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

<a NAME="E1:67_1_1"/><i><font color="Green">E32</font></i>: 
( 0 is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; 1 <a href="real_1.html#K6">/</a> 2 is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; 1 is   <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#T86">BORSUK_1:86</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A</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  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_4.html#T49">BORSUK_4:49</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</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  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_6.html#T9" target="_self">Th9</a>, <a class="ref" href="borsuk_4.html#T49">BORSUK_4:49</a></i>;<br/>
<b>set </b><font color="Maroon">T1</font> = <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">T2</font> = <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font>;<br/>
<b>set </b><font color="Maroon">T</font> =  <a href="topmetr.html#K5">I[01]</a> ;<br/>
<a NAME="E4:67_1_1"/>
<a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1
 
<b>by </b><i><a class="ref" href="topmetr.html#T31">TOPMETR:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> =  <a href="borsuk_6.html#K1" target="_self">L[01]</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,0,1 as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a>  <b>by </b><i>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> <a href="borsuk_1.html#K3">--&gt;</a> <a href="borsuk_1.html#K21">0[01]</a>  as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a>  ;<br/>
<a NAME="E7:67_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> <a href="borsuk_1.html#K1">--&gt;</a> 0
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D2">BORSUK_1:def 2</a>, <a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a></i>;<br/>
<i><font color="Green">E34</font></i>: <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> = 
<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="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</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> <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="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
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="treal_1.html#T2">TREAL_1:2</a></i>
<br/>.= 
 <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="E9:67_1_1"/>
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:67_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:67_1_1_2"/><b>then </b>
<font color="Maroon">p</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">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E3:67_1_1_2"/><b>then </b>
<font color="Maroon">p</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="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E4:67_1_1_2"/><b>then </b>
<font color="Maroon">p</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><a class="ref" href="topmetr2.html#T1">TOPMETR2:1</a></i>;<br/>
<a NAME="E5:67_1_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">p</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="E6:67_1_1_2"/><b>then </b>
<font color="Maroon">p</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="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E7:67_1_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="borsuk_6.html#T13" target="_self">Th13</a>, <a class="txt" href="borsuk_6.html#E7:67_1_1_2"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="borsuk_6.html#T10" target="_self">Th10</a>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E9:67_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E11:67_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="ref" href="borsuk_6.html#T12" target="_self">Th12</a>, <a class="ref" href="borsuk_2.html#T1">BORSUK_2:1</a></i>;<br/>
<a NAME="E12:67_1_1"/>
 <a href="borsuk_6.html#K4" target="_self">2RP</a>  <a href="funct_2.html#R2">=</a> <font color="Maroon">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:67_1_1_3"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K5">I[01]</a>  holds  <a href="borsuk_6.html#K4" target="_self">2RP</a>  <a href="funct_2.html#K3">.</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="funct_4.html#K1">+*</a> <font color="Maroon">f</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="67_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<i><font color="Green">E62</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<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="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_3_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_1_3_1_2"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2 or <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_3_1_2_1"><b>suppose </b></a><a NAME="E1:67_1_1_3_1_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:67_1_1_3_1_2_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E3:67_1_1_3_1_2_1"/><b>then </b><i><font color="Green">E64</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>
 <b>by </b><i>, <a class="ref" href="borsuk_6.html#T14" target="_self">Th14</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E4:67_1_1_3_1_2_1"/>
1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><b>then </b><i><font color="Green">E65</font></i>: <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i><a class="ref" href="borsuk_6.html#T14" target="_self">Th14</a>, </i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1

;<br/><b>thus </b><a href="borsuk_6.html#K4" target="_self">2RP</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="borsuk_6.html#T14" target="_self">Th14</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_3_1_2_2"><b>suppose </b></a><a NAME="E1:67_1_1_3_1_2_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:67_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E88</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>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>thus </b><a href="borsuk_6.html#K4" target="_self">2RP</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="borsuk_6.html#T14" target="_self">Th14</a>, </i>
<br/>.= 
<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="borsuk_6.html#T14" target="_self">Th14</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_3_1_2_3"><b>suppose </b></a><a NAME="E1:67_1_1_3_1_2_3"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:67_1_1_3_1_2_3"/><b>then </b><i><font color="Green">E91</font></i>: 
not <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#T48">RCOMP_1:48</a></i>;<br/><a NAME="E3:67_1_1_3_1_2_3"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E4:67_1_1_3_1_2_3"/><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>
 <b>by </b><i><a class="ref" href="borsuk_6.html#T14" target="_self">Th14</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E5:67_1_1_3_1_2_3"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><b>thus </b><a href="borsuk_6.html#K4" target="_self">2RP</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
0
<b>by </b><i><a class="ref" href="borsuk_6.html#T14" target="_self">Th14</a>, </i>
<br/>.= 
<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="borsuk_6.html#T10" target="_self">Th10</a>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:67_1_1_3"/>
 <a href="borsuk_6.html#K4" target="_self">2RP</a>  <a href="funct_2.html#R2">=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="borsuk_6.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:67_1_1"/>
 <a href="borsuk_6.html#K4" target="_self">2RP</a>  is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="borsuk_6.html#T13" target="_self">Th13</a></i>;<br/>


</div>
