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

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">d</font>, <font color="Maroon" title="c4">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E37">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c4">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 )
 ;<br/>

<a NAME="E2:38"/><b>then </b>
<font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T67" title="XREAL_1:th.67">XREAL_1:67</a></i>;<br/>
<a NAME="E3:38"/><b>then </b><i><font color="Green" title="E38">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">X</font> = 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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">Y</font> = 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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font>;<br/>
<a NAME="E4:38"/><i><font color="Green" title="E39">A3</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">a</font>,<font color="Maroon" title="c2">b</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">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/>
<a NAME="E5:38"/><i><font color="Green" title="E40">A4</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:38"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/>
<a NAME="E6:38"/><i><font color="Green" title="E41">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c7">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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">r1</font> = <font color="Maroon" title="c7">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jgraph_5.html#E4:38"><i><font color="Green" title="E39">A3</font></i></a></i>;<br/>
<a NAME="E3:38_1"/><i><font color="Green" title="E43">A7</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">r1</font> &amp; <font color="Maroon" title="c8">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:38"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E1:38_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E4:38_1"/><b>then </b><i><font color="Green" title="E44">A8</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">r1</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="xreal_1.html#T67" title="XREAL_1:th.67">XREAL_1:67</a></i>;<br/>
<a NAME="E5:38_1"/><i><font color="Green" title="E45">A9</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E3:38_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="xreal_1.html#T67" title="XREAL_1:th.67">XREAL_1:67</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">y1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">r1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font>;<br/>
<a NAME="E6:38_1"/><i><font color="Green" title="E46">A10</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">r1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font>
 
;<br/>
<a NAME="E7:38_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">r1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">r1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:38_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="jgraph_5.html#E5:38_1"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E8:38_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">r1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E5:38"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E9:38_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>] )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E6:38_1"><i><font color="Green" title="E46">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:38"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></span>)</span> st <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="jgraph_5.html#E6:38"><i><font color="Green" title="E41">A5</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c7">f1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E9:38"/><i><font color="Green" title="E42">A11</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">f2</font> = <font color="Maroon" title="c7">f1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></span>)</span> ;<br/>
<a NAME="E11:38"/><i><font color="Green" title="E43">A12</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/><font color="Maroon" title="c8">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:38"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E9:38"><i><font color="Green" title="E42">A11</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">S</font> =  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>;<br/>
<b>set </b><font color="Maroon" title="c10">T</font> =  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>;<br/>
<a NAME="E12:38"/><i><font color="Green" title="E44">A13</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:38"/><b>then </b><i><font color="Green" title="E45">A14</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
;<br/>
<a NAME="E14:38"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f2</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f2</font> &amp; <font color="Maroon" title="c8">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x1</font>, <font color="Maroon" title="c12">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_2"/><i><font color="Green" title="E46">A15</font></i>: 
( <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f2</font> &amp; <font color="Maroon" title="c12">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f2</font> &amp; <font color="Maroon" title="c8">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">r1</font> = <font color="Maroon" title="c11">x1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jgraph_5.html#E4:38"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E12:38"><i><font color="Green" title="E44">A13</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">r2</font> = <font color="Maroon" title="c12">x2</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jgraph_5.html#E4:38"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E12:38"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="jgraph_5.html#E1:38_2"><i><font color="Green" title="E46">A15</font></i></a></i>;<br/>
<a NAME="E4:38_2"/>
<font color="Maroon" title="c8">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E9:38"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="jgraph_5.html#E1:38_2"><i><font color="Green" title="E46">A15</font></i></a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">d</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c14">r2</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">d</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E9:38"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="jgraph_5.html#E1:38_2"><i><font color="Green" title="E46">A15</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c14">r2</font>

;<br/>
<a NAME="E6:38_2"/><b>then </b>
<font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c14">r2</font>
 
;<br/>
<a NAME="E7:38_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">e</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">r2</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/>
<b>hence </b><a NAME="E8:38_2"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x2</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:38"/><b>then </b><i><font color="Green" title="E46">A16</font></i>: 
<font color="Maroon" title="c8">f2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E16:38"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">f2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">y</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="E1:38_3"/><i><font color="Green" title="E47">A17</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:38_3"/><b>then </b>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E5:38"><i><font color="Green" title="E40">A4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">ry</font> = <font color="Maroon" title="c11">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E4:38_3"/><i><font color="Green" title="E48">A18</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">ry</font> &amp; <font color="Maroon" title="c12">ry</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E5:38"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E1:38_3"><i><font color="Green" title="E47">A17</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E5:38_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">ry</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E6:38_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">e</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c4">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">ry</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c4">e</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="real_1.html#T74" title="REAL_1:th.74">REAL_1:74</a></i>;<br/>
<a NAME="E7:38_3"/><b>then </b><i><font color="Green" title="E49">A19</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">ry</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c4">e</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E8:38_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">ry</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:38_3"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E9:38_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">e</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c4">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">ry</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c4">e</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="real_1.html#T74" title="REAL_1:th.74">REAL_1:74</a></i>;<br/>
<a NAME="E10:38_3"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">ry</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c4">e</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E11:38_3"/><b>then </b><i><font color="Green" title="E50">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">ry</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E7:38_3"><i><font color="Green" title="E49">A19</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<b>then </b><font color="Maroon" title="c8">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ry</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c4">e</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ry</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c4">e</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:38"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E9:38"><i><font color="Green" title="E42">A11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">ry</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
<br/>.= 
<font color="Maroon" title="c12">ry</font>

;<br/>
<b>hence </b><a NAME="E13:38_3"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">f2</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E4:38"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E12:38"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="jgraph_5.html#E11:38_3"><i><font color="Green" title="E50">A20</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:38"/><b>then </b><i><font color="Green" title="E47">A21</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E18:38"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">f3</font> = <font color="Maroon" title="c7">f1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="jgraph_5.html#E5:38"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E12:38"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font>;<br/>
<a NAME="E20:38"/><i><font color="Green" title="E48">A22</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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 <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</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 <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_4"/>
<font color="Maroon" title="c12">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 <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">rx</font> = <font color="Maroon" title="c12">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">ry</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">rx</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E4:38_4"/>
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> holds <br/><font color="Maroon" title="c14">ry</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font>
 
;<br/>
<b>hence </b><a NAME="E5:38_4"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</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 <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Olive" title="b1">y</font>] )
 <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:38"/>
 ex <font color="Olive" title="b1">f4</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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 <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="jgraph_5.html#E20:38"><i><font color="Green" title="E48">A22</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c12">f4</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E23:38"/><i><font color="Green" title="E49">A23</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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 <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c12">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">f5</font> = <font color="Maroon" title="c12">f4</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  ;<br/>
<a NAME="E25:38"/>
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Maroon" title="c13">f5</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E23:38"><i><font color="Green" title="E49">A23</font></i></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E26:38"/><b>then </b><i><font color="Green" title="E50">A24</font></i>: 
<font color="Maroon" title="c13">f5</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="topmetr.html#T28" title="TOPMETR:th.28">TOPMETR:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">B</font> = 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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="jgraph_5.html#E4:38"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E28:38"/><i><font color="Green" title="E51">A25</font></i>: 
<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E4:38"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="topmetr.html#T26" title="TOPMETR:th.26">TOPMETR:26</a></i>;<br/>
<a NAME="E29:38"/><i><font color="Green" title="E52">A26</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green" title="E53">A27</font></i>: <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">f5</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c14">B</font> = 
<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c14">B</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>
<br/>.= 
<font color="Maroon" title="c14">B</font>
<b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E31:38"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f3</font> holds <br/><font color="Maroon" title="c11">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f5</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_6"/><i><font color="Green" title="E54">A28</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f3</font>
 ;<br/>

<a NAME="E2:38_6"/><b>then </b><i><font color="Green" title="E55">A29</font></i>: 
<font color="Maroon" title="c15">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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
;<br/>
<b>reconsider </b><font color="Maroon" title="c16">rx</font> = <font color="Maroon" title="c15">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jgraph_5.html#E4:38"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E12:38"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="jgraph_5.html#E1:38_6"><i><font color="Green" title="E54">A28</font></i></a></i>;<br/>
<a NAME="E4:38_6"/>
<font color="Maroon" title="c12">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c16">rx</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E23:38"><i><font color="Green" title="E49">A23</font></i></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>hence </b><a NAME="E5:38_6"/>
<font color="Maroon" title="c11">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f5</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E9:38"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="jgraph_5.html#E2:38_6"><i><font color="Green" title="E55">A29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:38"/><b>then </b>
<font color="Maroon" title="c11">f3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f5</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c14">B</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E29:38"><i><font color="Green" title="E52">A26</font></i></a>, <a class="txt" href="jgraph_5.html#E30:38"><i><font color="Green" title="E53">A27</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E33:38"/><b>then </b><i><font color="Green" title="E54">A30</font></i>: 
<font color="Maroon" title="c11">f3</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E26:38"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="jgraph_5.html#E28:38"><i><font color="Green" title="E51">A25</font></i></a>, <a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">C</font> = 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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="jgraph_5.html#E5:38"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E35:38"/>
<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:38"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="jgraph_5.html#E5:38"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="topmetr.html#T26" title="TOPMETR:th.26">TOPMETR:26</a></i>;<br/>
<a NAME="E36:38"/><b>then </b><i><font color="Green" title="E55">A31</font></i>: 
<font color="Maroon" title="c8">f2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E33:38"><i><font color="Green" title="E54">A30</font></i></a>, <a class="ref" href="topmetr.html#T9" title="TOPMETR:th.9">TOPMETR:9</a></i>;<br/>
<a NAME="E37:38"/><i><font color="Green" title="E56">A32</font></i>: 
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="heine.html#T11" title="HEINE:th.11">HEINE:11</a></i>;<br/>
<a NAME="E38:38"/>
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#D8" title="TOPMETR:def.8">TOPMETR:def 8</a></i>;<br/>
<a NAME="E39:38"/><b>then </b>
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="compts_1.html#NR2" title="COMPTS_1:NR.2">is_T2</a> 
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T38" title="PCOMPS_1:th.38">PCOMPS_1:38</a></i>;<br/>
<a NAME="E40:38"/><b>then </b>
<font color="Maroon" title="c8">f2</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E13:38"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="jgraph_5.html#E15:38"><i><font color="Green" title="E46">A16</font></i></a>, <a class="txt" href="jgraph_5.html#E17:38"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jgraph_5.html#E36:38"><i><font color="Green" title="E55">A31</font></i></a>, <a class="txt" href="jgraph_5.html#E37:38"><i><font color="Green" title="E56">A32</font></i></a>, <a class="ref" href="compts_1.html#T26" title="COMPTS_1:th.26">COMPTS_1:26</a></i>;<br/>
<b>hence </b><a NAME="E41:38"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; ( for <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b2">r</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">d</font> ) )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E11:38"><i><font color="Green" title="E43">A12</font></i></a></i>;<br/>


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