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<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="osalg_1.html#V8" title="OSALG_1:attr.8">monotone</a> <a href="osalg_1.html#V10" title="OSALG_1:attr.10">regular</a> <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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">S</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">S</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/> <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">w1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Olive" title="b1">o</font></span><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">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Olive" title="b1">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> holds <br/> <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">w1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c2">o</font></span><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">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> implies  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">w1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c2">o</font> )</span><br/>





<b>assume </b><a NAME="E1:93"/><i><font color="Green" title="E50">A1</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">w1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c2">o</font></span><br/>

<b>set </b><font color="Maroon" title="c4">lo</font> =  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">w1</font>;<br/>
<a NAME="E2:93"/>
 <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">w1</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#E1:93"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="osalg_1.html#T14" target="_self" title="OSALG_1:th.14">Th14</a></i>;<br/>
<a NAME="E3:93"/><b>then </b>
(  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">w1</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> &amp;  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">w1</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> )
 
<b>by </b><i><a class="ref" href="osalg_1.html#D12" target="_self" title="OSALG_1:def.12">Def12</a></i>;<br/>
<a NAME="E4:93"/><b>then </b>
( <font color="Maroon" title="c2">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a>  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">w1</font> &amp;  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">w1</font></span>)</span> <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> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">w1</font></span>)</span> <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="c2">o</font> )
 
<b>by </b><i><a class="txt" href="osalg_1.html#E1:93"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a>, <a class="ref" href="osalg_1.html#D11" target="_self" title="OSALG_1:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E5:93"/>
 <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">w1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c2">o</font>
 <b>by </b><i><a class="ref" href="osalg_1.html#D22" target="_self" title="OSALG_1:def.22">Def22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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