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<b>let </b><font color="Maroon">SM</font> be  <a href="osalg_1.html#V8" target="_self">monotone</a> <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:41_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">SM</font> is <a href="osalg_1.html#V10" target="_self">regular</a>
 ;<br/><b>let </b><font color="Maroon">o</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">SM</font>;<br/><b>let </b><font color="Maroon">w1</font> be    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SM</font> <a href="finseq_2.html#K3">*</a> ;<br/><a NAME="E2:41_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">o</font> is <a href="osalg_1.html#V9" target="_self">regular</a>
 <b>by </b><i>, <a class="txt" href="osalg_1.html#E1"><i><font color="Green">Lemma23</font></i></a></i>;<br/><b>assume </b><a NAME="E3:41_1"/>
<font color="Maroon">w1</font> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font>
 ;<br/><b>then consider </b><font color="Maroon">o1</font> being   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">SM</font><b> such that </b><br/><a NAME="E5:41_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">o1</font> <a href="osalg_1.html#R3" target="_self">has_least_args_for</a> <font color="Maroon">o</font>,<font color="Maroon">w1</font>
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">o1</font> = <font color="Maroon">o1</font>;<br/><a NAME="E6:41_1"/>
<font color="Maroon">o1</font> <a href="osalg_1.html#R4" target="_self">has_least_sort_for</a> <font color="Maroon">o</font>,<font color="Maroon">w1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:41_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">o</font> <a href="osalg_1.html#R1" target="_self">~=</a> <font color="Maroon">o1</font> &amp; <font color="Maroon">w1</font> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font> )
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D11" target="_self">OSALG_1:def 11</a><br/>

<b>let </b><font color="Maroon">o2</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">SM</font>;<br/>

<b>assume </b><a NAME="E2:41_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">o</font> <a href="osalg_1.html#R1" target="_self">~=</a> <font color="Maroon">o2</font> &amp; <font color="Maroon">w1</font> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font> )
 ;<br/>

<a NAME="E3:41_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:41_1_1"/>
<font color="Maroon">o1</font> <a href="osalg_1.html#R1" target="_self">~=</a> <font color="Maroon">o2</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E5:41_1_1"/>
 <a href="osalg_1.html#K1" target="_self">the_result_sort_of</a> <font color="Maroon">o1</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="osalg_1.html#K1" target="_self">the_result_sort_of</a> <font color="Maroon">o2</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:41_1"/>
<font color="Maroon">o1</font> <a href="osalg_1.html#R5" target="_self">has_least_rank_for</a> <font color="Maroon">o</font>,<font color="Maroon">w1</font>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:41"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">o</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">SM</font><br/> for <font color="Olive">w1</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SM</font> <a href="finseq_2.html#K3">*</a>   st <font color="Olive">w1</font> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">o</font> holds <br/> ex <font color="Olive">o1</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">SM</font> st <font color="Olive">o1</font> <a href="osalg_1.html#R5" target="_self">has_least_rank_for</a> <font color="Olive">o</font>,<font color="Olive">w1</font>
 ;<br/>

<b>let </b><font color="Maroon">o</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">SM</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D14" target="_self">OSALG_1:def 14</a><br/>

<b>thus </b><a NAME="E3:41"/>
<font color="Maroon">o</font> is <a href="osalg_1.html#V7" target="_self">monotone</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D13" target="_self">OSALG_1:def 13</a><br/>

<b>let </b><font color="Maroon">w1</font> be    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SM</font> <a href="finseq_2.html#K3">*</a> ;<br/>

<b>assume </b><a NAME="E4:41"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">w1</font> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font>
 ;<br/>

<b>consider </b><font color="Maroon">o1</font> being   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">SM</font><b> such that </b><br/><a NAME="E6:41"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">o1</font> <a href="osalg_1.html#R5" target="_self">has_least_rank_for</a> <font color="Maroon">o</font>,<font color="Maroon">w1</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">o1</font>
;<br/>

<b>thus </b><a NAME="E7:41"/>
<font color="Maroon">o1</font> <a href="osalg_1.html#R3" target="_self">has_least_args_for</a> <font color="Maroon">o</font>,<font color="Maroon">w1</font>
 <b>by </b><i>, </i>;<br/>


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