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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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="E1:24"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></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="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM7">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">c<sub>4</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:24"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="tops_2.html#V3">being_homeomorphism</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E1:24"><i><font color="Green">E22</font></i></a>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E5:24"/><i><font color="Green">E24</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></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">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E4:24"><i><font color="Green">E23</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E7:24"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E8:24"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="jordan7.html#E4:24"><i><font color="Green">E23</font></i></a>, <a class="ref" href="jordan7.html#T15" target="_self">Th15</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E9:24"/>
( <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="jordan7.html#E8:24"><i><font color="Green">E26</font></i></a></i>;<br/>

<b>thus </b><a NAME="E10:24"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:24"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan7.html#E7:24"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>

<b>thus </b><a NAME="E11:24"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E4:24"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan7.html#E7:24"><i><font color="Green">E25</font></i></a></i>;<br/>


</div>
