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<b>let </b><font color="Maroon">I</font> be    <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">I</font>, <font color="Maroon">B</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:23"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">A</font> <a href="pzfmisc1.html#R1">is_transformable_to</a> <font color="Maroon">B</font>
 ;<br/>

<b>let </b><font color="Maroon">H</font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">H1</font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">B</font>,<font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E2:23"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">H</font> is <a href="msualg_3.html#V1">"1-1"</a> &amp; <font color="Maroon">H</font> is <a href="msualg_3.html#V2">"onto"</a> &amp; <font color="Maroon">H1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">H</font> <a href="msualg_3.html#K4">""</a>  )
 ;<br/>

<a NAME="E3:23"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="altcat_2.html#K3">**</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </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="E1:23_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:23_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <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>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">H1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E5:23_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i>, , <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>;<br/>
<a NAME="E6:23_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, , <a class="ref" href="msualg_3.html#D2">MSUALG_3:def 2</a></i>;<br/>
<a NAME="E7:23_1"/>
( <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, , <a class="ref" href="pzfmisc1.html#D3">PZFMISC1:def 3</a></i>;<br/>
<a NAME="E8:23_1"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E9:23_1"/><b>then </b>
<font color="Maroon">h1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>;<br/>
<b>hence </b><a NAME="E10:23_1"/>
<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="altcat_2.html#K3">**</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">H1</font> <a href="altcat_2.html#K3">**</a> <font color="Maroon">H</font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">I</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><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="E1:23_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:23_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="altcat_2.html#K3">**</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:23_2"/>
<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="altcat_2.html#K3">**</a> <font color="Maroon">H</font></span>)</span> <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>,<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">A</font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E7:23"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">F</font> <a href="pboole.html#R6">=</a>  <a href="msualg_3.html#K2">id</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#D1">MSUALG_3:def 1</a></i>;<br/>
<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3"><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="E1:23_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:23_3"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <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>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><b>reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">H1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><a NAME="E5:23_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> 
 <b>by </b><i>, , <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>;<br/><a NAME="E6:23_3"/>
<font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, , <a class="ref" href="msualg_3.html#D2">MSUALG_3:def 2</a></i>;<br/><a NAME="E7:23_3"/><b>then </b>
<font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>;<br/><a NAME="E8:23_3"/><b>then </b>
<font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/><b>hence </b><a NAME="E9:23_3"/>
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="altcat_2.html#K3">**</a> <font color="Maroon">H1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">H</font> <a href="altcat_2.html#K3">**</a> <font color="Maroon">H1</font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">I</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4"><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="E1:23_4"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:23_4"/><b>then </b>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:23_4"/>
<font color="Maroon">F1</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">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E11:23"/><b>then </b>
<font color="Maroon">F1</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">B</font>,<font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>hence </b><a NAME="E12:23"/>
( <font color="Maroon">H</font> <a href="altcat_2.html#K3">**</a> <font color="Maroon">H1</font> <a href="pboole.html#R6">=</a>  <a href="msualg_3.html#K2">id</a> <font color="Maroon">B</font> &amp; <font color="Maroon">H1</font> <a href="altcat_2.html#K3">**</a> <font color="Maroon">H</font> <a href="pboole.html#R6">=</a>  <a href="msualg_3.html#K2">id</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , <a class="ref" href="msualg_3.html#D1">MSUALG_3:def 1</a></i>;<br/>


</div>
