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<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="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S</font>  st ( <font color="Maroon" title="c1">S</font> is  <a href="orders_3.html#V1" title="ORDERS_3:attr.1">discrete</a>  or <font color="Maroon" title="c1">S</font> is  <a href="osalg_1.html#V6" title="OSALG_1:attr.6">op-discrete</a>  ) holds <br/><font color="Olive" title="b1">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="c2">A</font> be   <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="c1">S</font> is  <a href="orders_3.html#V1" title="ORDERS_3:attr.1">discrete</a>  or <font color="Maroon" title="c1">S</font> is  <a href="osalg_1.html#V6" title="OSALG_1:attr.6">op-discrete</a>  ) implies <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">assume </span><a NAME="E1:76"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
( <font color="Maroon" title="c1">S</font> is  <a href="orders_3.html#V1" title="ORDERS_3:attr.1">discrete</a>  or <font color="Maroon" title="c1">S</font> is  <a href="osalg_1.html#V6" title="OSALG_1:attr.6">op-discrete</a>  )
 ; <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><span class="kw">that </span><br/><a NAME="E2:76"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<font color="Maroon" title="c3">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c4">o2</font>
 <span class="kw">and </span><br/><a NAME="E3:76"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
(  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c3">o1</font> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c4">o2</font> &amp;  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c3">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o2</font> )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="osalg_1.html#D20" target="_self" title="OSALG_1:def.20">OSALG_1:def 20</a> <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="E4:76"/>
<font color="Maroon" title="c3">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:76_1_1"/>
( <font color="Maroon" title="c1">S</font> is  <a href="orders_3.html#V1" title="ORDERS_3:attr.1">discrete</a>  or <font color="Maroon" title="c1">S</font> is  <a href="osalg_1.html#V6" title="OSALG_1:attr.6">op-discrete</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E44">A1</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:76_1_1_1"/>
<font color="Maroon" title="c1">S</font> is  <a href="orders_3.html#V1" title="ORDERS_3:attr.1">discrete</a> 
 ; <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:76_1_1_1"/>
<font color="Maroon" title="c3">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E3:76"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="osalg_1.html#T8" target="_self" title="OSALG_1:th.8">Th8</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:76_1_1_2"/>
<font color="Maroon" title="c1">S</font> is  <a href="osalg_1.html#V6" title="OSALG_1:attr.6">op-discrete</a> 
 ; <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:76_1_1_2"/>
<font color="Maroon" title="c3">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="osalg_1.html#T3" target="_self" title="OSALG_1:th.3">Th3</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:76"/>
<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="ref" href="relset_1.html#T19" title="RELSET_1:th.19">RELSET_1:19</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/>


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