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<b>let </b><font color="Maroon">S</font> be    <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">t</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="relat_1.html#NM1">Relation</a>;<br/><b>let </b><font color="Maroon">V</font> be    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">X</font>;<br/>







<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">V</font> <a href="pboole.html#R6">=</a> <font color="Maroon">X</font> <a href="msafree3.html#K3" target="_self">variables_in</a> <font color="Maroon">t</font>
 ;<br/><b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:19_1"/>
<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="E3:19_1"/><b>then </b>
( <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font> : <font color="Olive">a</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> </span>}</span>  )
 <b>by </b><i>, , <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>;<br/><b>hence </b><a NAME="E4:19_1"/>
<font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font> : <font color="Olive">a</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> </span>}</span> 
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:19"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</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">V</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font> : <font color="Olive">a</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Olive">s</font> </span>}</span> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_2"/><i><font color="Green">E48</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/><b>hence </b><font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> = 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font> : <font color="Olive">a</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> </span>}</span> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:19"/>
<font color="Maroon">V</font> <a href="pboole.html#R6">=</a> <font color="Maroon">X</font> <a href="msafree3.html#K3" target="_self">variables_in</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>;<br/>


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