>
 <a href="jordan5c.html#R1" target="_self">LE</a> <font color="Maroon">q</font>, <a href="jordan5c.html#K2" target="_self">Last_Point</a> <font color="Maroon">P</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">Q</font>,<font color="Maroon">P</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan5c.html#E1:36"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan5c.html#E1"><i><font color="Green">Lemma16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:36_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan5c.html#E5:36_1"><i><font color="Green">E33</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E5:36"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/>

<b>hence </b><a NAME="E6:36"/>
 <a href="jordan3.html#R2">LE</a> <font color="Maroon">q</font>, <a href="jordan5c.html#K2" target="_self">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">Q</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/>


</div>
