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<b>let </b><font color="Maroon">S1</font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">U1</font> be   <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S1</font>;<br/>



<b>set </b><font color="Maroon">U2</font> =  <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> #);<br/>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </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">S1</font>;<br/><b>thus </b> <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font> = 
the <a href="msualg_1.html#U5">Charact</a> of <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> #) <a href="msualg_3.html#K1">.</a> <font color="Maroon">o1</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> #)
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
;<br/><b>thus </b> <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> #) <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">S1</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/>.= 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> #)
<b>by </b><i><a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>
;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E2:25"/>
( <font color="Maroon">U1</font> is <a href="osalg_1.html#V13">monotone</a> implies  <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> #) is <a href="osalg_1.html#V13">monotone</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">U1</font> is <a href="osalg_1.html#V13">monotone</a>
 ;<br/>

<b>let </b><font color="Maroon">o1</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S1</font>, <font color="Maroon">o2</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D23">OSALG_1:def 23</a><br/>

<b>assume </b><a NAME="E2:25_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">o1</font> <a href="osalg_1.html#R6">&lt;=</a> <font color="Maroon">o2</font>
 ;<br/>

<a NAME="E3:25_2"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">U1</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font>
 
<b>by </b><i>, , <a class="ref" href="osalg_1.html#D23">OSALG_1:def 23</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> #)</span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> #)</span>)</span> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">U1</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> #)</span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> #)
<b>by </b><i/>
;<br/>


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

<b>assume </b><a NAME="E3:25"/><i><font color="Green">E40</font></i>: 
 <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> #) is <a href="osalg_1.html#V13">monotone</a>
 ;<br/>

<b>let </b><font color="Maroon">o1</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S1</font>, <font color="Maroon">o2</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D23">OSALG_1:def 23</a><br/>

<b>assume </b><a NAME="E4:25"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">o1</font> <a href="osalg_1.html#R6">&lt;=</a> <font color="Maroon">o2</font>
 ;<br/>

<a NAME="E5:25"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> #)</span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> #)</span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> #)
 
<b>by </b><i>, , <a class="ref" href="osalg_1.html#D23">OSALG_1:def 23</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">U1</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> #)</span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> #)
<b>by </b><i>, </i>
<br/>.= 
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font>
<b>by </b><i/>
;<br/>


</div>
