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<b>let </b><font color="Maroon">A</font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">X</font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">X</font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font>, <font color="Maroon">B</font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">X</font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">X</font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> holds  <font color="Maroon">A</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">o</font> <a href="funct_2.html#R2">=</a>  <a href="msafree.html#K9" target="_self">DenOp</a> <font color="Olive">o</font>,<font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:33_1_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> holds  <font color="Maroon">B</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">o</font> <a href="funct_2.html#R2">=</a>  <a href="msafree.html#K9" target="_self">DenOp</a> <font color="Olive">o</font>,<font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:33_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> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> holds <br/><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_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="E1:33_1_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</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="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E3:33_1_2_1"/>
( <font color="Maroon">A</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font> <a href="funct_2.html#R2">=</a>  <a href="msafree.html#K9" target="_self">DenOp</a> <font color="Maroon">s</font>,<font color="Maroon">X</font> &amp; <font color="Maroon">B</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font> <a href="funct_2.html#R2">=</a>  <a href="msafree.html#K9" target="_self">DenOp</a> <font color="Maroon">s</font>,<font color="Maroon">X</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:33_1_2_1"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:33_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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