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<b>consider </b><font color="Maroon">R</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E2:45_1"/><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">F<sub>1</sub></font>() holds <br/><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4">PBOOLE:sch 4</a>();<br/></i>
<a NAME="E3:45_1"/>
<font color="Maroon">R</font> is   <a href="msualg_4.html#NM3">ManySortedRelation</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_4.html#D2">MSUALG_4:def 2</a><br/>

<b>assume </b><a NAME="E1:45_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<a NAME="E2:45_1_1"/><b>then </b>
( <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">i</font>) is    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">i</font>) )
 
<b>by </b><i><a class="txt" href="msualg_6.html#E1:45"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E3:45_1_1"/>
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:45_1"/>
 ex <font color="Olive">R</font> being  <a href="msualg_4.html#NM3">ManySortedRelation</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>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">F<sub>1</sub></font>() holds <br/><font color="Olive">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">i</font>)
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">R1</font> be   <a href="msualg_4.html#NM3">ManySortedRelation</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Maroon">R2</font> be   <a href="msualg_4.html#NM3">ManySortedRelation</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>

<b>assume </b><b>that </b><br/><a NAME="E5:45_1"/><i><font color="Green">E43</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">F<sub>1</sub></font>() holds <br/><font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">i</font>)
 <b>and </b><br/><a NAME="E6:45_1"/><i><font color="Green">E46</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">F<sub>1</sub></font>() holds <br/><font color="Maroon">R2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">i</font>)
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_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="E1:45_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><a NAME="E2:45_1_2"/><b>then </b>
( <font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">i</font>) &amp; <font color="Maroon">R2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">i</font>) )
 <b>by </b><i><a class="txt" href="msualg_6.html#E2:45_1"><i><font color="Green">E34</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:45_1_2"/>
<font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:45_1"/>
<font color="Maroon">R1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">R2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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