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<b>let </b><font color="Maroon" title="c1">S</font> be   <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:76_1"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a>
 ;<br/><b>let </b><font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c4">o2</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>;<br/><b>assume </b><a NAME="E2:76_1"/><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c3">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c4">o2</font>
 ;<br/><a NAME="E3:76_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="osalg_1.html#E1:76_1"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="osalg_1.html#E2:76_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="osalg_1.html#D23" target="_self" title="OSALG_1:def.23">Def23</a></i>;<br/><b>hence </b><a NAME="E4:76_1"/>
 <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c2">A</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T88" title="RELAT_1:th.88">RELAT_1:88</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:76"/><i><font color="Green" title="E44">A3</font></i>: 
for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Olive" title="b2">o2</font> holds <br/> <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o1</font>,<font color="Maroon" title="c2">A</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b2">o2</font>,<font color="Maroon" title="c2">A</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">o1</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D23" target="_self" title="OSALG_1:def.23">OSALG_1:def 23</a><br/><b>let </b><font color="Maroon" title="c4">o2</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>;<br/>



<b>assume </b><a NAME="E3:76"/><i><font color="Green" title="E45">A4</font></i>: 
<font color="Maroon" title="c3">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c4">o2</font>
 ;<br/>

<a NAME="E4:76"/><i><font color="Green" title="E46">A5</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c2">A</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c2">A</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c2">A</font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_1.html#E2:76"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="osalg_1.html#E3:76"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="wellfnd1.html#T1" title="WELLFND1:th.1">WELLFND1:1</a></i>
<br/>.= 
 <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="txt" href="osalg_1.html#E4:76"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></i>
;<br/><br/>


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