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<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="altcat_2.html#K1" target="_self">~</a> <font color="Maroon">c<sub>4</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E2:9_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="pboole.html#M3">ManySortedFunction</a> of  <a href="altcat_2.html#K1" target="_self">~</a> <font color="Maroon">c<sub>2</sub></font>, <a href="altcat_2.html#K1" target="_self">~</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:9_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_1_1_1"/><i><font color="Green">E1</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> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E4:9_1_1_1"/><i><font color="Green">E2</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/>
<a NAME="E5:9_1_1_1"/><i><font color="Green">E3</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E3:9_1_1_1"><i><font color="Green">E1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:9_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:9_1_1_1"/><b>then </b><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:9_1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="altcat_2.html#E5:9_1_1_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>;<br/>
<a NAME="E8:9_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:9_1_1_1"/><b>then </b><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K1" target="_self">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:9_1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="altcat_2.html#E5:9_1_1_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>;<br/>
<a NAME="E10:9_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E11:9_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K1" target="_self">~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:9_1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="altcat_2.html#E5:9_1_1_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>;<br/>
<b>hence </b><a NAME="E12:9_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="altcat_2.html#K1" target="_self">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="altcat_2.html#K1" target="_self">~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="altcat_2.html#E5:9_1_1_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="altcat_2.html#E7:9_1_1_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="altcat_2.html#E9:9_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:9_1_1"/>
 <a href="funct_4.html#K2">~</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="pboole.html#M3">ManySortedFunction</a> of  <a href="altcat_2.html#K1" target="_self">~</a> <font color="Maroon">c<sub>2</sub></font>, <a href="altcat_2.html#K1" target="_self">~</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


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