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

<b>let </b><font color="Maroon">P</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">Q</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><a NAME="E1:28"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">P</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:28"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<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="E4:28"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</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="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</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#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan5a.html#T1" target="_self">Th1</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">P</font></span>)</span><b> such that </b><br/><a NAME="E7:28"/><i><font color="Green">E39</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/>
<b>reconsider </b><font color="Maroon">PP</font> = <font color="Maroon">P</font> as   <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/>
<a NAME="E9:28"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">PP</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:28"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E11:28"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i>, , <a class="ref" href="compts_1.html#T18">COMPTS_1:18</a></i>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> ;<br/>
<a NAME="E12:28"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T70">TOPS_2:70</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E14:28"/><i><font color="Green">E53</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> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:28"/><i><font color="Green">E54</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>  &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <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">P</font></span>)</span> &amp; <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="E16:28"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  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="E17:28"/>
 <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">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">PQ</font> = <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</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 <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">P</font></span>)</span> <b>by </b><i>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E19:28"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp; <font color="Maroon">PQ</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</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">P</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">P</font>, <font color="Maroon">Q1</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 <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">PQ1</font> = <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</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 <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="jordan5a.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E22:28"/><i><font color="Green">E59</font></i>: 
<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">P1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</font></span>)</span> <a href="hidden.html#R1">=</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> <a href="pre_topc.html#K3">|</a> <font color="Maroon">PQ1</font>
 
<b>by </b><i><a class="ref" href="goboard9.html#T4">GOBOARD9:4</a></i>;<br/>
<a NAME="E23:28"/>
<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">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span> is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i>, <a class="ref" href="jordan5a.html#T2" target="_self">Th2</a>, <a class="ref" href="compts_1.html#T12">COMPTS_1:12</a></i>;<br/>
<a NAME="E24:28"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">PQ</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i>, <a class="ref" href="compts_1.html#T12">COMPTS_1:12</a></i>;<br/>
<a NAME="E25:28"/><i><font color="Green">E61</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E26:28"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">PQ</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 
;<br/>
<a NAME="E27:28"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">PQ</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K3">R^1</a> 
 
<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">GPQ</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">PQ</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  ;<br/>
<a NAME="E29:28"/>
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>   st <font color="Olive">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">GPQ</font> holds <br/><font color="Olive">P</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E2"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="jordan5a.html#T3" target="_self">Th3</a>, <a class="ref" href="weierstr.html#T14">WEIERSTR:14</a></i>;<br/>
<a NAME="E30:28"/><b>then </b>
<font color="Maroon">GPQ</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i>, <a class="ref" href="compts_1.html#T11">COMPTS_1:11</a></i>;<br/>
<a NAME="E31:28"/><b>then </b><i><font color="Green">E64</font></i>: 
(  <a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font> is <a href="seq_4.html#V3">bounded</a> &amp;  <a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font> is <a href="rcomp_1.html#V2">closed</a> )
 
<b>by </b><i><a class="ref" href="weierstr.html#T17">WEIERSTR:17</a>, <a class="ref" href="weierstr.html#T18">WEIERSTR:18</a></i>;<br/>
<a NAME="E32:28"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">OO</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E34:28"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">OO</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PQ</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E35:28"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <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#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E36:28"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">OO</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GPQ</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E4:28"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>set </b><font color="Maroon">Ex</font> =  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font></span>)</span>;<br/>
<b>take </b><font color="Maroon">EX</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <span class="p2">(<span class="default"><a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font></span>)</span></span>)</span>;<br/>

<a NAME="E37:28"/>
(  <a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font> is <a href="rcomp_1.html#V2">closed</a> &amp;  <a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i>, <a class="txt" href="jordan5a.html#E7:28"><i><font color="Green">E39</font></i></a>, <a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a>, <a class="ref" href="weierstr.html#D3">WEIERSTR:def 3</a></i>;<br/>
<a NAME="E38:28"/><b>then </b>
 <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T31">RCOMP_1:31</a></i>;<br/>
<a NAME="E39:28"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">GPQ</font>
 
<b>by </b><i><a class="ref" href="weierstr.html#D3">WEIERSTR:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">EX</font> = <font color="Maroon">EX</font> as   <a href="pre_topc.html#NM2">Point</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">P</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E42:28"/><i><font color="Green">E78</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <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#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E43:28"/><i><font color="Green">E79</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <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">P</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E44:28"/>
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T63">TOPS_2:63</a></i>;<br/>
<a NAME="E45:28"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E46:28"/><b>then </b><i><font color="Green">E82</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K2">"</a>  &amp; <font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T63">TOPS_2:63</a>, <a class="ref" href="tops_2.html#T64">TOPS_2:64</a></i>;<br/>
<b>consider </b><font color="Maroon">pq</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E48:28"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">pq</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> &amp; <font color="Maroon">pq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PQ</font> &amp;  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">pq</font> )
 <b>by </b><i><a class="ref" href="jordan5a.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E49:28"/><i><font color="Green">E93</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> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <span class="p2">(<span class="default"><a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">EX</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font></span>)</span> &amp;  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="jordan5a.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E50:28"/>
for <font color="Olive">t</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font></span>)</span> holds <br/> not <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green">E94</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font></span>)</span> )
 ;<br/>

<a NAME="E2:28_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">t</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="E3:28_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">t</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="txt" href="jordan5a.html#E9:28"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<a NAME="E4:28_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">t</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="jordan5a.html#E1:28"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan5a.html#E9:28"><i><font color="Green">E46</font></i></a>, , </i>;<br/>
<b>assume </b><a NAME="E5:28_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E6:28_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</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> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E10:28"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E7:28_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E8:28_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PQ</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:28_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">GPQ</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E10:28_1"/>
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E11:28_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="txt" href="jordan5a.html#E11:28"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/>
<a NAME="E12:28_1"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3">[#]</a> <font color="Maroon">GPQ</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E12:28"><i><font color="Green">E51</font></i></a>, <a class="ref" href="weierstr.html#D3">WEIERSTR:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:28_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan5a.html#E9:28"><i><font color="Green">E46</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E51:28"/>
( <font color="Maroon">EX</font> is   <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> &amp; <font color="Maroon">EX</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> &amp;  ex <font color="Olive">g</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">P</font></span>)</span> ex <font color="Olive">s2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">g</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">EX</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s2</font> &amp; <font color="Olive">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; ( for <font color="Olive">t</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s2</font> holds <br/> not <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> ) ) )
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/>


</div>
