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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be  <a href="relat_1.html#V2">V5</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E1:39"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msaterm.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E2:39"/>
 <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font> ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="msafree.html#D12">MSAFREE:def 12</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></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">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:39"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E5:39"/><i><font color="Green">E24</font></i>: 
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></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">b<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="msaterm.html#E1:39"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>assume </b><a NAME="E6:39"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></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">b<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="msaterm.html#E5:39"><i><font color="Green">E24</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>suppose </b></a><a NAME="E1:39_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></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">b<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><a NAME="E4:39_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="msaterm.html#E4:39"><i><font color="Green">E23</font></i></a>, <a class="txt" href="msaterm.html#E6:39"><i><font color="Green">E25</font></i></a>, <a class="txt" href="msaterm.html#E3:39_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/><a NAME="E5:39_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E6:39_1_1"/>
 <a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="msaterm.html#E24"><i><font color="Green">Lemma17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2"><b>suppose </b></a><a NAME="E1:39_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:39_1_2"/><i><font color="Green">E26</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:39_1_2"/>
 <a href="msaterm.html#K7" target="_self">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="msaterm.html#E4:39"><i><font color="Green">E23</font></i></a>, <a class="txt" href="msaterm.html#E6:39"><i><font color="Green">E25</font></i></a>, <a class="txt" href="msaterm.html#E3:39_1_2"><i><font color="Green">E26</font></i></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>

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