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

<b>let </b><font color="Maroon">T</font> be  <a href="compts_1.html#V3">being_T2</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Q</font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E3:7"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E4:7"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">g</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E5:7"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E6:7"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <font color="Maroon">T</font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P'</font> = <font color="Maroon">P</font>, <font color="Maroon">Q'</font> = <font color="Maroon">Q</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<a NAME="E9:7"/>
<font color="Maroon">f</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P'</font></span>)</span>
 
;<br/>
<a NAME="E10:7"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:7"/>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E12:7"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E13:7"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">g</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Q'</font></span>)</span>
 
;<br/>
<a NAME="E14:7"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:7"/>
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E16:7"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E17:7"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E18:7"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<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">a</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">a</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> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">a</font></span>)</span>;<br/>
<a NAME="E19:7"/><i><font color="Green">E39</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> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_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="E1:7_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<a NAME="E2:7_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:7_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:7_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/>
<b>take </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span>
;<br/>

<b>thus </b><a NAME="E6:7_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E21:7"/><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E22:7"/><i><font color="Green">E42</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> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<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">a</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">a</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> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">a</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>;<br/>
<a NAME="E23:7"/><i><font color="Green">E43</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> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><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="E1:7_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<a NAME="E2:7_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:7_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:7_2"/>
( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<b>take </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
;<br/>

<b>thus </b><a NAME="E6:7_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">g2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E25:7"/><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> <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">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E26:7"/><i><font color="Green">E68</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> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">R</font> = <font color="Maroon">P'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q'</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">M</font> ;<br/>
<i><font color="Green">E70</font></i>: <font color="Maroon">R</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:7_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_4"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:7_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 <b>by </b><i>, <a class="txt" href="topmetr2.html#E3:7"><i><font color="Green">E23</font></i></a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:7_4"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><a NAME="E7:7_4"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="topmetr2.html#T1" target="_self">Th1</a>, <a class="txt" href="topmetr2.html#E3:7"><i><font color="Green">E23</font></i></a>, </i>;<br/><a NAME="E8:7_4"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font> &amp; 2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, <a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/><a NAME="E9:7_4"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 ;<br/><a NAME="E10:7_4"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>hence </b><a NAME="E11:7_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E30:7"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:7_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_5"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:7_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:7_5"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><a NAME="E7:7_5"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="topmetr2.html#E1:7"><i><font color="Green">E21</font></i></a>, , </i>;<br/><a NAME="E8:7_5"/>
( 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 2 )
 ;<br/><a NAME="E9:7_5"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font> &amp; 2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> 1 )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E10:7_5"/><b>then </b>
( 1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 1 &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E11:7_5"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 ;<br/><a NAME="E12:7_5"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>hence </b><a NAME="E13:7_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E32:7"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E33:7"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  )
 
;<br/>
<a NAME="E34:7"/><b>then </b>
( 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; 1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E35:7"/><b>then </b>
( <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/>
<a NAME="E36:7"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; 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">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  )
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">T1</font> =  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>, <font color="Maroon">T2</font> =  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1 as    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="topmetr.html#T4">TOPMETR:4</a></i>;<br/>
<a NAME="E38:7"/>
( <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E39:7"/><b>then </b><i><font color="Green">E79</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font></span>)</span> )
 
<b>by </b><i>, , <a class="txt" href="topmetr2.html#E2:7"><i><font color="Green">E22</font></i></a>, <a class="txt" href="topmetr2.html#E3:7"><i><font color="Green">E23</font></i></a>, , <a class="ref" href="topmetr.html#T25">TOPMETR:25</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f2</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T1</font>,<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g2</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T2</font>,<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E42:7"/>
1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">pp</font> = 1 <a href="real_1.html#K6">/</a> 2 as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E44:7"/>
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E45:7"/><b>then </b><i><font color="Green">E83</font></i>: 
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font> <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">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</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="E46:7"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="heine.html#T2">HEINE:2</a></i>;<br/>
<a NAME="E47:7"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E48:7"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">pp</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E49:7"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">T1</font> is <a href="compts_1.html#V2">compact</a> &amp; <font color="Maroon">T2</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="E50:7"/>
 <a href="pcomps_1.html#K4">TopSpaceMetr</a> <a href="metric_1.html#K8">RealSpace</a>  <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="E51:7"/><b>then </b><i><font color="Green">E87</font></i>: 
 <a href="topmetr.html#K5">I[01]</a>  <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T3">TOPMETR:3</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = 2 <a href="real_1.html#K4">*</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">f3</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E53:7"/><i><font color="Green">E90</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">f3</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f3</font> = <font color="Maroon">f3</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>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E55:7"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">f3</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">2 <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="txt" href="topmetr2.html#E5:7"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E56:7"/><b>then </b><i><font color="Green">E91</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#T28">TOPMETR:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">PP</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></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/>
<b>reconsider </b><font color="Maroon">PP</font> = <font color="Maroon">PP</font> as  non <a href="xboole_0.html#V1">empty</a> <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="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E59:7"/><i><font color="Green">E93</font></i>: 
<a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">PP</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T26">TOPMETR:26</a></i>;<br/>
<i><font color="Green">E94</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f3</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f3</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>
;<br/>
<a NAME="E61:7"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f3</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="tarski.html#R1">c=</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">f3'</font> = <font color="Maroon">f3</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">PP</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T1</font>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topmetr2.html#E13:7"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E63:7"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">f3'</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="topmetr2.html#E10:7"><i><font color="Green">E32</font></i></a>, <a class="txt" href="topmetr2.html#E12:7"><i><font color="Green">E33</font></i></a>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/>
<a NAME="E64:7"/><i><font color="Green">E97</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f3'</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E65:7"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f3'</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_7"><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="E1:7_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f3'</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_7"/><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> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f3'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:7_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="topmetr2.html#E16:7"><i><font color="Green">E36</font></i></a>, <a class="txt" href="topmetr2.html#E17:7"><i><font color="Green">E37</font></i></a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E5:7_7"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:7_7"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/>
<i><font color="Green">E100</font></i>: <font color="Maroon">y</font> = 
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topmetr2.html#E17:7"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
<b>by </b><i><a class="txt" href="topmetr2.html#E5:7"><i><font color="Green">E31</font></i></a>, <a class="txt" href="topmetr2.html#E18:7"><i><font color="Green">E38</font></i></a></i>
;<br/>
<a NAME="E9:7_7"/>
( 2 <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font> &amp; 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 
<b>by </b><i><a class="txt" href="topmetr2.html#E18:7"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E10:7_7"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">rr</font> where <font color="Olive">rr</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">rr</font> &amp; <font color="Olive">rr</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="topmetr2.html#E19:7"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:7_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f4</font> = <font color="Maroon">f3'</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T1</font>,<a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topmetr2.html#E16:7"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E67:7"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">f4</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="topmetr2.html#E14:7"><i><font color="Green">E35</font></i></a>, <a class="ref" href="topmetr.html#T8">TOPMETR:8</a></i>;<br/>
<a NAME="E68:7"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E69:7"/>
<font color="Maroon">f</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P'</font></span>)</span>
 
;<br/>
<a NAME="E70:7"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font></span>)</span> )
 
<b>by </b><i>, <a class="txt" href="topmetr2.html#E2:7"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E72:7"/>
<font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P'</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T5">TOPMETR:5</a></i>;<br/>
<a NAME="E73:7"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">ff</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="topmetr2.html#E18:7"><i><font color="Green">E38</font></i></a>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/>
<a NAME="E74:7"/>
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 <font color="Maroon">T1</font> holds <br/><font color="Maroon">f1</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">ff</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f4</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_8"><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="E1:7_8"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>
 ;<br/>

<a NAME="E2:7_8"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E4:7_8"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</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/>
<b>hence </b><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i><a class="ref" href="topmetr2.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i><a class="txt" href="topmetr2.html#E5:7"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="topmetr2.html#E13:7"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f4</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E75:7"/><b>then </b>
<font color="Maroon">f1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">ff</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f4</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>
<a NAME="E76:7"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="topmetr2.html#E17:7"><i><font color="Green">E37</font></i></a>, <a class="txt" href="topmetr2.html#E19:7"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1;<br/>
<b>consider </b><font color="Maroon">g3</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E78:7"/><i><font color="Green">E111</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">g3</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">g3</font> = <font color="Maroon">g3</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>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E80:7"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">g3</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">2 <a href="real_1.html#K4">*</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_9"><b>proof </b></a><div class="add">

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

<b>thus </b><font color="Maroon">g3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="txt" href="topmetr2.html#E21:7"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E81:7"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">g3</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">PP</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></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/>
<b>reconsider </b><font color="Maroon">PP</font> = <font color="Maroon">PP</font> as  non <a href="xboole_0.html#V1">empty</a> <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="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E84:7"/><i><font color="Green">E113</font></i>: 
<a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">PP</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T26">TOPMETR:26</a></i>;<br/>
<i><font color="Green">E114</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g3</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font>
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>
;<br/>
<a NAME="E86:7"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="tarski.html#R1">c=</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">g3'</font> = <font color="Maroon">g3</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">PP</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T2</font>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E88:7"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">g3'</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="topmetr2.html#E22:7"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topmetr2.html#E23:7"><i><font color="Green">E43</font></i></a>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/>
<a NAME="E89:7"/><i><font color="Green">E117</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g3'</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E90:7"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g3'</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_11"><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="E1:7_11"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g3'</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_11"/><i><font color="Green">E118</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">g3'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g3'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:7_11"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="topmetr2.html#E3"><i><font color="Green">Lemma46</font></i></a>, , <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E5:7_11"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:7_11"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<i><font color="Green">E120</font></i>: <font color="Maroon">y</font> = 
<font color="Maroon">g3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="txt" href="topmetr2.html#E21:7"><i><font color="Green">E41</font></i></a>, </i>
;<br/>
<a NAME="E9:7_11"/>
( 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font> &amp; 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> 1 &amp; 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E10:7_11"/><b>then </b>
( 1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1 &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E11:7_11"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">rr</font> where <font color="Olive">rr</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">rr</font> &amp; <font color="Olive">rr</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:7_11"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">g4</font> = <font color="Maroon">g3'</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T2</font>,<a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topmetr2.html#E3"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E92:7"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">g4</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="topmetr2.html#E25:7"><i><font color="Green">E45</font></i></a>, <a class="ref" href="topmetr.html#T8">TOPMETR:8</a></i>;<br/>
<a NAME="E93:7"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E94:7"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font></span>)</span> )
 
<b>by </b><i>, , <a class="txt" href="topmetr2.html#E2:7"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E96:7"/>
<font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Q'</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T5">TOPMETR:5</a></i>;<br/>
<a NAME="E97:7"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">g'</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/>
<a NAME="E98:7"/>
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 <font color="Maroon">T2</font> holds <br/><font color="Maroon">g1</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">g'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g4</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_12"><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="E1:7_12"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font>
 ;<br/>

<a NAME="E2:7_12"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font> <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">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E4:7_12"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font> <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">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</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/>
<b>hence </b><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x'</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="topmetr2.html#E1:7"><i><font color="Green">E21</font></i></a>, </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i><a class="txt" href="topmetr2.html#E21:7"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g4</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E99:7"/><b>then </b>
<font color="Maroon">g1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g4</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>
<a NAME="E100:7"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">g1</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, , <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E101:7"/><i><font color="Green">E128</font></i>: 
( 1 <a href="real_1.html#K6">/</a> 2 <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">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; 1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<b>then </b><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">pp</font> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="topmetr2.html#T1" target="_self">Th1</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">pp</font>
<b>by </b><i><a class="txt" href="topmetr2.html#E1:7"><i><font color="Green">E21</font></i></a>, <a class="ref" href="topmetr2.html#T4" target="_self">Th4</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">f2</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g2</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i>, , , <a class="txt" href="topmetr2.html#E4:7"><i><font color="Green">E30</font></i></a>, , , </i>;<br/>
<i><font color="Green">E129</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="heine.html#T2">HEINE:2</a></i>
;<br/>
<a NAME="E105:7"/><i><font color="Green">E130</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E106:7"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="txt" href="topmetr2.html#E3:7"><i><font color="Green">E23</font></i></a>, , <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E107:7"/><b>then </b><i><font color="Green">E131</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h'</font> = <font color="Maroon">h</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font></span>)</span></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">h'</font>
;<br/>

<a NAME="E109:7"/><i><font color="Green">E133</font></i>: 
 <a href="topmetr.html#K5">I[01]</a>  is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="ref" href="heine.html#T11">HEINE:11</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E110:7"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T3">TOPMETR:3</a></i>;<br/>
<a NAME="E111:7"/><i><font color="Green">E135</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="funct_4.html#T19">FUNCT_4:19</a></i>;<br/>
<a NAME="E112:7"/>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:7_15"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_15"/><i><font color="Green">E136</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">g</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:7_15"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">x'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:7_15"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<a NAME="E7:7_15"/>
( 0 <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font> <a href="real_1.html#K3">+</a> 1 &amp; <font color="Maroon">x'</font> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K3">+</a> 1 &amp; 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:7_15"/><b>then </b>
( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K6">/</a> 2 &amp; 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E9:7_15"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<a NAME="E10:7_15"/><b>then </b><i><font color="Green">E138</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<font color="Maroon">a</font> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x'</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
<b>by </b><i>, <a class="txt" href="topmetr2.html#E1:7"><i><font color="Green">E21</font></i></a>, </i>
;<br/>
<b>hence </b><a NAME="E12:7_15"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E113:7"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E114:7"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:7_16"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_16"/><i><font color="Green">E141</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">f</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:7_16"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">x'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:7_16"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<a NAME="E7:7_16"/>
( 0 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font> <a href="real_1.html#K6">/</a> 2 &amp; <font color="Maroon">x'</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 &amp; 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E8:7_16"/><b>then </b>
<font color="Maroon">x'</font> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 
;<br/>
<a NAME="E9:7_16"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">x'</font> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<font color="Maroon">a</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="topmetr2.html#T1" target="_self">Th1</a>, </i>
;<br/>
<b>hence </b><a NAME="E11:7_16"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E115:7"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:7_17"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_17"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_17_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_17_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_17_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_17_1_1_1"><b>suppose </b></a><a NAME="E1:7_17_1_1_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><div class="add"><a NAME="E2:7_17_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:7_17_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E4:7_17_1_1_1"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:7_17_1_1_1"/><i><font color="Green">E147</font></i>: 
1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><i><font color="Green">E148</font></i>: <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="topmetr2.html#E1:7"><i><font color="Green">E21</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
<b>by </b><i>, , <a class="ref" href="topmetr2.html#T1" target="_self">Th1</a>, </i>
;<br/><a NAME="E7:7_17_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:7_17_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><b>hence </b><a NAME="E9:7_17_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_17_1_1_2"><b>suppose </b></a><a NAME="E1:7_17_1_1_2"/><i><font color="Green">E149</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><div class="add"><a NAME="E2:7_17_1_1_2"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:7_17_1_1_2"/><b>then </b><i><font color="Green">E151</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">h</font>
 <b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E4:7_17_1_1_2"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><b>hence </b><a NAME="E5:7_17_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:7_17"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E116:7"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E117:7"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E118:7"/><b>then </b><i><font color="Green">E152</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E119:7"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E120:7"/><b>then </b><i><font color="Green">E153</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<div><i><font color="Green">E154</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_18"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:7_18"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font>
 ;<br/><a NAME="E2:7_18"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:7_18"/><i><font color="Green">E155</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><a NAME="E5:7_18"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font> &amp; 2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, <a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/><a NAME="E6:7_18"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 ;<br/><b>hence </b><a NAME="E7:7_18"/>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E156</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_19"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:7_19"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font>
 ;<br/><a NAME="E2:7_19"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:7_19"/><i><font color="Green">E157</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><a NAME="E5:7_19"/>
( 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 2 )
 ;<br/><a NAME="E6:7_19"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font> &amp; 2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> 1 )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E7:7_19"/><b>then </b>
( 1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 1 &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:7_19"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 ;<br/><b>hence </b><a NAME="E9:7_19"/>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_20"><b>now </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><b>that </b><br/><a NAME="E1:7_20"/><i><font color="Green">E158</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">h</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">h</font> )
 <b>and </b><br/><a NAME="E2:7_20"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_20_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_20_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_20_1_1"/>
( ( not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> &amp; not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> ) or ( not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> &amp; not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_20_1_1_1"><b>suppose </b></a><a NAME="E1:7_20_1_1_1"/><i><font color="Green">E160</font></i>: 
( not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> &amp; not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> )
 ;<br/><div class="add"><a NAME="E2:7_20_1_1_1"/><i><font color="Green">E161</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E3:7_20_1_1_1"/><b>then </b><i><font color="Green">E162</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:7_20_1_1_1"/><i><font color="Green">E163</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> )
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:7_20_1_1_1"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">x1'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:7_20_1_1_1"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">x1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>and </b><br/><a NAME="E8:7_20_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1'</font> &amp; <font color="Maroon">x1'</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><a NAME="E9:7_20_1_1_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">x2'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E11:7_20_1_1_1"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">x2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>and </b><br/><a NAME="E12:7_20_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2'</font> &amp; <font color="Maroon">x2'</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><a NAME="E13:7_20_1_1_1"/><i><font color="Green">E168</font></i>: 
( 2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; 2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="txt" href="topmetr2.html#E26:7"><i><font color="Green">E68</font></i></a>, <a class="txt" href="topmetr2.html#E28:7"><i><font color="Green">E70</font></i></a>, </i>;<br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font></span>)</span> = 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i>, <a class="ref" href="topmetr2.html#T1" target="_self">Th1</a>, <a class="txt" href="topmetr2.html#E26:7"><i><font color="Green">E68</font></i></a>, <a class="txt" href="topmetr2.html#E28:7"><i><font color="Green">E70</font></i></a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="topmetr2.html#T1" target="_self">Th1</a>, <a class="txt" href="topmetr2.html#E26:7"><i><font color="Green">E68</font></i></a>, </i>
;<br/><a NAME="E15:7_20_1_1_1"/><b>then </b>
( 2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font> &amp; 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E16:7_20_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="txt" href="topmetr2.html#E28:7"><i><font color="Green">E70</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_20_1_1_2"><b>suppose </b></a><a NAME="E1:7_20_1_1_2"/><i><font color="Green">E169</font></i>: 
( not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> )
 ;<br/><div class="add"><a NAME="E2:7_20_1_1_2"/><b>then </b>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">x2'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:7_20_1_1_2"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">x2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>and </b><br/><a NAME="E5:7_20_1_1_2"/>
( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2'</font> &amp; <font color="Maroon">x2'</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><a NAME="E6:7_20_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E7:7_20_1_1_2"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:7_20_1_1_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">x1'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:7_20_1_1_2"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">x1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>and </b><br/><a NAME="E11:7_20_1_1_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1'</font> &amp; <font color="Maroon">x1'</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><i><font color="Green">E173</font></i>: <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font></span>)</span> = 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i>, <a class="ref" href="topmetr2.html#T1" target="_self">Th1</a>, <a class="txt" href="topmetr2.html#E26:7"><i><font color="Green">E68</font></i></a>, </i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="topmetr2.html#E1:7"><i><font color="Green">E21</font></i></a>, , </i>
;<br/><a NAME="E13:7_20_1_1_2"/><i><font color="Green">E174</font></i>: 
( 2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 <b>by </b><i>, , , , <a class="txt" href="topmetr2.html#E26:7"><i><font color="Green">E68</font></i></a>, </i>;<br/><a NAME="E14:7_20_1_1_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E15:7_20_1_1_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="txt" href="topmetr2.html#E28:7"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E16:7_20_1_1_2"/><b>then </b><i><font color="Green">E175</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E17:7_20_1_1_2"/>
( 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E18:7_20_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R1">=</a> 0 <a href="real_1.html#K3">+</a> 1 )
 <b>by </b><i>, , , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E19:7_20_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 1 &amp; 2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font> <a href="hidden.html#R1">=</a> 1 &amp; 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><b>hence </b><a NAME="E20:7_20_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_20_1_1_3"><b>suppose </b></a><a NAME="E1:7_20_1_1_3"/><i><font color="Green">E176</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> &amp; not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> )
 ;<br/><div class="add"><a NAME="E2:7_20_1_1_3"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">x1'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:7_20_1_1_3"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">x1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>and </b><br/><a NAME="E5:7_20_1_1_3"/>
( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1'</font> &amp; <font color="Maroon">x1'</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><a NAME="E6:7_20_1_1_3"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E7:7_20_1_1_3"/><b>then </b><i><font color="Green">E178</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:7_20_1_1_3"/><b>then </b>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">x2'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:7_20_1_1_3"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">x2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>and </b><br/><a NAME="E11:7_20_1_1_3"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2'</font> &amp; <font color="Maroon">x2'</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><i><font color="Green">E180</font></i>: <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font></span>)</span> = 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
<b>by </b><i>, <a class="ref" href="topmetr2.html#T1" target="_self">Th1</a>, <a class="txt" href="topmetr2.html#E26:7"><i><font color="Green">E68</font></i></a>, </i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="topmetr2.html#E1:7"><i><font color="Green">E21</font></i></a>, , </i>
;<br/><a NAME="E13:7_20_1_1_3"/><i><font color="Green">E181</font></i>: 
( 2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 <b>by </b><i>, , , , <a class="txt" href="topmetr2.html#E26:7"><i><font color="Green">E68</font></i></a>, </i>;<br/><a NAME="E14:7_20_1_1_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E15:7_20_1_1_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="txt" href="topmetr2.html#E28:7"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E16:7_20_1_1_3"/><b>then </b><i><font color="Green">E182</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E17:7_20_1_1_3"/>
( 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E18:7_20_1_1_3"/><b>then </b>
( <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R1">=</a> 0 <a href="real_1.html#K3">+</a> 1 )
 <b>by </b><i>, , , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E19:7_20_1_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 1 &amp; 2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font> <a href="hidden.html#R1">=</a> 1 &amp; 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><b>hence </b><a NAME="E20:7_20_1_1_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_20_1_1_4"><b>suppose </b></a><a NAME="E1:7_20_1_1_4"/><i><font color="Green">E183</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> )
 ;<br/><div class="add"><a NAME="E2:7_20_1_1_4"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">x1'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:7_20_1_1_4"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">x1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>and </b><br/><a NAME="E5:7_20_1_1_4"/>
( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1'</font> &amp; <font color="Maroon">x1'</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><a NAME="E6:7_20_1_1_4"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">x2'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:7_20_1_1_4"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">x2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>and </b><br/><a NAME="E9:7_20_1_1_4"/>
( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2'</font> &amp; <font color="Maroon">x2'</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><a NAME="E10:7_20_1_1_4"/><i><font color="Green">E186</font></i>: 
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 <b>by </b><i>, , , <a class="txt" href="topmetr2.html#E26:7"><i><font color="Green">E68</font></i></a></i>;<br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> = 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i>, <a class="txt" href="topmetr2.html#E1:7"><i><font color="Green">E21</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="topmetr2.html#E1:7"><i><font color="Green">E21</font></i></a>, , <a class="txt" href="topmetr2.html#E26:7"><i><font color="Green">E68</font></i></a></i>
;<br/><a NAME="E12:7_20_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E13:7_20_1_1_4"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1'</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">x2'</font>
 ;<br/><b>hence </b><a NAME="E14:7_20_1_1_4"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="txt" href="topmetr2.html#E26:7"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:7_20"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E124:7"/><b>then </b>
<font color="Maroon">h</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/>
<b>hence </b><a NAME="E125:7"/>
<font color="Maroon">h'</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, , , , <a class="ref" href="compts_1.html#T26">COMPTS_1:26</a></i>;<br/>

<a NAME="E126:7"/><i><font color="Green">E187</font></i>: 
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_21"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_21"/>
0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 ;<br/><a NAME="E2:7_21"/><b>then </b>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> 0 &amp; 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>hence </b><a NAME="E3:7_21"/>
contradiction
 ;<br/></div><b>end;</b></div>
<a NAME="E128:7"/><b>then </b>
( 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> &amp; not 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> )
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>hence </b><font color="Maroon">h'</font> <a href="funct_1.html#K1">.</a> 0 = 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="topmetr2.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0

;<br/><br/>

<a NAME="E130:7"/><i><font color="Green">E188</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<b>hence </b><font color="Maroon">h'</font> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="topmetr2.html#E1:7"><i><font color="Green">E21</font></i></a>, </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1

;<br/><br/>


</div>
