<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">A</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a>  iff  for <font color="Olive" title="b2">o1</font>, <font color="Olive" title="b3">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="b2">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Olive" title="b3">o2</font> holds <br/> <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<font color="Olive" title="b2">o1</font>,<font color="Olive" title="b1">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="b3">o2</font>,<font color="Olive" title="b1">A</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a>  iff  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>) )</span><br/>



<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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>) ) implies <font color="Maroon" title="c2">A</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:75_1"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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>)</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies  <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>) )</span><br/><span class="kw">assume </span><a NAME="E2:75_1"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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>)</span><br/><a NAME="E3:75_1"/><span class="kw">then </span>
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="osalg_1.html#D21" target="_self" title="OSALG_1:def.21">Def23</a></span>;<br/><span class="kw">hence </span><a NAME="E4:75_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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T59" title="RELAT_1:th.59">RELAT_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:75"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
 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>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A</font> is  <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> </span><br/>

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="osalg_1.html#D21" target="_self" title="OSALG_1:def.21">OSALG_1:def 21</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="Maroon" title="c3">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Olive" title="b1">o2</font> holds <br/><span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> (<font color="Olive" title="b1">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>)</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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>) )</span><br/>



<span class="kw">assume </span><a NAME="E3:75"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/>

<a NAME="E4:75"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="grfunc_1.html#T27" title="GRFUNC_1:th.27">GRFUNC_1:27</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:75"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="relat_1.html#T69" title="RELAT_1:th.69">RELAT_1:69</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
