<?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">U0</font> be    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><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>assume </b><a NAME="E1:11"/><i><font color="Green">E38</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>thus </b><a NAME="E2:11"/>
 <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">U0</font>,<font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Olive">o</font>,<font color="Maroon">U0</font></span>)</span> where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> : (  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> 
  <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="11_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="E1:11_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">U0</font>,<font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:11_1"/>
 ex <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp;  <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">U0</font>,<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Olive">A</font> :  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">U0</font></span>)</span> ) </span>}</span>  )
 
<b>by </b><i>, <a class="ref" href="msualg_2.html#D4">MSUALG_2:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:11_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E5:11_1"/><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"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> :  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">U0</font></span>)</span> ) </span>}</span> 
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E7:11_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E8:11_1"/><i><font color="Green">E45</font></i>: 
 ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">U0</font></span>)</span> )
 <b>by </b><i/>;<br/>
<b>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="E10:11_1"/><i><font color="Green">E68</font></i>: 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E11:11_1"/><i><font color="Green">E69</font></i>: 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E12:11_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U0</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:11_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U0</font></span>)</span>
 <b>and </b><br/><a NAME="E15:11_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E16:11_1"/><i><font color="Green">E76</font></i>: 
 <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">s</font>
 
<b>by </b><i>, <a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>;<br/>
<a NAME="E17:11_1"/><i><font color="Green">E77</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/>
<a NAME="E18:11_1"/><b>then </b>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>;<br/>
<a NAME="E19:11_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o1</font>,<font color="Maroon">U0</font>
 
<b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E20:11_1"/>
<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="pralg_3.html#K2" target="_self">const</a> <font color="Olive">o</font>,<font color="Maroon">U0</font></span>)</span> where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> : (  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> 
 <b>by </b><i>, </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="E3:11"/>
<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="pralg_3.html#K2" target="_self">const</a> <font color="Olive">o</font>,<font color="Maroon">U0</font></span>)</span> where <font color="Olive">o</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> : (  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">o</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:11"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font>
 <b>and </b><br/><a NAME="E6:11"/><i><font color="Green">E79</font></i>: 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E7:11"/><i><font color="Green">E80</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>
<a NAME="E8:11"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>
 
;<br/>
<a NAME="E9:11"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E82</font></i>:  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
<b>by </b><i>, <a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
;<br/>
<b>thus </b><a NAME="E11:11"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">U0</font>,<font color="Maroon">s</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:11_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E3:11_3"/><i><font color="Green">E84</font></i>: 
 <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">U0</font>,<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> :  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">U0</font></span>)</span> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="msualg_2.html#D4">MSUALG_2:def 4</a></i>;<br/>
<a NAME="E4:11_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E5:11_3"/>
 ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">U0</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">o</font>
;<br/>

<a NAME="E1:11_3_1"/><i><font color="Green">E86</font></i>: 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:11_3_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>;<br/>
<a NAME="E3:11_3_1"/><b>then </b><i><font color="Green">E87</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:11_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:11_3_1"/>
( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a>, <a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:11_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">U0</font>,<font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/>


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


</div>
