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

<b>set </b><font color="Maroon">X</font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:16_1"/><i><font color="Green">E46</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">a</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:16_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><a NAME="E2:16_1_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>) &amp; <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>() )
 <b>by </b><i>, </i>;<br/><a NAME="E3:16_1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E4:16_1_1"/>
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>(), <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<a NAME="E5:16_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
;<br/>
<a NAME="E6:16_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:16_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E1:16_1_2"/>
<font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">A</font>)
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:16_1_2"/>
<font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E8:16_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E1:16_1_3"/>
( <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">A</font>) &amp; <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">B</font>) &amp; <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span>) <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:16_1_3"/>
<font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E9:16_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E1:16_1_4"/>
( <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">A</font>) &amp; <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span>) <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:16_1_4"/>
<font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">T</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><a href="topgen_3.html#K1" target="_self">rng</a> <font color="Maroon">c</font></span>)</span> #) as  <a href="pre_topc.html#V1">strict</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i>, , , , </i>;<br/>
<b>take </b>
<font color="Maroon">T</font>
;<br/>

<b>thus </b><a NAME="E11:16_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>thus </b> <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> = 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
<b>by </b><i>, , , , </i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">A</font>)
<b>by </b><i/>
;<br/>


</div>
