<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">r</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">s</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">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>set </b><font color="Maroon">L</font> =  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>;<br/>
<b>assume </b><a NAME="E1:70"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 ;<br/>

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:70"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_2.html#K2">.]</a></span>
 <b>and </b><br/><a NAME="E3:70"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:70"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<b>set </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2;<br/>
<a NAME="E5:70"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i>, <a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/>
<a NAME="E6:70"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/>
<b>set </b><font color="Maroon">C1</font> =  <a href="toprealb.html#K5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T39">PRE_TOPC:39</a></i>;<br/>
<a NAME="E8:70"/><i><font color="Green">E48</font></i>: 
 <a href="toprealb.html#K5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="toprealb.html#D3">TOPREALB:def 3</a></i>;<br/>
<a NAME="E9:70"/><i><font color="Green">E49</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> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E9:70"><i><font color="Green">E49</font></i></a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E10:70"/><i><font color="Green">E50</font></i>: 
 <a href="toprealb.html#K5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</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> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:70_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="toprealb.html#K5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="rcomp_3.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E3:70_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 )
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T5" target="_self">Th5</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E4:70_1"/><b>then </b>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E1:70_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rcomp_3.html#T4" target="_self">Th4</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:70_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T6" target="_self">Th6</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:70"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="toprealb.html#K5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:70_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="rcomp_3.html#E10:70"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E3:70_2"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="txt" href="rcomp_3.html#E10:70"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<a NAME="E4:70_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:70_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="toprealb.html#K5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T5" target="_self">Th5</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:70"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T7" target="_self">Th7</a>, <a class="ref" href="tops_3.html#T57">TOPS_3:57</a></i>;<br/>
<b>hence </b><a NAME="E13:70"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E10:70"><i><font color="Green">E50</font></i></a>, <a class="txt" href="rcomp_3.html#E4:70"><i><font color="Green">E45</font></i></a></i>;<br/>


</div>
