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

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">d</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:37"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:37"/><b>then </b>
<font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E3:37"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>set </b><font color="Maroon">X</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> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">Y</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font>;<br/>
<a NAME="E4:37"/><i><font color="Green">E32</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">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E5:37"/><i><font color="Green">E33</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E6:37"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E45</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>
 ;<br/>

<b>then reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E3:37_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E4:37_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E5:37_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<b>set </b><font color="Maroon">y1</font> = <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font>;<br/>
<a NAME="E6:37_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font>
 
;<br/>
<a NAME="E7:37_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font> )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E8:37_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E9:37_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:37_1"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:37"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of 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>,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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span>)</span> st <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<b>then consider </b><font color="Maroon">f1</font> being   <a href="funct_2.html#NM1">Function</a> of 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>,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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E9:37"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 ;<br/>
<b>reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">f1</font> as   <a href="struct_0.html#NM6">Function</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>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span>)</span> ;<br/>
<a NAME="E11:37"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</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> holds <br/><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font>
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">T</font> =  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>;<br/>
<a NAME="E12:37"/><i><font color="Green">E77</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</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="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:37"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <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>
 
;<br/>
<a NAME="E14:37"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:37_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">x1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">x2</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, , </i>;<br/>
<a NAME="E4:37_2"/>
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font>
 
<b>by </b><i>, </i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font>

;<br/>
<a NAME="E6:37_2"/><b>then </b>
<font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font>
 
;<br/>
<a NAME="E7:37_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<b>hence </b><a NAME="E8:37_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:37"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">f2</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E16:37"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:37_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:37_3"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">ry</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:37_3"/><i><font color="Green">E83</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ry</font> &amp; <font color="Maroon">ry</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font> )
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E5:37_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ry</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E6:37_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">ry</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E7:37_3"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">ry</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E8:37_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">ry</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E9:37_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">e</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">ry</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E10:37_3"/><b>then </b>
<font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">ry</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E11:37_3"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">ry</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">e</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="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>then </b><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ry</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">e</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">ry</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ry</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<font color="Maroon">ry</font>

;<br/>
<b>hence </b><a NAME="E13:37_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:37"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E18:37"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">f3</font> = <font color="Maroon">f1</font> as   <a href="struct_0.html#NM6">Function</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="topmetr.html#K3">R^1</a>  <b>by </b><i>, , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font>;<br/>
<a NAME="E20:37"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:37_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">rx</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">ry</font> = <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">rx</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:37_4"/>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> holds <br/><font color="Maroon">ry</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font>
 
;<br/>
<b>hence </b><a NAME="E5:37_4"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:37"/>
 ex <font color="Olive">f4</font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  st <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">f4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<b>then consider </b><font color="Maroon">f4</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E23:37"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 ;<br/>
<b>reconsider </b><font color="Maroon">f5</font> = <font color="Maroon">f4</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a>  ;<br/>
<a NAME="E25:37"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">f5</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E26:37"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">f5</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="topmetr.html#T28">TOPMETR:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</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> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E28:37"/><i><font color="Green">E94</font></i>: 
<a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="topmetr.html#T26">TOPMETR:26</a></i>;<br/>
<a NAME="E29:37"/><i><font color="Green">E95</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E97</font></i>: <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f5</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> = 
<a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>
<br/>.= 
<font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E31:37"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f3</font> holds <br/><font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f5</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:37_6"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f3</font>
 ;<br/>

<a NAME="E2:37_6"/><b>then </b><i><font color="Green">E99</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>
 
;<br/>
<b>reconsider </b><font color="Maroon">rx</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, , </i>;<br/>
<a NAME="E4:37_6"/>
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">rx</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>hence </b><a NAME="E5:37_6"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f5</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="jgraph_5.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:37"/><b>then </b>
<font color="Maroon">f3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f5</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="jgraph_5.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E33:37"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">f3</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, , <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E35:37"/>
<a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="topmetr.html#T26">TOPMETR:26</a></i>;<br/>
<a NAME="E36:37"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">f2</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T9">TOPMETR:9</a></i>;<br/>
<a NAME="E37:37"/><i><font color="Green">E102</font></i>: 
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i>, <a class="ref" href="heine.html#T11">HEINE:11</a></i>;<br/>
<a NAME="E38:37"/>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#D8">TOPMETR:def 8</a></i>;<br/>
<a NAME="E39:37"/><b>then </b>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T38">PCOMPS_1:38</a></i>;<br/>
<a NAME="E40:37"/><b>then </b>
<font color="Maroon">f2</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i>, , , <a class="ref" href="jgraph_5.html#T8" target="_self">Th8</a>, , <a class="ref" href="compts_1.html#T26">COMPTS_1:26</a></i>;<br/>
<b>hence </b><a NAME="E41:37"/>
 ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</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>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font></span>)</span></span>)</span> st <br/>( <font color="Olive">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</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> holds <br/><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">d</font> ) )
 <b>by </b><i/>;<br/>


</div>
