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<div class="add">

<b>let </b><font color="Maroon">S</font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">z</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">z</font>;<br/>





<b>set </b><font color="Maroon">A</font> =  <a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font>;<br/>
<b>thus </b><a NAME="E1:79"/>
 <a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font> is <a href="msualg_1.html#V5">non-empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<a NAME="E1:79_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font></span>)</span> <a href="pboole.html#R6">=</a>  <a href="osalg_1.html#K5" target="_self">ConstOSSet</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="osalg_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E2:79_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font></span>)</span> is <a href="relat_1.html#V2">non-empty</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:79_1"/>
 <a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font> is <a href="msualg_1.html#V5">non-empty</a>
 <b>by </b><i><a class="ref" href="msualg_1.html#D8">MSUALG_1:def 8</a></i>;<br/>


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

<b>then reconsider </b><font color="Maroon">A1</font> =  <a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font> as  <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E3:79"/>
for <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>  st <font color="Olive">o1</font> <a href="osalg_1.html#R6" target="_self">&lt;=</a> <font color="Olive">o2</font> holds <br/> <a href="msualg_1.html#K5">Den</a> <font color="Olive">o1</font>,<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Olive">o2</font>,<font color="Maroon">A1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o1</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">o2</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/>



<b>assume </b><a NAME="E1:79_2"/>
<font color="Maroon">o1</font> <a href="osalg_1.html#R6" target="_self">&lt;=</a> <font color="Maroon">o2</font>
 ;<br/>

<a NAME="E2:79_2"/><b>then </b><i><font color="Green">E24</font></i>: 
( <font color="Maroon">o1</font> <a href="osalg_1.html#R1" target="_self">~=</a> <font color="Maroon">o2</font> &amp;  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font> &amp;  <a href="osalg_1.html#K1" target="_self">the_result_sort_of</a> <font color="Maroon">o1</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="osalg_1.html#K1" target="_self">the_result_sort_of</a> <font color="Maroon">o2</font> )
 
<b>by </b><i><a class="ref" href="osalg_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E3:79_2"/><i><font color="Green">E40</font></i>: 
(  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<span class="p2">(<span class="default"><a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">z1</font> &amp;  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<span class="p1">(<span class="default"><a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o2</font>,<span class="p2">(<span class="default"><a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">z1</font> )
 
<b>by </b><i><a class="ref" href="osalg_1.html#D4" target="_self">Def4</a></i>;<br/>
<i><font color="Green">E50</font></i>:  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">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></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>
;<br/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o2</font>,<span class="p1">(<span class="default"><a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">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">o2</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>
;<br/>
<a NAME="E6:79_2"/><b>then </b>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o2</font>,<span class="p1">(<span class="default"><a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="osalg_1.html#T3" target="_self">Th3</a>, , </i>;<br/>
<b>hence </b><a NAME="E7:79_2"/>
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">A1</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T5">FUNCT_4:5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:79"/>
 <a href="osalg_1.html#K8" target="_self">TrivialOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">z1</font> is <a href="osalg_1.html#V13" target="_self">monotone</a>
 <b>by </b><i/>;<br/>


</div>
