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<b>let </b><font color="Maroon" title="c1">SR</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">op1</font>, <font color="Olive" title="b2">op2</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">SR</font><br/> for <font color="Olive" title="b3">w</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">SR</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Olive" title="b1">op1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Olive" title="b2">op2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">op1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b2">op2</font></span>)</span> &amp; <font color="Olive" title="b3">w</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">op1</font> &amp; <font color="Olive" title="b3">w</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="b2">op2</font> holds <br/> <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Olive" title="b1">op1</font>,<font color="Olive" title="b3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Olive" title="b2">op2</font>,<font color="Olive" title="b3">w</font></span><br/><b>let </b><font color="Maroon" title="c2">op1</font>, <font color="Maroon" title="c3">op2</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">SR</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">w</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">SR</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Maroon" title="c2">op1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c3">op2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c2">op1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c3">op2</font></span>)</span> &amp; <font color="Olive" title="b1">w</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">op1</font> &amp; <font color="Olive" title="b1">w</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="c3">op2</font> holds <br/> <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">op1</font>,<font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c3">op2</font>,<font color="Olive" title="b1">w</font></span><br/><b>let </b><font color="Maroon" title="c4">w</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">SR</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="c2">op1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c3">op2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c2">op1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c3">op2</font></span>)</span> &amp; <font color="Maroon" title="c4">w</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">op1</font> &amp; <font color="Maroon" title="c4">w</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="c3">op2</font> implies  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">op1</font>,<font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c3">op2</font>,<font color="Maroon" title="c4">w</font> )</span><br/>





<b>assume </b><a NAME="E1:91"/><i><font color="Green" title="E50">A1</font></i>: 
( <font color="Maroon" title="c2">op1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c3">op2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c2">op1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c3">op2</font></span>)</span> &amp; <font color="Maroon" title="c4">w</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">op1</font> &amp; <font color="Maroon" title="c4">w</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="c3">op2</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">op1</font>,<font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c3">op2</font>,<font color="Maroon" title="c4">w</font></span><br/>

<b>set </b><font color="Maroon" title="c5">Lo1</font> =  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">op1</font>,<font color="Maroon" title="c4">w</font>;<br/>
<b>set </b><font color="Maroon" title="c6">Lo2</font> =  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c3">op2</font>,<font color="Maroon" title="c4">w</font>;<br/>
<a NAME="E2:91"/><i><font color="Green" title="E51">A2</font></i>: 
(  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">op1</font>,<font color="Maroon" title="c4">w</font> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c2">op1</font>,<font color="Maroon" title="c4">w</font> &amp;  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c3">op2</font>,<font color="Maroon" title="c4">w</font> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c3">op2</font>,<font color="Maroon" title="c4">w</font> )
 
<b>by </b><i><a class="txt" href="osalg_1.html#E1:91"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="osalg_1.html#D15" target="_self" title="OSALG_1:def.15">Def15</a></i>;<br/>
<a NAME="E3:91"/><b>then </b><i><font color="Green" title="E52">A3</font></i>: 
( <font color="Maroon" title="c2">op1</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">op1</font>,<font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c4">w</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> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">op1</font>,<font color="Maroon" title="c4">w</font></span>)</span> &amp; ( 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">SR</font>  st <font color="Maroon" title="c2">op1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Olive" title="b1">o2</font> &amp; <font color="Maroon" title="c4">w</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">o2</font> holds <br/> <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">op1</font>,<font color="Maroon" title="c4">w</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="Olive" title="b1">o2</font> ) )
 
<b>by </b><i><a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a></i>;<br/>
<a NAME="E4:91"/><i><font color="Green" title="E53">A4</font></i>: 
( <font color="Maroon" title="c3">op2</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="c3">op2</font>,<font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c4">w</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> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c3">op2</font>,<font color="Maroon" title="c4">w</font></span>)</span> &amp; ( 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">SR</font>  st <font color="Maroon" title="c3">op2</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Olive" title="b1">o2</font> &amp; <font color="Maroon" title="c4">w</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">o2</font> holds <br/> <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="c3">op2</font>,<font color="Maroon" title="c4">w</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="Olive" title="b1">o2</font> ) )
 
<b>by </b><i><a class="txt" href="osalg_1.html#E2:91"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a></i>;<br/>
<a NAME="E5:91"/><b>then </b><i><font color="Green" title="E54">A5</font></i>: 
( <font color="Maroon" title="c2">op1</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="c3">op2</font>,<font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c3">op2</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">op1</font>,<font color="Maroon" title="c4">w</font> )
 
<b>by </b><i><a class="txt" href="osalg_1.html#E1:91"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="osalg_1.html#E3:91"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="osalg_1.html#T2" target="_self" title="OSALG_1:th.2">Th2</a></i>;<br/>
<a NAME="E6:91"/><b>then </b><i><font color="Green" title="E55">A6</font></i>: 
 <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">op1</font>,<font color="Maroon" title="c4">w</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="c3">op2</font>,<font color="Maroon" title="c4">w</font>
 
<b>by </b><i><a class="txt" href="osalg_1.html#E3:91"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="osalg_1.html#T2" target="_self" title="OSALG_1:th.2">Th2</a></i>;<br/>
<a NAME="E7:91"/><i><font color="Green" title="E56">A7</font></i>: 
(  <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">op1</font>,<font color="Maroon" title="c4">w</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> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c3">op2</font>,<font color="Maroon" title="c4">w</font></span>)</span> &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="c3">op2</font>,<font color="Maroon" title="c4">w</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> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">op1</font>,<font color="Maroon" title="c4">w</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osalg_1.html#E3:91"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="osalg_1.html#E4:91"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="osalg_1.html#E5:91"><i><font color="Green" title="E54">A5</font></i></a></i>;<br/>
<a NAME="E8:91"/><b>then </b><i><font color="Green" title="E57">A8</font></i>: 
 <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">op1</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c3">op2</font>,<font color="Maroon" title="c4">w</font></span>)</span>
 
<b>by </b><i><a class="ref" href="osalg_1.html#T6" target="_self" title="OSALG_1:th.6">Th6</a></i>;<br/>
<a NAME="E9:91"/>
(  <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">op1</font>,<font color="Maroon" title="c4">w</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> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c3">op2</font>,<font color="Maroon" title="c4">w</font></span>)</span> &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="c3">op2</font>,<font color="Maroon" title="c4">w</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> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">op1</font>,<font color="Maroon" title="c4">w</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osalg_1.html#E6:91"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="osalg_1.html#E7:91"><i><font color="Green" title="E56">A7</font></i></a>, <a class="ref" href="osalg_1.html#D8" target="_self" title="OSALG_1:def.8">Def8</a></i>;<br/>
<a NAME="E10:91"/><b>then </b>
 <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">op1</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c3">op2</font>,<font color="Maroon" title="c4">w</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>;<br/>
<b>hence </b><a NAME="E11:91"/>
 <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c2">op1</font>,<font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c3">op2</font>,<font color="Maroon" title="c4">w</font>
 <b>by </b><i><a class="txt" href="osalg_1.html#E6:91"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="osalg_1.html#E8:91"><i><font color="Green" title="E57">A8</font></i></a>, <a class="ref" href="osalg_1.html#D4" target="_self" title="OSALG_1:def.4">Def4</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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