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<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>;<br/><a NAME="E2:24_1_1_1"/><i><font color="Green">E43</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>
 ;<br/><a NAME="E3:24_1_1_1"/>
 <a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E4:24_1_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> &amp;  <a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E5:24_1_1_1"/>
<span class="p1">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T45">FUNCT_2:45</a></i>;<br/><b>hence </b><a NAME="E6:24_1_1_1"/>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> is <a href="connsp_1.html#V1">connected</a>
 <b>by </b><i>, , , <a class="ref" href="connsp_1.html#T15">CONNSP_1:15</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:24_1_1_2"/><b>then </b>
( <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="seq_4.html#K1">}</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><a NAME="E3:24_1_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/><a NAME="E4:24_1_1_2"/><b>then </b>
(  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> is <a href="connsp_1.html#V2">connected</a> )
 <b>by </b><i><a class="ref" href="connsp_1.html#T29">CONNSP_1:29</a></i>;<br/><b>hence </b><a NAME="E5:24_1_1_2"/>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> is <a href="connsp_1.html#V1">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#T28">CONNSP_1:28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:24"/>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> is <a href="connsp_1.html#V1">connected</a>
 ;<br/>


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