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<b>consider </b><font color="Maroon" title="c4">op</font> being    <a href="osalg_1.html#M1" title="OSALG_1:mode.1">Element</a> of <font color="Maroon" title="c2">on</font><b> such that </b><br/><a NAME="E2:92_1_1"/><i><font color="Green" title="E51">A2</font></i>: 
<font color="Maroon" title="c3">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="c4">op</font>
 <b>by </b><i><a class="txt" href="osalg_1.html#E1:92"><i><font color="Green" title="E50">A1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">Lo</font> =  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c4">op</font>,<font color="Maroon" title="c3">w</font>;<br/>
<a NAME="E3:92_1_1"/>
 <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c4">op</font>,<font color="Maroon" title="c3">w</font> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c4">op</font>,<font color="Maroon" title="c3">w</font>
 
<b>by </b><i><a class="txt" href="osalg_1.html#E2:92_1_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="osalg_1.html#D15" target="_self" title="OSALG_1:def.15">Def15</a></i>;<br/>
<a NAME="E4:92_1_1"/><b>then </b>
( <font color="Maroon" title="c4">op</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="c4">op</font>,<font color="Maroon" title="c3">w</font> &amp; <font color="Maroon" title="c3">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="c4">op</font>,<font color="Maroon" title="c3">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="c4">op</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="c3">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="c4">op</font>,<font color="Maroon" title="c3">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="E5:92_1_1"/><b>then </b><i><font color="Green" title="E52">A3</font></i>: 
 <a href="osalg_1.html#K10" title="OSALG_1:func.10">Name</a> <font color="Maroon" title="c4">op</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K10" title="OSALG_1:func.10">Name</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c4">op</font>,<font color="Maroon" title="c3">w</font></span>)</span>
 
<b>by </b><i><a class="ref" href="osalg_1.html#T31" target="_self" title="OSALG_1:th.31">Th31</a></i>;<br/>
<a NAME="E6:92_1_1"/><b>then </b><i><font color="Green" title="E53">A4</font></i>: 
 <a href="osalg_1.html#K10" title="OSALG_1:func.10">Name</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c4">op</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">on</font>
 
<b>by </b><i><a class="ref" href="osalg_1.html#T33" target="_self" title="OSALG_1:th.33">Th33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">Lo</font> =  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c4">op</font>,<font color="Maroon" title="c3">w</font> as    <a href="osalg_1.html#M1" title="OSALG_1:mode.1">Element</a> of <font color="Maroon" title="c2">on</font> <b>by </b><i><a class="ref" href="osalg_1.html#T33" target="_self" title="OSALG_1:th.33">Th33</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">Lo</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">op</font> being   <a href="osalg_1.html#M1" title="OSALG_1:mode.1">Element</a> of <font color="Maroon" title="c2">on</font>  st <font color="Maroon" title="c3">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">op</font> holds <br/><font color="Maroon" title="c6">Lo</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="b1">op</font>,<font color="Maroon" title="c3">w</font></span><br/>

<b>let </b><font color="Maroon" title="c7">op1</font> be    <a href="osalg_1.html#M1" title="OSALG_1:mode.1">Element</a> of <font color="Maroon" title="c2">on</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">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="c7">op1</font> implies <font color="Maroon" title="c6">Lo</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="c7">op1</font>,<font color="Maroon" title="c3">w</font> )</span><br/>

<b>assume </b><a NAME="E8:92_1_1"/><i><font color="Green" title="E54">A5</font></i>: 
<font color="Maroon" title="c3">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="c7">op1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">Lo</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="c7">op1</font>,<font color="Maroon" title="c3">w</font></span><br/>

<a NAME="E9:92_1_1"/>
 <a href="osalg_1.html#K10" title="OSALG_1:func.10">Name</a> <font color="Maroon" title="c7">op1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">on</font>
 
<b>by </b><i><a class="ref" href="osalg_1.html#T33" target="_self" title="OSALG_1:th.33">Th33</a></i>;<br/>
<a NAME="E10:92_1_1"/><b>then </b><i><font color="Green" title="E55">A6</font></i>: 
<font color="Maroon" title="c7">op1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c4">op</font>
 
<b>by </b><i><a class="txt" href="osalg_1.html#E5:92_1_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="osalg_1.html#E6:92_1_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="ref" href="osalg_1.html#T31" target="_self" title="OSALG_1:th.31">Th31</a></i>;<br/>
<a NAME="E11:92_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w</font> <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="c7">op1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w</font> <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="c4">op</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osalg_1.html#E2:92_1_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="txt" href="osalg_1.html#E8:92_1_1"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="osalg_1.html#D7" target="_self" title="OSALG_1:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E12:92_1_1"/>
<font color="Maroon" title="c6">Lo</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="c7">op1</font>,<font color="Maroon" title="c3">w</font>
 <b>by </b><i><a class="txt" href="osalg_1.html#E2:92_1_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="txt" href="osalg_1.html#E8:92_1_1"><i><font color="Green" title="E54">A5</font></i></a>, <a class="txt" href="osalg_1.html#E10:92_1_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="ref" href="osalg_1.html#T34" target="_self" title="OSALG_1:th.34">Th34</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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