<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">X</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>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="msafree.html#K15">Reverse</a> <font color="Maroon">X</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><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="E1:12_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">i</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="E2:12_1"/><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><b>thus </b><a NAME="E4:12_1"/>
<span class="p1">(<span class="default"><a href="extens_1.html#K2" target="_self">rngs</a> <span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:12_1_1"/>
<span class="p1">(<span class="default"><a href="extens_1.html#K2" target="_self">rngs</a> <span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="extens_1.html#K2" target="_self">rngs</a> <span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:12_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/>
<a NAME="E3:12_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>hence </b><a NAME="E4:12_1_1_1"/>
<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">i</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:12_1_1"/><i><font color="Green">E53</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">i</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">s0</font> = <font color="Maroon">i</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">D</font> =  <a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">X</font>;<br/>
<a NAME="E4:12_1_1"/><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s0</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="msafree.html#T7">MSAFREE:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">t</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s0</font></span><a href="tarski.html#K4">]</a></span> as   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">X</font></span>)</span> ;<br/>
<a NAME="E6:12_1_1"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s0</font> <a href="funct_2.html#R1">=</a>  <a href="msafree.html#K14">Reverse</a> <font color="Maroon">s0</font>,<font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D20">MSAFREE:def 20</a></i>;<br/>
<a NAME="E7:12_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">s0</font>,<font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:12_1_1"/>
<font color="Maroon">t</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E9:12_1_1"/><b>then </b>
 <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">tt</font></span>)</span> where <font color="Olive">tt</font> is   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">X</font></span>)</span> : ( <font color="Olive">tt</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Olive">tt</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E10:12_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">s0</font>,<font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="msafree.html#T14">MSAFREE:14</a></i>;<br/>
<a NAME="E11:12_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s0</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s0</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s0</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">t</font></span>)</span> = 
<font color="Maroon">t</font> <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/>
<b>hence </b><a NAME="E13:12_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="extens_1.html#K2" target="_self">rngs</a> <span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:12"/>
 <a href="extens_1.html#K2" target="_self">rngs</a> <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">X</font></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
