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<b>let </b><font color="Maroon">I</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>, <font color="Maroon">M</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">SF</font> be  <a href="relat_1.html#V2">V5</a> <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">M</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:76"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">A</font> <a href="pboole.html#R1">in</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E2:76"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">B</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>  st <font color="Olive">B</font> <a href="pboole.html#R1">in</a> <font color="Maroon">SF</font> holds <br/><font color="Maroon">A</font> <a href="pboole.html#R1">in</a> <font color="Olive">B</font>
 ;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D4">PBOOLE:def 4</a><br/>

<b>assume </b><a NAME="E3:76"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Q</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span><b> such that </b><br/><a NAME="E5:76"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E6:76"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/>
<a NAME="E7:76"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">T</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E9:76"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">T</font> <a href="pboole.html#R1">in</a> <font color="Maroon">SF</font>
 <b>by </b><i><a class="ref" href="pboole.html#T146">PBOOLE:146</a></i>;<br/>
<a NAME="E10:76"/>
for <font color="Olive">B'</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> holds <br/><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">B'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B'</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:76_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">B'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E2:76_1"/><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">B'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E3:76_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">B'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i>, <a class="ref" href="pzfmisc1.html#T1">PZFMISC1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">K</font> = <font color="Maroon">T</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">B'</font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:76_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E67</font></i>: <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">B'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">B'</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E7:76_1"/>
<font color="Maroon">K</font> <a href="pboole.html#R1">in</a> <font color="Maroon">SF</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D4">PBOOLE:def 4</a><br/>

<b>assume </b><a NAME="E1:76_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_2_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> or <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_1_1"><b>suppose </b></a><a NAME="E1:76_1_2_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:76_1_2_1_1"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_1_2"><b>suppose </b></a><a NAME="E1:76_1_2_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:76_1_2_1_2"/><b>then </b>
not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">B'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:76_1_2_1_2"/><b>then </b>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E4:76_1_2_1_2"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
 <b>by </b><i>, , <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E8:76_1"/><b>then </b>
<font color="Maroon">A</font> <a href="pboole.html#R1">in</a> <font color="Maroon">K</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:76_1"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font>
 <b>by </b><i>, , <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:76"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>


</div>
