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

<b>let </b><font color="Maroon">P1</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">P2</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">P3</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">P1</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">P2</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E3:13"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">P3</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E4:13"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P3</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span>
 <b>and </b><br/><a NAME="E5:13"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font>
 ;<br/>

<b>set </b><font color="Maroon">P</font> = <font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font>;<br/>
<a NAME="E6:13"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">P2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E7:13"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">P3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:13"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i>, <a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E10:13"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P3</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E11:13"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P3</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span><b> such that </b><br/><a NAME="E13:13"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<i><font color="Green">E76</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E15:13"/><i><font color="Green">E77</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="E16:13"/><i><font color="Green">E78</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="E17:13"/><i><font color="Green">E79</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E18:13"/><b>then </b><i><font color="Green">E80</font></i>: 
 <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> 
 
;<br/>
<a NAME="E19:13"/><i><font color="Green">E81</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></i>;<br/>
<i><font color="Green">E82</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">P1</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">U2</font> = <font color="Maroon">P2</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E22:13"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">U2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E23:13"/>
<font color="Maroon">P2</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, <a class="ref" href="jordan6.html#T12">JORDAN6:12</a></i>;<br/>
<a NAME="E24:13"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">U2</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, <a class="ref" href="jordan6.html#T3">JORDAN6:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">U3</font> = <font color="Maroon">P3</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E26:13"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">U3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P3</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E27:13"/>
<font color="Maroon">P3</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, <a class="ref" href="jordan6.html#T12">JORDAN6:12</a></i>;<br/>
<a NAME="E28:13"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">U3</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="jordan16.html#E1:13"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan6.html#T3">JORDAN6:3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_3"/>
( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 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 holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P3</font> or  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 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 &amp; not <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P3</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3_1"><b>suppose </b></a><a NAME="E1:13_3_1"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 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 holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P3</font>
 ;<br/><div class="add"><a NAME="E2:13_3_1"/>
<font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_1_1"><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="E1:13_3_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</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="E3:13_3_1_1"/><i><font color="Green">E92</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">y</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/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, </i>;<br/>
<a NAME="E5:13_3_1_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 )
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E6:13_3_1_1"/><b>then </b>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 1 or ( 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 ) )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E7:13_3_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P3</font>
 <b>by </b><i>, , , , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:13_3_1"/>
( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font> or <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P3</font> )
 <b>by </b><i>, , <a class="ref" href="jordan6.html#T59">JORDAN6:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3_2"><b>suppose </b></a><a NAME="E1:13_3_2"/><i><font color="Green">E93</font></i>: 
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 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 &amp; not <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P3</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_3_2_1_1"/>
( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 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 holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> or  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 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 &amp; not <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3_2_1_1_1"><b>case </b></a><a NAME="E1:13_3_2_1_1_1"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 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 holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font>
 ;<br/><div class="add"><a NAME="E2:13_3_2_1_1_1"/>
<font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_2_1_1_1_1"><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="E1:13_3_2_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</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="E3:13_3_2_1_1_1_1"/><i><font color="Green">E95</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">y</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/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, </i>;<br/>
<a NAME="E5:13_3_2_1_1_1_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 )
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E6:13_3_2_1_1_1_1"/><b>then </b>
( ( 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 ) or <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E7:13_3_2_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font>
 <b>by </b><i>, , , , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:13_3_2_1_1_1"/>
( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font> or <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P3</font> )
 <b>by </b><i>, , <a class="ref" href="jordan6.html#T59">JORDAN6:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3_2_1_1_2"><b>case </b></a><a NAME="E1:13_3_2_1_1_2"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 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 &amp; not <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:13_3_2_1_1_2"/><i><font color="Green">E97</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;</a> 1 &amp; not <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> )
 ;<br/><b>consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:13_3_2_1_1_2"/><i><font color="Green">E99</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> 1 &amp; not <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P3</font> )
 <b>by </b><i/>;<br/><a NAME="E6:13_3_2_1_1_2"/>
<font color="Maroon">r2</font> <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#T48">RCOMP_1:48</a></i>;<br/><a NAME="E7:13_3_2_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:13_3_2_1_1_2"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P3</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:13_3_2_1_1_2"/>
<font color="Maroon">r1</font> <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#T48">RCOMP_1:48</a></i>;<br/><a NAME="E10:13_3_2_1_1_2"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E11:13_3_2_1_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_3_2_1_1_2_1_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> or <font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:13_3_2_1_1_2_1_1_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_2_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3_2_1_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_3_2_1_1_2_1_1_1_1_1"/>
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> or <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> )
 </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="13_3_2_1_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:13_3_2_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:13_3_2_1_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3_2_1_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:13_3_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><i><font color="Green">E105</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font>
<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"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span>

<br/>.= 
<font color="Maroon">P1</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E4:13_3_2_1_1_2_1_1_1_1_1_2"/><b>then </b>
 <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"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">g</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"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E6:13_3_2_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><a NAME="E7:13_3_2_1_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T5">TOPMETR:5</a></i>;<br/><a NAME="E8:13_3_2_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E106</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="E9:13_3_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">U2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">U3</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#D2">TOPMETR:def 2</a></i>;<br/><a NAME="E10:13_3_2_1_1_2_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T20">TOPMETR:20</a></i>;<br/><b>then consider </b><font color="Maroon">r0</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E12:13_3_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r0</font> &amp; <font color="Maroon">r0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U3</font> )
 <b>by </b><i>, <a class="txt" href="jordan16.html#E2:13"><i><font color="Green">E29</font></i></a>, , , , , , , , <a class="ref" href="topmetr3.html#T17">TOPMETR3:17</a></i>;<br/><a NAME="E13:13_3_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> or <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E14:13_3_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green">E111</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r0</font>
 <b>by </b><i>, </i>;<br/><a NAME="E15:13_3_2_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon">r0</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E16:13_3_2_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">r0</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="jordan16.html#T6" target="_self">Th6</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E17:13_3_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green">E113</font></i>: 
0 <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="E18:13_3_2_1_1_2_1_1_1_1_1_2"/>
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#T48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E19:13_3_2_1_1_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="jordan16.html#T5" target="_self">Th5</a>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:13_3_2_1_1_2_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_3_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:13_3_2_1_1_2_1_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><i><font color="Green">E115</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font>
<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"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span>

<br/>.= 
<font color="Maroon">P1</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E4:13_3_2_1_1_2_1_1_2"/><b>then </b>
 <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"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">g</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"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E6:13_3_2_1_1_2_1_1_2"/>
<font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><a NAME="E7:13_3_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T5">TOPMETR:5</a></i>;<br/><a NAME="E8:13_3_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E116</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="E9:13_3_2_1_1_2_1_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">U2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">U3</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#D2">TOPMETR:def 2</a></i>;<br/><a NAME="E10:13_3_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P3</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T20">TOPMETR:20</a></i>;<br/><b>then consider </b><font color="Maroon">r0</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E12:13_3_2_1_1_2_1_1_2"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r0</font> &amp; <font color="Maroon">r0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">U3</font> )
 <b>by </b><i>, <a class="txt" href="jordan16.html#E2:13"><i><font color="Green">E29</font></i></a>, , , , , , , , <a class="ref" href="topmetr3.html#T17">TOPMETR3:17</a></i>;<br/><a NAME="E13:13_3_2_1_1_2_1_1_2"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> or <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E14:13_3_2_1_1_2_1_1_2"/><i><font color="Green">E120</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r0</font>
 <b>by </b><i>, </i>;<br/><a NAME="E15:13_3_2_1_1_2_1_1_2"/>
<font color="Maroon">r0</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E16:13_3_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">r0</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="jordan16.html#T5" target="_self">Th5</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E17:13_3_2_1_1_2_1_1_2"/>
( 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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#T48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E18:13_3_2_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, , , , , , <a class="ref" href="jordan16.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:13_3_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:13_3_2"/>
( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font> or <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P3</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
