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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1"><b>suppose </b></a><a NAME="E1:31_1_1_1_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>set </b><font color="Maroon">M</font> =  <a href="pboole.html#K1">[0]</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><a NAME="E2:31_1_1_1_1"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pboole.html#K1">[0]</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</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="E3:31_1_1_1_1"/>
 <a href="pboole.html#K1">[0]</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> is <a href="funcop_1.html#V1">Function-yielding</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a><br/>

<b>thus </b><a NAME="E1:31_1_1_1_1_1"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pboole.html#K1">[0]</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> or <span class="p1">(<span class="default"><a href="pboole.html#K1">[0]</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is    <a href="hidden.html#M1">set</a>  )
 <b>by </b><i><a class="ref" href="altcat_1.html#T2" target="_self">Th2</a>, <a class="ref" href="altcat_1.html#T3" target="_self">Th3</a>, <a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">M</font> =  <a href="pboole.html#K1">[0]</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/><b>take </b>
<font color="Maroon">M</font>
;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D11" target="_self">ALTCAT_1:def 11</a><br/><b>thus </b><a NAME="E5:31_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M</font> implies  ex <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">f</font> ) )
 <b>by </b><i><a class="ref" href="altcat_1.html#T2" target="_self">Th2</a>, <a class="ref" href="altcat_1.html#T3" target="_self">Th3</a>, <a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2"><b>suppose </b></a><a NAME="E1:31_1_1_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A</font>, <font color="Maroon">B1</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A1</font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B1</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">a<sub>2</sub></font>;<br/><b>consider </b><font color="Maroon">M</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A1</font>,<font color="Maroon">B1</font></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E4:31_1_1_1_2"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A1</font><br/> for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B1</font> holds  <font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>,<font color="Olive">j</font>)
 <b>from </b><i><a class="ref" href="altcat_1.html#S2" target="_self">ALTCAT_1:sch 2</a>();<br/></i><a NAME="E5:31_1_1_1_2"/>
<font color="Maroon">M</font> is <a href="funcop_1.html#V1">Function-yielding</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:31_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:31_1_1_1_2_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A1</font>,<font color="Maroon">B1</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="E3:31_1_1_1_2_1"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">g</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a></i>;<br/>
<a NAME="E5:31_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="altcat_1.html#T3" target="_self">Th3</a>, <a class="ref" href="mcart_1.html#T24">MCART_1:24</a></i>;<br/>
<b>then </b><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>

<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="altcat_1.html#T2" target="_self">Th2</a>, </i>
;<br/>
<b>hence </b><a NAME="E7:31_1_1_1_2_1"/>
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/><b>take </b>
<font color="Maroon">M</font>
;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D11" target="_self">ALTCAT_1:def 11</a><br/><b>assume </b><a NAME="E7:31_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E8:31_1_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A1</font>,<font color="Maroon">B1</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="E9:31_1_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">g</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a></i>;<br/><b>take </b>
<font color="Maroon">g</font>
;<br/><b>take </b>
<font color="Maroon">f</font>
;<br/><b>thus </b><a NAME="E11:31_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="altcat_1.html#T3" target="_self">Th3</a>, <a class="ref" href="mcart_1.html#T24">MCART_1:24</a></i>;<br/><a NAME="E12:31_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="altcat_1.html#T3" target="_self">Th3</a>, <a class="ref" href="mcart_1.html#T24">MCART_1:24</a></i>;<br/><b>hence </b><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>

<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="altcat_1.html#T2" target="_self">Th2</a>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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