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

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">a<sub>1</sub></font>,<a href="arytm_3.html#K13">{}</a> ,<font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:53_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="arytm_3.html#K13">{}</a> ) <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/>
<a NAME="E2:53_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> holds <br/> ( <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><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">X</font>;<br/>

<a NAME="E1:53_1_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  )
 
;<br/>
<a NAME="E2:53_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> or <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:53_1_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font>)
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>

<a NAME="E4:53_1_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E5:53_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:53_1_1"/><i><font color="Green">E44</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">X</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">B</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_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">X</font>, <font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_2_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or ( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_2_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_2_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">B</font>)
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_2_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:53_1_1_2_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/><a NAME="E3:53_1_1_2_1_2"/><b>then </b>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">B</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E4:53_1_1_2_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">B</font>)
 <b>by </b><i><a class="ref" href="xboole_1.html#T5">XBOOLE_1:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:53_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>)) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_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">X</font>;<br/>

<a NAME="E1:53_1_1_3"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  )
 
;<br/>
<a NAME="E2:53_1_1_3"/><b>then </b>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font>) <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or ( <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/>
<a NAME="E3:53_1_1_3"/><b>then </b>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font>)) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font>) or ( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font>)) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:53_1_1_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font>)) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font>)
 <b>by </b><i><a class="ref" href="xboole_1.html#T6">XBOOLE_1:6</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">T</font> being  <a href="pre_topc.html#V1">strict</a> <a href="pre_topc.html#NM1">TopSpace</a><b> such that </b><br/><a NAME="E6:53_1_1"/><i><font color="Green">E48</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> holds   <a href="pre_topc.html#K4">Cl</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) ) )
 <b>from </b><i><a class="ref" href="topgen_3.html#S2" target="_self">TOPGEN_3:sch 2</a>(, <a class="ref" href="topgen_3.html#T2" target="_self">Th2</a>, , );<br/></i>
<b>take </b>
<font color="Maroon">T</font>
;<br/>

<b>thus </b><a NAME="E7:53_1_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">X</font>
 <b>by </b><i/>;<br/>

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

<b>thus </b><a NAME="E8:53_1_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">F</font>,<a href="arytm_3.html#K13">{}</a> ,<font color="Maroon">F</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div>
