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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="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/>



<b>assume </b><a NAME="E1:77"/><i><font color="Green" title="E44">A1</font></i>: 
( <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> )
 ;<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="E2:77"/><i><font color="Green" title="E45">A2</font></i>: 
( <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> &amp;  <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="osalg_1.html#K1" title="OSALG_1:func.1">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="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c4">o2</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D22" target="_self" title="OSALG_1:def.22">OSALG_1:def 22</a><br/>

<a NAME="E3:77"/>
<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="77_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_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><b>by </b><i><a class="txt" href="osalg_1.html#E1:77"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_1_1"/>
<font color="Maroon" title="c1">S</font> is <a href="orders_3.html#V1" title="ORDERS_3:attr.1">discrete</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_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>
 <b>by </b><i><a class="txt" href="osalg_1.html#E2:77"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="osalg_1.html#T8" target="_self" title="OSALG_1:th.8">Th8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_2"><b>suppose </b></a><a NAME="E1:77_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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_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>
 <b>by </b><i><a class="txt" href="osalg_1.html#E2:77"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="osalg_1.html#T3" target="_self" title="OSALG_1:th.3">Th3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:77"/>
<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="ref" href="funct_2.html#T40" title="FUNCT_2:th.40">FUNCT_2:40</a></i>;<br/>


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