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<b>let </b><font color="Maroon">I1</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">I2</font> be  non <a href="xboole_0.html#V1">empty</a>  <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">I1</font>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I2</font>;<br/><b>let </b><font color="Maroon">o</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I2</font>;<br/>







<b>assume </b><a NAME="E1:29"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I1</font>,<font color="Maroon">I2</font>;<br/>



<b>assume </b><a NAME="E2:29"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">I1</font> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">o</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span> = <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">m</font>;<br/>
<b>reconsider </b><font color="Maroon">Xm</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o'</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">o'</font>)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">o'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I1</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">o'</font>] </span>}</span>  as   <a href="funct_1.html#NM1">Function</a> <b>from </b><i><a class="ref" href="altcat_2.html#S1">ALTCAT_2:sch 1</a>();<br/></i>
<a NAME="E4:29"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">Xm</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o'</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">o'</font>)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">o'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I1</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">o'</font>] </span>}</span> 
 
;<br/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">Xm</font> = 
<span class="p1">{<span class="default"> <font color="Olive">o'</font> where <font color="Olive">o'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I1</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">o'</font>] </span>}</span> 
<b>from </b><i><a class="ref" href="altcat_2.html#S2">ALTCAT_2:sch 2</a>()</i>
<br/>.= 
<font color="Maroon">I1</font>
<b>by </b><i><a class="ref" href="domain_1.html#T48">DOMAIN_1:48</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">Xm</font> = <font color="Maroon">Xm</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I1</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span> = <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">m</font>;<br/>
<a NAME="E7:29"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">Xm</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o'</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Olive">o'</font>)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">o'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I1</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">o'</font>] </span>}</span> 
 
;<br/>
<a NAME="E8:29"/>
<font color="Maroon">Xm</font> is    <a href="functor0.html#M1" target="_self">MSUnTrans</a> of <font color="Maroon">f</font>,<font color="Maroon">A</font>,<font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_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>assume </b><a NAME="E1:29_2_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">o'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I1</font> ;<br/><a NAME="E3:29_2_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">o'</font>]
 ;<br/><a NAME="E4:29_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="functor0.html#E2:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="functor0.html#E4:29"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E5:29_2_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="functor0.html#E2:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="functor0.html#E4:29"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E6:29_2_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:29_2_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E8:29_2_1"/>
<font color="Maroon">Xm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">o'</font>)
 <b>from </b><i><a class="ref" href="altcat_2.html#S3">ALTCAT_2:sch 3</a>(, );<br/></i><b>hence </b><a NAME="E9:29_2_1"/>
<font color="Maroon">Xm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:29_2"/>
<font color="Maroon">Xm</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D5" target="_self">FUNCTOR0:def 5</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:29"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o'</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o'</font></span>)</span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">m</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">o'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I1</font> : verum </span>}</span>  is    <a href="functor0.html#M1" target="_self">MSUnTrans</a> of <font color="Maroon">f</font>,<font color="Maroon">A</font>,<font color="Maroon">B</font>
 ;<br/>


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