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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>;<br/><b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">o3</font>, <font color="Maroon" title="c4">o4</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">SR</font>;<br/><b>let </b><font color="Maroon" title="c5">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">SR</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ;<br/>









<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c5">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> &amp; <font color="Maroon" title="c3">o3</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="c5">w1</font> &amp; <font color="Maroon" title="c4">o4</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="c5">w1</font> )
 ;<br/>

<a NAME="E2:46"/><b>then </b><i><font color="Green" title="E28">A2</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c3">o3</font> &amp; <font color="Maroon" title="c5">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="c3">o3</font> &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">o</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="c5">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">o2</font> holds <br/> <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c3">o3</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> ) )
 
<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="E3:46"/>
( <font color="Maroon" title="c2">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c4">o4</font> &amp; <font color="Maroon" title="c5">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="c4">o4</font> &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">o</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="c5">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">o2</font> holds <br/> <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c4">o4</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> ) )
 
<b>by </b><i><a class="txt" href="osalg_1.html#E1:46"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a></i>;<br/>
<a NAME="E4:46"/><b>then </b><i><font color="Green" title="E29">A3</font></i>: 
( <font color="Maroon" title="c3">o3</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c4">o4</font> &amp;  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c4">o4</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">o3</font> &amp;  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c3">o3</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">o4</font> )
 
<b>by </b><i><a class="txt" href="osalg_1.html#E2:46"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="osalg_1.html#T2" target="_self" title="OSALG_1:th.2">Th2</a></i>;<br/>
<a NAME="E5:46"/><b>then </b><i><font color="Green" title="E30">A4</font></i>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c3">o3</font> <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> <font color="Maroon" title="c4">o4</font>
 
<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="E6:46"/>
(  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c3">o3</font> <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="c4">o4</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c4">o4</font> <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="c3">o3</font> )
 
<b>by </b><i><a class="txt" href="osalg_1.html#E4:46"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="osalg_1.html#D8" target="_self" title="OSALG_1:def.8">Def8</a></i>;<br/>
<a NAME="E7:46"/><b>then </b>
 <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c3">o3</font> <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> <font color="Maroon" title="c4">o4</font>
 
<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="E8:46"/>
<font color="Maroon" title="c3">o3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o4</font>
 <b>by </b><i><a class="txt" href="osalg_1.html#E4:46"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="osalg_1.html#E5:46"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="osalg_1.html#D4" target="_self" title="OSALG_1:def.4">Def4</a></i>;<br/>


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