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<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="msualg_1.html#V2">void</a> <a href="catalg_1.html#NM1">Signature</a>;<br/><b>let </b><font color="Maroon">Y</font> be  <a href="relat_1.html#V2">V3</a>  <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">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">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/>









<a NAME="E1:6"/><i><font color="Green">E42</font></i>: 
( <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">Y</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K14">Reverse</a> <font color="Maroon">s</font>,<font color="Maroon">Y</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K14">Reverse</a> <font color="Maroon">s</font>,<font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">s</font>,<font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="msafree.html#D20">MSAFREE:def 20</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 ;<br/><a NAME="E2:6_1"/><b>then </b><i><font color="Green">E48</font></i>: 
(  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">s</font>,<font color="Maroon">Y</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">Y</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="msafree.html#D17">MSAFREE:def 17</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">Y</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span></span>)</span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="msafree.html#D19">MSAFREE:def 19</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><a NAME="E4:6_1"/><b>then </b>
 <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">Y</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="relset_1.html#K11">"</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>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E5:6_1"/>
 <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">Y</font></span>)</span> <a href="equation.html#K1">""</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="equation.html#D1">EQUATION:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:6"/>
 <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">Y</font></span>)</span> <a href="equation.html#K1">""</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E4:6"/><b>then </b>
 <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">Y</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="relset_1.html#K11">"</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>
 
<b>by </b><i><a class="ref" href="equation.html#D1">EQUATION:def 1</a></i>;<br/>
<a NAME="E5:6"/><b>then </b><i><font color="Green">E49</font></i>: 
(  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">s</font>,<font color="Maroon">Y</font> &amp; <span class="p1">(<span class="default"><a href="msafree.html#K14">Reverse</a> <font color="Maroon">s</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:6"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp;  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="msafree.html#D17">MSAFREE:def 17</a></i>;<br/>
<a NAME="E8:6"/><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="trees_4.html#T4">TREES_4:4</a></i>;<br/>
<a NAME="E9:6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="msafree.html#D19">MSAFREE:def 19</a></i>;<br/>
<b>hence </b><a NAME="E11:6"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div>
