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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="fin_topo.html#L1">FT_Space_Str</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E1:4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="fintopo5.html#R1" target="_self">is_homeomorphism</a> 
 ;<br/>

<a NAME="E2:4"/><b>then </b><i><font color="Green">E4</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="funct_2.html#V2">onto</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default">the <a href="fin_topo.html#U1">Nbd</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fin_topo.html#U1">Nbd</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="fintopo5.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:4"/><b>then </b><i><font color="Green">E5</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K2">"</a>  as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="fintopo5.html#E2:4"><i><font color="Green">E4</font></i></a>, <a class="ref" href="funct_2.html#T31">FUNCT_2:31</a></i>;<br/>
<a NAME="E5:4"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#T62">FUNCT_1:62</a>, <a class="txt" href="fintopo5.html#E2:4"><i><font color="Green">E4</font></i></a></i>;<br/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T55">FUNCT_1:55</a>, <a class="txt" href="fintopo5.html#E2:4"><i><font color="Green">E4</font></i></a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E7:4"/><b>then </b><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E8:4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>4</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default">the <a href="fin_topo.html#U1">Nbd</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fin_topo.html#U1">Nbd</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E1:4_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E2:4"><i><font color="Green">E4</font></i></a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a>, <a class="txt" href="fintopo5.html#E3:4"><i><font color="Green">E5</font></i></a></i>;<br/>
<a NAME="E2:4_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default">the <a href="fin_topo.html#U1">Nbd</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fin_topo.html#U1">Nbd</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E2:4"><i><font color="Green">E4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:4_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default">the <a href="fin_topo.html#U1">Nbd</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fin_topo.html#U1">Nbd</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo5.html#E2:4"><i><font color="Green">E4</font></i></a>, <a class="ref" href="fintopo5.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:4"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="fintopo5.html#R1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="fintopo5.html#E5:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="fintopo5.html#E7:4"><i><font color="Green">E7</font></i></a>, <a class="ref" href="fintopo5.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E10:4"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K2">"</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="fintopo5.html#R1" target="_self">is_homeomorphism</a>  )
 ;<br/>


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