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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">P</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> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:63"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span>  )
 ;<br/>

<a NAME="E2:63"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">P</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> <font color="Maroon">n</font></span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E4:63"/>
for <font color="Olive">w1</font>, <font color="Olive">w7</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> <font color="Maroon">n</font></span>)</span>  st <font color="Olive">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Olive">w7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Olive">w1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">w7</font> holds <br/> ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Q</font></span>)</span> st <br/>( <font color="Olive">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">w1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Olive">w7</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w1</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> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w7</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> <font color="Maroon">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:63_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">w7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">w1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">w7</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1_1"/>
( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( not <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> &amp; not <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font> ) or  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> or <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1"><b>suppose </b></a><a NAME="E1:63_1_1_1"/>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( not <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> &amp; not <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">w2</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> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w3</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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:63_1_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">w3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w2</font>,<font color="Maroon">w3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w3</font>,<font color="Maroon">w7</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, <a class="txt" href="jordan2c.html#E1:63_1"><i><font color="Green">E56</font></i></a>, </i>;<br/><b>thus </b><a NAME="E4:63_1_1_1"/>
 ex <font color="Olive">h</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Q</font></span>)</span> st <br/>( <font color="Olive">h</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:63_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan2c.html#E3:63_1_1_1"><i><font color="Green">E57</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_2"><b>suppose </b></a><a NAME="E1:63_1_1_2"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> or <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">w2</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> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w3</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> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w4</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> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w5</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> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w6</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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:63_1_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">w3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">w4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">w5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">w6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w2</font>,<font color="Maroon">w3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w3</font>,<font color="Maroon">w4</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w4</font>,<font color="Maroon">w5</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w5</font>,<font color="Maroon">w6</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w6</font>,<font color="Maroon">w7</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, <a class="txt" href="jordan2c.html#E1:63_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma51</font></i></a></i>;<br/><b>thus </b><a NAME="E4:63_1_1_2"/>
 ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Q</font></span>)</span> st <br/>( <font color="Olive">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:63_1"><i><font color="Green">E56</font></i></a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:63"/>
<font color="Maroon">P</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan1.html#T5">JORDAN1:5</a></i>;<br/>


</div>
