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

<b>assume </b><a NAME="E1:124"/><i><font color="Green">E34</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">sn</font> &amp; <font color="Maroon">sn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/>

<a NAME="E2:124"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">sn</font> <a href="jgraph_4.html#K7" target="_self">-FanMorphE</a>  is   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">sn</font> <a href="jgraph_4.html#K7" target="_self">-FanMorphE</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"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="jgraph_4.html#D1" target="_self">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">sn</font> <a href="jgraph_4.html#K7" target="_self">-FanMorphE</a>  as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:124"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">h</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">sn</font> <a href="jgraph_4.html#K7" target="_self">-FanMorphE</a>  &amp; <font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i>, <a class="ref" href="jgraph_4.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E6:124"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:124"/><i><font color="Green">E40</font></i>: 
 <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"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_4.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E8:124"/>
for <font color="Olive">p2</font> being  <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>  ex <font color="Olive">K</font> being 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 <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">K</font> &amp;  ex <font color="Olive">V2</font> being  <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> st <br/>( <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">V2</font> &amp; <font color="Olive">V2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">V2</font> <a href="tarski.html#R1">c=</a> <font color="Olive">K</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">V2</font> ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:124"/><b>then </b>
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="jgraph_4.html#T19" target="_self">Th19</a>, , , </i>;<br/>
<b>hence </b><a NAME="E10:124"/>
 ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">sn</font> <a href="jgraph_4.html#K7" target="_self">-FanMorphE</a>  &amp; <font color="Olive">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  )
 ;<br/>


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