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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_1.html#NM1">Function</a> =  <a href="pralg_3.html#K3" target="_self">proj</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">i</font>;<br/>
<b>consider </b><font color="Maroon">F</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E2:30_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">s</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">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">s</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i>
<a NAME="E3:30_1_1"/>
<font color="Maroon">F</font> is   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span>
 
<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">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:30_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:30_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">s'</font> = <font color="Maroon">s</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E2:30_1_1_1_1"/>
<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="pralg_3.html#K3" target="_self">proj</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span>,<font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E4:30_1_1_1_1"/><i><font color="Green">E43</font></i>: 
not the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<i><font color="Green">E44</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F'</font> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" target="_self">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>
;<br/>
<a NAME="E6:30_1_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F'</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:30_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F'</font>
 ;<br/>

<a NAME="E2:30_1_1_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" target="_self">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:30_1_1_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" target="_self">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E5:30_1_1_1_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" target="_self">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:30_1_1_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i/>;<br/>
<a NAME="E8:30_1_1_1_1_2"/><i><font color="Green">E73</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:30_1_1_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">U0</font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E11:30_1_1_1_1_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font> )
 <b>by </b><i><a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>;<br/>
<b>thus </b><a NAME="E12:30_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, , , , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:30_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</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#T11">RELSET_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:30_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> is    <a href="relset_1.html#M2">Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F</font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">F'</font>
;<br/>

<b>thus </b><a NAME="E5:30_1_1"/>
for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</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="hidden.html#R1">=</a>  <a href="pralg_3.html#K3" target="_self">proj</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Olive">s</font></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i/>;<br/>


</div>
