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<b>let </b><font color="Maroon">M1</font> be  <a href="altcat_1.html#V7" target="_self">compositional</a> <a href="pboole.html#NM2">ManySortedFunction</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <font color="Maroon">M2</font> be  <a href="altcat_1.html#V7" target="_self">compositional</a> <a href="pboole.html#NM2">ManySortedFunction</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><a NAME="E2:33_1_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</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">B</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:33_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M1</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:33_1_1_1"/><i><font color="Green">E27</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">f1</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E6:33_1_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font>
 <b>by </b><i/>;<br/><a NAME="E7:33_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="ref" href="altcat_1.html#T3" target="_self">Th3</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">f2</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:33_1_1_1"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g2</font>,<font color="Maroon">f2</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E10:33_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f2</font>
 <b>by </b><i/>;<br/><a NAME="E11:33_1_1_1"/>
( <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> )
 <b>by </b><i>, <a class="ref" href="altcat_1.html#T4" target="_self">Th4</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E12:33_1_1_1"/>
<font color="Maroon">M1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:33_1_1"/>
<font color="Maroon">M1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">M2</font>
 <b>by </b><i/>;<br/>


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