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

<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> , <font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">d</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="E1:47"/><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/>

<b>assume </b><a NAME="E2:47"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 ;<br/>

<b>let </b><font color="Maroon">x</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="E3:47"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 ;<br/>

<b>assume </b><a NAME="E4:47"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 ;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="borsuk_6.html#K1" target="_self">L[01]</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>;<br/>
<b>set </b><font color="Maroon">F</font> =  <a href="treal_1.html#K3">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="treal_1.html#K2">(#)</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">G</font> =  <a href="treal_1.html#K4">P[01]</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>;<br/>
<a NAME="E5:47"/><i><font color="Green">E36</font></i>: 
(  <a href="treal_1.html#K1">(#)</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="treal_1.html#K2">(#)</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 
<b>by </b><i>, <a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a>, <a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>;<br/>
<a NAME="E6:47"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <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>
 
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T1" target="_self">Th1</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E7:47"/><b>then </b><i><font color="Green">E37</font></i>: 
<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">a</font>,<font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E7:47"><i><font color="Green">E37</font></i></a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E8:47"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:47"/><i><font color="Green">E63</font></i>: 
( 0 <a href="hidden.html#R1">=</a>  <a href="treal_1.html#K1">(#)</a> 0,1 &amp; 1 <a href="hidden.html#R1">=</a> 0,1 <a href="treal_1.html#K2">(#)</a>  )
 
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a>, <a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>;<br/>
<a NAME="E10:47"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <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> 0,1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T1" target="_self">Th1</a>, </i>;<br/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K1" target="_self">L[01]</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E7:47"><i><font color="Green">E37</font></i></a>, , , <a class="ref" href="treal_1.html#D4">TREAL_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i>, <a class="txt" href="borsuk_6.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="borsuk_6.html#T5" target="_self">Th5</a>, <a class="ref" href="treal_1.html#D3">TREAL_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E12:47"/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K1" target="_self">L[01]</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font>
 ;<br/>


</div>
