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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="fintopo2.html#L1" target="_self">FMT_Space_Str</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:59"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="fintopo2.html#V2" target="_self">Fo_filled</a>
 ;<br/>

<b>assume </b><a NAME="E2:59"/><i><font color="Green">E25</font></i>: 
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 <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<a NAME="E3:59"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> <a href="fintopo2.html#K10" target="_self">^Foi</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:59_1"/>
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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="fintopo2.html#K10" target="_self">^Foi</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><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>1</sub></font>;<br/>

<a NAME="E1:59_1_1"/><i><font color="Green">E27</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo2.html#E2:59"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:59_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:59_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="subset_1.html#T63">SUBSET_1:63</a></i>;<br/>
<b>hence </b><a NAME="E4:59_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="fintopo2.html#K10" target="_self">^Foi</a> 
 <b>by </b><i><a class="txt" href="fintopo2.html#E1:59_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:59_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="fintopo2.html#K10" target="_self">^Foi</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:59"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="fintopo2.html#K10" target="_self">^Foi</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:59_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="fintopo2.html#K10" target="_self">^Foi</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="fintopo2.html#K10" target="_self">^Foi</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="fintopo2.html#E1:59"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fintopo2.html#E3:59"><i><font color="Green">E26</font></i></a>, <a class="ref" href="fintopo2.html#T37" target="_self">Th37</a></i>;<br/>
<b>hence </b><a NAME="E2:59_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="fintopo2.html#K10" target="_self">^Foi</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:59"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fintopo2.html#K10" target="_self">^Foi</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="fintopo2.html#K10" target="_self">^Foi</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="fintopo2.html#K10" target="_self">^Foi</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E6:59"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fintopo2.html#K10" target="_self">^Foi</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fintopo2.html#K10" target="_self">^Foi</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fintopo2.html#K10" target="_self">^Foi</a> 
 ;<br/>


</div>
