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

<b>set </b><font color="Maroon" title="c1">X</font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>consider </b><font color="Maroon" title="c2">c</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:13_1"/><i><font color="Green" title="E13">A5</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">a</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:13_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><a NAME="E2:13_1_1"/><b>then </b>
( <font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>) &amp; <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>() )
 <b>by </b><i><a class="txt" href="topgen_3.html#E2:13"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="topgen_3.html#E2:13_1"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E3:13_1_1"/>
<font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">c</font> = <font color="Maroon" title="c2">c</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon">F<sub>1</sub></font>(), <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="topgen_3.html#E2:13_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/>
<a NAME="E5:13_1"/>
 <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E6:13_1"/><b>then </b><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:13_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="topgen_3.html#E1:13"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<a NAME="E7:13_1"/><i><font color="Green" title="E15">A7</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E1:13_1_2"/>
<font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">A</font>)
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:13_1"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:13_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E2:13"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:13_1"/><i><font color="Green" title="E16">A8</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">B</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E1:13_1_3"/>
( <font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">A</font>) &amp; <font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">B</font>) &amp; <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">B</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:13_1"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:13_1_3"/>
<font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:13"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:13_1"/><i><font color="Green" title="E17">A9</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E1:13_1_4"/>
( <font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">A</font>) &amp; <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">A</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:13_1"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:13_1_4"/>
<font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E4:13"><i><font color="Green" title="E12">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c4">T</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <span class="p2">(<span class="default"><a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span> #) as  <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> <b>by </b><i><a class="txt" href="topgen_3.html#E6:13_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="topgen_3.html#E7:13_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="topgen_3.html#E8:13_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="topgen_3.html#E9:13_1"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="topgen_3.html#T7" target="_self" title="TOPGEN_3:th.7">Th7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">T</font>
;<br/>

<b>thus </b><a NAME="E11:13_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>let </b><font color="Maroon" title="c5">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T</font>;<br/>

<b>thus </b> <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">A</font> = 
<font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="txt" href="topgen_3.html#E6:13_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="topgen_3.html#E7:13_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="topgen_3.html#E8:13_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="topgen_3.html#E9:13_1"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="topgen_3.html#T7" target="_self" title="TOPGEN_3:th.7">Th7</a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">A</font>)
<b>by </b><i><a class="txt" href="topgen_3.html#E2:13_1"><i><font color="Green" title="E13">A5</font></i></a></i>
;<br/>


</div>
