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<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">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">V</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">S</font>;<br/>





<a NAME="E1:1"/><i><font color="Green">E37</font></i>: 
 <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10">FreeOper</a> <font color="Maroon">V</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msafree.html#D16">MSAFREE:def 16</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:1_1"/>
<font color="Maroon">x</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">V</font>
 ;<br/><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">V</font> ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="trees_4.html#M1">FinSequence</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">V</font></span>)</span> <b>by </b><i><a class="ref" href="msaterm.html#D1">MSATERM:def 1</a></i>;<br/><a NAME="E4:1_1"/>
 <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">V</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="msaterm.html#T21">MSATERM:21</a></i>;<br/><b>hence </b><a NAME="E5:1_1"/>
<font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="msafree.html#T10">MSAFREE:10</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:1"/>
<font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">V</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">V</font></span>)</span> ;<br/>
<a NAME="E5:1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">V</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">x</font>
 ;<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="E3:1_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E5:1_2"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_6.html#T2">MSUALG_6:2</a></i>;<br/>
<a NAME="E6:1_2"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span> &amp;  <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">V</font> )
 
<b>by </b><i>, , <a class="ref" href="msafree.html#D13">MSAFREE:def 13</a>, <a class="ref" href="msaterm.html#T12">MSATERM:12</a>, <a class="ref" href="msualg_6.html#T2">MSUALG_6:2</a></i>;<br/>
<b>hence </b><a NAME="E7:1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">V</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="trees_4.html#M1">FinSequence</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">V</font></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E7:1"/>
(  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">V</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">x</font> &amp;  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">V</font> )
 
<b>by </b><i>, <a class="ref" href="msafree.html#T10">MSAFREE:10</a>, <a class="ref" href="msaterm.html#D1">MSATERM:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:1"/>
<font color="Maroon">x</font> is    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="msaterm.html#T21">MSATERM:21</a></i>;<br/>


</div>
