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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>thus </b><a NAME="E1:7"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> implies for <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>thus </b><a NAME="E2:7_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:7_1_1_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:7_1_1_1"/><b>then </b><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</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="pzfmisc1.html#E1:7_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/><a NAME="E3:7_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="pzfmisc1.html#E1:7_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="pzfmisc1.html#E1:7_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="pzfmisc1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E4:7_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <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="pzfmisc1.html#E2:7_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:7_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></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="E4:7_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E5:7_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="pzfmisc1.html#E1:7_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="pzfmisc1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E6:7_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</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="pzfmisc1.html#E3:7_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:7"/><i><font color="Green">E4</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E3:7"/><b>then </b><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:7_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:7_2_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:7_2_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_2_1_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:7_2_1_1_1"/><i><font color="Green">E8</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></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="E3:7_2_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="pzfmisc1.html#E1:7_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="pzfmisc1.html#T1" target="_self">Th1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:7_2_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></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">E9</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="pzfmisc1.html#E2:7_2_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E7:7_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></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="E1:7_2_1_1_1_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_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="E1:7_2_1_1_1_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:7_2_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_2_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="pzfmisc1.html#E1:7_2_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pzfmisc1.html#E6:7_2_1_1_1"><i><font color="Green">E9</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:7_2_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:7_2_1_1_1_2_1_2"/><b>then </b>
not <font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pzfmisc1.html#E2:7_2_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:7_2_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E4:7_2_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="pzfmisc1.html#E3:7"><i><font color="Green">E5</font></i></a>, <a class="txt" href="pzfmisc1.html#E1:7_2_1_1_1_2"><i><font color="Green">E10</font></i></a>, <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>
<b>hence </b><a NAME="E8:7_2_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pzfmisc1.html#E2:7"><i><font color="Green">E4</font></i></a>, <a class="txt" href="pzfmisc1.html#E6:7_2_1_1_1"><i><font color="Green">E9</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:7_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>hence </b><a NAME="E3:7_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pzfmisc1.html#E3:7"><i><font color="Green">E5</font></i></a>, <a class="txt" href="pzfmisc1.html#E1:7_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:7_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></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>hence </b><a NAME="E4:7_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pzfmisc1.html#E1:7_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="pzfmisc1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:7"/>
<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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