<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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; <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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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; <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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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; <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/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:91"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:91"/><span class="lab"><font color="Green" title="E51">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:91"/><span class="lab"><font color="Green" title="E52">A3</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">set </span><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/>
<span class="kw">set </span><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="E4:91"/><span class="lab"><font color="Green" title="E53">A4</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="ref" href="osalg_1.html#D15" target="_self" title="OSALG_1:def.15">Def15</a></span>;<br/>
<a NAME="E5:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A5</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a></span>;<br/>
<a NAME="E6:91"/><span class="lab"><font color="Green" title="E55">A6</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="ref" href="osalg_1.html#D15" target="_self" title="OSALG_1:def.15">Def15</a></span>;<br/>
<a NAME="E7:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A7</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a></span>;<br/>
<a NAME="E8:91"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:91"><span class="lab"><font color="Green" title="E55">A6</font></span></a>, <a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a></span>;<br/>
<a NAME="E9:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A8</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="ref" href="osalg_1.html#T2" target="_self" title="OSALG_1:th.2">Th2</a></span>;<br/>
<a NAME="E10:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A9</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:91"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="ref" href="osalg_1.html#T2" target="_self" title="OSALG_1:th.2">Th2</a></span>;<br/>
<a NAME="E11:91"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91"><span class="lab"><font color="Green" title="E53">A4</font></span></a>, <a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a></span>;<br/>
<a NAME="E12:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A10</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E5:91"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="txt" href="#E7:91"><span class="lab"><font color="Green" title="E56">A7</font></span></a>, <a class="ref" href="osalg_1.html#T2" target="_self" title="OSALG_1:th.2">Th2</a></span>;<br/>
<a NAME="E13:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A11</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:91"><span class="lab"><font color="Green" title="E58">A9</font></span></a>, <a class="ref" href="osalg_1.html#D8" target="_self" title="OSALG_1:def.8">Def8</a></span>;<br/>
<a NAME="E14:91"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:91"><span class="lab"><font color="Green" title="E55">A6</font></span></a>, <a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a></span>;<br/>
<a NAME="E15:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A12</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91"><span class="lab"><font color="Green" title="E53">A4</font></span></a>, <a class="txt" href="#E9:91"><span class="lab"><font color="Green" title="E57">A8</font></span></a>, <a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a></span>;<br/>
<a NAME="E16:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A13</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:91"><span class="lab"><font color="Green" title="E59">A10</font></span></a>, <a class="ref" href="osalg_1.html#T6" target="_self" title="OSALG_1:th.6">Th6</a></span>;<br/>
<a NAME="E17:91"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:91"><span class="lab"><font color="Green" title="E58">A9</font></span></a>, <a class="txt" href="#E15:91"><span class="lab"><font color="Green" title="E61">A12</font></span></a>, <a class="ref" href="osalg_1.html#D8" target="_self" title="OSALG_1:def.8">Def8</a></span>;<br/>
<a NAME="E18:91"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:91"><span class="lab"><font color="Green" title="E60">A11</font></span></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:91"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:91"><span class="lab"><font color="Green" title="E58">A9</font></span></a>, <a class="txt" href="#E16:91"><span class="lab"><font color="Green" title="E62">A13</font></span></a>, <a class="ref" href="osalg_1.html#D4" target="_self" title="OSALG_1:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
