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<b>let </b><font color="Maroon">P</font> be  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>;<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>, <font color="Maroon">q1</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">q2</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>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">s1</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green">E29</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E3:26"/>
 ex <font color="Olive">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> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Olive">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  )
 
<b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, , </i>;<br/>
<b>hence </b><a NAME="E4:26"/>
(  not <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> or  not <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> or  not <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> or  not 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s1</font> or  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 or  not <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> or  not 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 or  not  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> or <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> )
 <b>by </b><i><a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>


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