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

<b>let </b><font color="Maroon">FT1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">FT2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">FT1</font>,<font color="Maroon">FT2</font>;<br/>



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

<a NAME="E2:4"/><b>then </b><i><font color="Green">E24</font></i>: 
( <font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">h</font> is <a href="funct_2.html#V2">onto</a> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font> holds  <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT1</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E3:4"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT2</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">g2</font> = <font color="Maroon">h</font> <a href="funct_1.html#K2">"</a>  as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">FT2</font>,<font color="Maroon">FT1</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T31">FUNCT_2:31</a></i>;<br/>
<a NAME="E5:4"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">g2</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>, </i>;<br/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="funct_1.html#T55">FUNCT_1:55</a>, </i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT1</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">E36</font></i>: 
<font color="Maroon">g2</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">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font> holds  <font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT2</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K1">}</a></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">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font>;<br/>

<a NAME="E1:4_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a>, </i>;<br/>
<a NAME="E2:4_2"/><b>then </b>
<font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT1</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:4_2"/>
<font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT2</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:4"/><b>then </b>
<font color="Maroon">g2</font> <a href="fintopo5.html#R1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E10:4"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">FT2</font>,<font color="Maroon">FT1</font> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K2">"</a>  &amp; <font color="Olive">g</font> <a href="fintopo5.html#R1" target="_self">is_homeomorphism</a>  )
 ;<br/>


</div>
