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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="osalg_1.html#V8" target="_self">monotone</a> <a href="osalg_1.html#V10" target="_self">regular</a> <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> ;<br/>





<b>assume </b><a NAME="E1:91"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="osalg_1.html#R1" target="_self">~=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E2:91"/><i><font color="Green">E51</font></i>: 
(  <a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="osalg_1.html#R3" target="_self">has_least_args_for</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="osalg_1.html#R3" target="_self">has_least_args_for</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="osalg_1.html#E1:91"><i><font color="Green">E50</font></i></a>, <a class="ref" href="osalg_1.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E3:91"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="osalg_1.html#R1" target="_self">~=</a>  <a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="osalg_1.html#R1" target="_self">~=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="ref" href="osalg_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E4:91"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="osalg_1.html#R1" target="_self">~=</a>  <a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>3</sub></font> <a href="osalg_1.html#R1" target="_self">~=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="osalg_1.html#E2:91"><i><font color="Green">E51</font></i></a>, <a class="ref" href="osalg_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E5:91"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="osalg_1.html#R1" target="_self">~=</a>  <a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="osalg_1.html#R1" target="_self">~=</a>  <a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="osalg_1.html#E1:91"><i><font color="Green">E50</font></i></a>, <a class="txt" href="osalg_1.html#E3:91"><i><font color="Green">E52</font></i></a>, <a class="ref" href="osalg_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E6:91"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="osalg_1.html#R1" target="_self">~=</a>  <a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_1.html#E3:91"><i><font color="Green">E52</font></i></a>, <a class="ref" href="osalg_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E7:91"/><i><font color="Green">E56</font></i>: 
(  <a href="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="osalg_1.html#R2" target="_self">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osalg_1.html#E3:91"><i><font color="Green">E52</font></i></a>, <a class="txt" href="osalg_1.html#E4:91"><i><font color="Green">E53</font></i></a>, <a class="txt" href="osalg_1.html#E5:91"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E8:91"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="osalg_1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E9:91"/>
(  <a href="osalg_1.html#K1" target="_self">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="osalg_1.html#K1" target="_self">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="osalg_1.html#K1" target="_self">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="osalg_1.html#K1" target="_self">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osalg_1.html#E6:91"><i><font color="Green">E55</font></i></a>, <a class="txt" href="osalg_1.html#E7:91"><i><font color="Green">E56</font></i></a>, <a class="ref" href="osalg_1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E10:91"/><b>then </b>
 <a href="osalg_1.html#K1" target="_self">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="osalg_1.html#K1" target="_self">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>
<b>hence </b><a NAME="E11:91"/>
 <a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="osalg_1.html#K3" target="_self">LBound</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="osalg_1.html#E6:91"><i><font color="Green">E55</font></i></a>, <a class="txt" href="osalg_1.html#E8:91"><i><font color="Green">E57</font></i></a>, <a class="ref" href="osalg_1.html#D4" target="_self">Def4</a></i>;<br/>


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