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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">A</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">V</font> be    <a href="msaterm.html#M4">Variables</a> of <font color="Maroon">A</font>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">t'</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font><br/> for <font color="Olive">T</font> being  <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</font>  st <font color="Olive">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">T</font> <a href="hidden.html#R1">=</a> <font color="Olive">t'</font> holds <br/> <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">T</font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t'</font>;<br/>
<a NAME="E1:1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font><br/> for <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">v</font>,<font color="Olive">s</font></span><a href="tarski.html#K4">]</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">v</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">T</font> be   <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</font>;<br/>







<b>assume </b><a NAME="E1:1_1"/>
( <font color="Maroon">t</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">v</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> )
 ;<br/>

<a NAME="E2:1_1"/><b>then </b>
(  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="msaterm.html#T14">MSATERM:14</a>, <a class="ref" href="msaterm.html#T16">MSATERM:16</a></i>;<br/>
<b>hence </b><a NAME="E3:1_1"/>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><br/> for <font color="Olive">p</font> being   <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Olive">o</font>,<font color="Maroon">V</font>  st ( for <font color="Olive">t'</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font>  st <font color="Olive">t'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t'</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</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="trees_4.html#K4">-tree</a> <font color="Olive">p</font>]
 
<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">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">p</font> be    <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>assume </b><a NAME="E1:1_2"/>
for <font color="Olive">t'</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font>  st <font color="Olive">t'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> holds <br/>for <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font><br/> for <font color="Olive">T</font> being  <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</font>  st <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">t'</font> &amp; <font color="Olive">T</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> holds <br/> <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">T</font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t</font>
 ;<br/>

<b>let </b><font color="Maroon">t</font> be   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">T</font> be   <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</font>;<br/>



<b>assume </b><a NAME="E2:1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</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="trees_4.html#K4">-tree</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E3:1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">t</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">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<a NAME="E4:1_2"/><b>then </b>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="msaterm.html#T17">MSATERM:17</a></i>;<br/>
<b>hence </b><a NAME="E5:1_2"/>
(  not <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> or  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font> )
 <b>by </b><i>, <a class="ref" href="msaterm.html#T17">MSATERM:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:1"/>
for <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>]
 
<b>from </b><i><a class="ref" href="msaterm.html#S1">MSATERM:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:1"/>
for <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">V</font><br/> for <font color="Olive">T</font> being  <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">A</font>,<font color="Maroon">V</font>  st <font color="Olive">T</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> holds <br/> <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">T</font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t</font>
 ;<br/>


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