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

<b>let </b><font color="Maroon" title="c1">r</font>, <font color="Maroon" title="c2">s</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>







<b>set </b><font color="Maroon" title="c5">L</font> =  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font>;<br/>
<b>assume </b><a NAME="E1:60"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:60"/><i><font color="Green" title="E47">A2</font></i>: 
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 <b>and </b><br/><a NAME="E3:60"/><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">a</font>
 <b>and </b><br/><a NAME="E4:60"/><i><font color="Green" title="E49">A4</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c7">c</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2;<br/>
<a NAME="E5:60"/><i><font color="Green" title="E50">A5</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E3:60"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="xreal_1.html#T228" title="XREAL_1:th.228">XREAL_1:228</a></i>;<br/>
<a NAME="E6:60"/><i><font color="Green" title="E51">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E3:60"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="xreal_1.html#T228" title="XREAL_1:th.228">XREAL_1:228</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">C1</font> =  <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">A</font> = <font color="Maroon" title="c6">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T39" title="PRE_TOPC:th.39">PRE_TOPC:39</a></i>;<br/>
<a NAME="E8:60"/><i><font color="Green" title="E52">A7</font></i>: 
 <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="toprealb.html#D3" title="TOPREALB:def.3">TOPREALB:def 3</a></i>;<br/>
<a NAME="E9:60"/><i><font color="Green" title="E53">A8</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E1:60"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/>
<a NAME="E10:60"/><i><font color="Green" title="E54">A9</font></i>: 
 <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:60_1"/><i><font color="Green" title="E55">A10</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c10">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="rcomp_3.html#E8:60"><i><font color="Green" title="E52">A7</font></i></a></i>;<br/>
<a NAME="E3:60_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c11">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E8:60"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="rcomp_3.html#E1:60_1"><i><font color="Green" title="E55">A10</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<a NAME="E4:60_1"/><b>then </b>
( <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c11">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> )
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E4:60"><i><font color="Green" title="E49">A4</font></i></a>, <a class="txt" href="rcomp_3.html#E5:60"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:60_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E9:60"><i><font color="Green" title="E53">A8</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:60"/>
<font color="Maroon" title="c9">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:60_2"/><i><font color="Green" title="E55">A11</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c10">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="rcomp_3.html#E2:60"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/>
<a NAME="E3:60_2"/>
( <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c11">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E2:60"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="rcomp_3.html#E1:60_2"><i><font color="Green" title="E55">A11</font></i></a>, <a class="ref" href="rcomp_2.html#T3" title="RCOMP_2:th.3">RCOMP_2:3</a></i>;<br/>
<a NAME="E4:60_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c11">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E6:60"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:60_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E8:60"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:60"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c6">X</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E10:60"><i><font color="Green" title="E54">A9</font></i></a>, <a class="ref" href="tops_3.html#T57" title="TOPS_3:th.57">TOPS_3:57</a></i>;<br/>
<b>hence </b><a NAME="E13:60"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E2:60"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="rcomp_3.html#T50" target="_self" title="RCOMP_3:th.50">Th50</a></i>;<br/>


</div>
