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

<b>assume </b><b>that </b><br/><a NAME="E1:31_1_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> holds <br/> ex <font color="Olive">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> &amp; <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> )
 <b>and </b><br/><a NAME="E2:31_1_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> holds <br/> ex <font color="Olive">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> &amp; <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> )
 ;<br/>

<a NAME="E3:31_1_2"/>
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">J</font> holds <br/><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_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="E1:31_1_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/>

<a NAME="E2:31_1_2_1"/><b>then </b>
(  ex <font color="Olive">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</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; <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> ) &amp;  ex <font color="Olive">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</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; <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> ) )
 
<b>by </b><i>, <a class="ref" href="pralg_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E3:31_1_2_1"/>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:31_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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