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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>) <b>-&gt; </b>  <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K4" target="_self">Class</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> =  <a href="msualg_4.html#K15" target="_self">MSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:30_1_1"/><i><font color="Green">E28</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; ( for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">d</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S4">FUNCT_1:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:30_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> is   <a href="funct_1.html#NM1">Function</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_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> ;<br/>

<b>assume </b><a NAME="E1:30_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">f</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E3:30_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K15" target="_self">MSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font>,<font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:30_1_1_1"/>
<font color="Maroon">f</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">f</font> = <font color="Maroon">f</font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/>
<a NAME="E6:30_1_1"/>
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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K4" target="_self">Class</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><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="E1:30_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E3:30_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K15" target="_self">MSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font>,<font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:30_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K4" target="_self">Class</a> <font color="Maroon">R</font></span>)</span> <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 the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>, <a href="msualg_4.html#K4" target="_self">Class</a> <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E9:30_1_1"/>
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">s</font> <a href="funct_2.html#R1">=</a>  <a href="msualg_4.html#K15" target="_self">MSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font>,<font color="Olive">s</font>
 <b>by </b><i/>;<br/>


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