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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:16_1"/><i><font color="Green">E15</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:16_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_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<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) &amp; <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>() )
 <b>by </b><i><a class="txt" href="topgen_3.html#E2:16"><i><font color="Green">E12</font></i></a>, <a class="txt" href="topgen_3.html#E2:16_1"><i><font color="Green">E15</font></i></a></i>;<br/><a NAME="E3:16_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></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<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_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<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>(), <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="topgen_3.html#E2:16_1"><i><font color="Green">E15</font></i></a>, <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">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></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><a class="txt" href="topgen_3.html#E2:16_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="topgen_3.html#E1:16"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E7:16_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></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<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>4</sub></font>)
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:16_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:16_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E2:16"><i><font color="Green">E12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:16_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</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> <a href="subset_1.html#K9">/\</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>2</sub></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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></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<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>4</sub></font>) &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>5</sub></font>) &amp; <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>) <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="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:16_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:16_1_3"/>
<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="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</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="topgen_3.html#E3:16"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:16_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() holds <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>3</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> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></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<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>4</sub></font>) &amp; <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>) <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>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:16_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:16_1_4"/>
<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>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E4:16"><i><font color="Green">E14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></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<sub>3</sub></font></span>)</span> #) as  <a href="pre_topc.html#V1">strict</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i><a class="txt" href="topgen_3.html#E6:16_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="topgen_3.html#E7:16_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="topgen_3.html#E8:16_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="topgen_3.html#E9:16_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="topgen_3.html#T9" target="_self">Th9</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E11:16_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>

<b>thus </b> <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="topgen_3.html#E6:16_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="topgen_3.html#E7:16_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="topgen_3.html#E8:16_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="topgen_3.html#E9:16_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="topgen_3.html#T9" target="_self">Th9</a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>5</sub></font>)
<b>by </b><i><a class="txt" href="topgen_3.html#E2:16_1"><i><font color="Green">E15</font></i></a></i>
;<br/>


</div>
