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<b>let </b><font color="Maroon" title="c1">SM</font> be  <a href="osalg_1.html#V8" title="OSALG_1:attr.8">monotone</a> <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:41_1"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c1">SM</font> is <a href="osalg_1.html#V10" title="OSALG_1:attr.10">regular</a>
 ;<br/><b>let </b><font color="Maroon" title="c2">o</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">SM</font>;<br/><b>let </b><font color="Maroon" title="c3">w1</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">SM</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ;<br/><a NAME="E2:41_1"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c2">o</font> is <a href="osalg_1.html#V9" title="OSALG_1:attr.9">regular</a>
 <b>by </b><i><a class="txt" href="osalg_1.html#E1:41_1"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="osalg_1.html#D14" target="_self" title="OSALG_1:def.14">Def14</a></i>;<br/><b>assume </b><a NAME="E3:41_1"/>
<font color="Maroon" title="c3">w1</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="c2">o</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">o1</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">SM</font><b> such that </b><br/><a NAME="E5:41_1"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c4">o1</font> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">w1</font>
 <b>by </b><i><a class="txt" href="osalg_1.html#E2:41_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="osalg_1.html#D13" target="_self" title="OSALG_1:def.13">Def13</a></i>;<br/><b>take </b><font color="Maroon" title="c5">o1</font> = <font color="Maroon" title="c4">o1</font>;<br/><a NAME="E6:41_1"/>
<font color="Maroon" title="c5">o1</font> <a href="osalg_1.html#R4" title="OSALG_1:pred.4">has_least_sort_for</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">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" title="E28">A4</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c5">o1</font> &amp; <font color="Maroon" title="c3">w1</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="c5">o1</font> )
 <b>by </b><i><a class="txt" href="osalg_1.html#E5:41_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D11" target="_self" title="OSALG_1:def.11">OSALG_1:def 11</a><br/>

<b>let </b><font color="Maroon" title="c6">o2</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">SM</font>;<br/>

<b>assume </b><a NAME="E2:41_1_1"/><i><font color="Green" title="E29">A5</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c6">o2</font> &amp; <font color="Maroon" title="c3">w1</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="c6">o2</font> )
 ;<br/>

<a NAME="E3:41_1_1"/><b>then </b><i><font color="Green" title="E30">A6</font></i>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c5">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="c6">o2</font>
 
<b>by </b><i><a class="txt" href="osalg_1.html#E5:41_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a></i>;<br/>
<a NAME="E4:41_1_1"/>
<font color="Maroon" title="c5">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c6">o2</font>
 
<b>by </b><i><a class="txt" href="osalg_1.html#E1:41_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="osalg_1.html#E2:41_1_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="osalg_1.html#T2" target="_self" title="OSALG_1:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E5:41_1_1"/>
 <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c5">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="c6">o2</font>
 <b>by </b><i><a class="txt" href="osalg_1.html#E3:41_1_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="osalg_1.html#D8" target="_self" title="OSALG_1:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:41_1"/>
<font color="Maroon" title="c5">o1</font> <a href="osalg_1.html#R5" title="OSALG_1:pred.5">has_least_rank_for</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">w1</font>
 <b>by </b><i><a class="txt" href="osalg_1.html#E5:41_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="osalg_1.html#D12" target="_self" title="OSALG_1:def.12">Def12</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:41"/><i><font color="Green" title="E25">A7</font></i>: 
for <font color="Olive" title="b1">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">SM</font><br/> for <font color="Olive" title="b2">w1</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">SM</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Olive" title="b2">w1</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="Olive" title="b1">o</font> holds <br/> ex <font color="Olive" title="b3">o1</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">SM</font> st <font color="Olive" title="b3">o1</font> <a href="osalg_1.html#R5" title="OSALG_1:pred.5">has_least_rank_for</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">w1</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">o</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">SM</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D14" target="_self" title="OSALG_1:def.14">OSALG_1:def 14</a><br/>

<b>thus </b><a NAME="E3:41"/>
<font color="Maroon" title="c2">o</font> is <a href="osalg_1.html#V7" title="OSALG_1:attr.7">monotone</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D13" target="_self" title="OSALG_1:def.13">OSALG_1:def 13</a><br/>

<b>let </b><font color="Maroon" title="c3">w1</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">SM</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ;<br/>

<b>assume </b><a NAME="E4:41"/><i><font color="Green" title="E26">A8</font></i>: 
<font color="Maroon" title="c3">w1</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="c2">o</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">o1</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">SM</font><b> such that </b><br/><a NAME="E6:41"/><i><font color="Green" title="E27">A9</font></i>: 
<font color="Maroon" title="c4">o1</font> <a href="osalg_1.html#R5" title="OSALG_1:pred.5">has_least_rank_for</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">w1</font>
 <b>by </b><i><a class="txt" href="osalg_1.html#E2:41"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="osalg_1.html#E4:41"><i><font color="Green" title="E26">A8</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">o1</font>
;<br/>

<b>thus </b><a NAME="E7:41"/>
<font color="Maroon" title="c4">o1</font> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">w1</font>
 <b>by </b><i><a class="txt" href="osalg_1.html#E6:41"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="osalg_1.html#D12" target="_self" title="OSALG_1:def.12">Def12</a></i>;<br/>


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