<?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>let </b><font color="Maroon">s1</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">s2</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s2</font>
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E33</font></i>: 
<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="funct_1.html#K1">.</a> <font color="Maroon">s1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <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="funct_1.html#K1">.</a> <font color="Maroon">s2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:31"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <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="funct_1.html#K1">.</a> <font color="Maroon">s1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <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="funct_1.html#K1">.</a> <font color="Maroon">s2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:31"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:31"/><i><font color="Green">E45</font></i>: 
( <span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K7" target="_self">FreeSort</a> <font color="Maroon">X</font>,<font color="Maroon">s1</font> &amp; <span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K7" target="_self">FreeSort</a> <font color="Maroon">X</font>,<font color="Maroon">s2</font> )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">D</font> =  <a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">A1</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">X</font></span>)</span> : (  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Olive">a</font> <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="Olive">x</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o1</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> ) ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">A2</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">X</font></span>)</span> : (  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> &amp; <font color="Olive">a</font> <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="Olive">x</font>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o1</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> ) ) </span>}</span> ;<br/>
<a NAME="E7:31"/><i><font color="Green">E46</font></i>: 
( <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">X</font></span>)</span> : (  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Olive">a</font> <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="Olive">x</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o1</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> ) ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">X</font></span>)</span> : (  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> &amp; <font color="Olive">a</font> <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="Olive">x</font>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o1</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> ) ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E9:31"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E10:31"/><i><font color="Green">E49</font></i>: 
(  ex <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">a</font> <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="Olive">x1</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o1</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E12:31"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E13:31"/><i><font color="Green">E51</font></i>: 
(  ex <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">b</font> <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="Olive">x2</font>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o2</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> ) )
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1"/>
(  ex <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">a</font> <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="Olive">x1</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o1</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1"><b>suppose </b></a><a NAME="E1:31_1_1"/>
 ex <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">a</font> <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="Olive">x1</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">a</font> <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">x1</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1_1_1"/>
(  ex <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">b</font> <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="Olive">x2</font>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o2</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1_1"><b>case </b></a><a NAME="E1:31_1_1_1_1_1"/>
 ex <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">b</font> <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="Olive">x2</font>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_1_1_1_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">b</font> <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">x2</font>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><a NAME="E4:31_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">s1</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">x2</font>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="msafree.html#T1" target="_self">Th1</a>, , , , <a class="ref" href="trees_4.html#T4">TREES_4:4</a></i>;<br/><b>hence </b><a NAME="E5:31_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1_2"><b>case </b></a><a NAME="E1:31_1_1_1_1_2"/>
 ex <font color="Olive">o2</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">o2</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:31_1_1_1_1_2"/><i><font color="Green">E55</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> )
 ;<br/><a NAME="E4:31_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o2</font>,the <a href="struct_0.html#U1">carrier</a> of <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">x1</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="msafree.html#T1" target="_self">Th1</a>, , , , <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/><a NAME="E5:31_1_1_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E6:31_1_1_1_1_2"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 ;<br/><b>hence </b><a NAME="E7:31_1_1_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:31_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2"><b>suppose </b></a><a NAME="E1:31_1_2"/>
 ex <font color="Olive">o1</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">o1</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:31_1_2"/><i><font color="Green">E62</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_2_1_1"/>
(  ex <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">b</font> <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="Olive">x2</font>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o2</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1_1_1"><b>case </b></a><a NAME="E1:31_1_2_1_1_1"/>
 ex <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">b</font> <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="Olive">x2</font>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_1_2_1_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">b</font> <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">x2</font>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><a NAME="E4:31_1_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <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">x2</font>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="msafree.html#T1" target="_self">Th1</a>, , , , <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/><a NAME="E5:31_1_2_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E6:31_1_2_1_1_1"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 ;<br/><b>hence </b><a NAME="E7:31_1_2_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1_1_2"><b>case </b></a><a NAME="E1:31_1_2_1_1_2"/>
 ex <font color="Olive">o2</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">o2</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:31_1_2_1_1_2"/><i><font color="Green">E89</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> )
 ;<br/><b>thus </b><a NAME="E4:31_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="msafree.html#T1" target="_self">Th1</a>, , , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:31_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
