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<span class="kw">let </span><font color="Maroon" title="c1">SM</font> be   <a href="osalg_1.html#V8" title="OSALG_1:attr.8">monotone</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">SM</font> is  <a href="osalg_1.html#V6" title="OSALG_1:attr.6">op-discrete</a>  implies <font color="Maroon" title="c1">SM</font> is  <a href="osalg_1.html#V10" title="OSALG_1:attr.10">regular</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:41"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c1">SM</font> is  <a href="osalg_1.html#V6" title="OSALG_1:attr.6">op-discrete</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">SM</font> is  <a href="osalg_1.html#V10" title="OSALG_1:attr.10">regular</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">om</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">SM</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="osalg_1.html#D13" target="_self" title="OSALG_1:def.13">OSALG_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">om</font> is  <a href="osalg_1.html#V9" title="OSALG_1:attr.9">regular</a> </span><br/>

<span class="kw">thus </span><a NAME="E2:41"/>
<font color="Maroon" title="c2">om</font> is  <a href="osalg_1.html#V7" title="OSALG_1:attr.7">monotone</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="osalg_1.html#D12" target="_self" title="OSALG_1:def.12">OSALG_1:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">w1</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">SM</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Olive" title="b1">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">om</font> holds <br/> ex <font color="Olive" title="b2">o1</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">SM</font> st <font color="Olive" title="b2">o1</font> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c2">om</font>,<font color="Olive" title="b1">w1</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">wm1</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">SM</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">wm1</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">om</font> implies  ex <font color="Olive" title="b1">o1</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">SM</font> st <font color="Olive" title="b1">o1</font> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c2">om</font>,<font color="Maroon" title="c3">wm1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:41"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c3">wm1</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">om</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o1</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">SM</font> st <font color="Olive" title="b1">o1</font> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c2">om</font>,<font color="Maroon" title="c3">wm1</font></span><br/>

<a NAME="E4:41"/>
<font color="Maroon" title="c2">om</font> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c2">om</font>,<font color="Maroon" title="c3">wm1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:41_1"/>
( <font color="Maroon" title="c2">om</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c2">om</font> &amp; <font color="Maroon" title="c3">wm1</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">om</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="osalg_1.html#D9" target="_self" title="OSALG_1:def.9">OSALG_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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">SM</font>  st <font color="Maroon" title="c2">om</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">wm1</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="c2">om</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></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">om2</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">SM</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">om</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c4">om2</font> &amp; <font color="Maroon" title="c3">wm1</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">om2</font> implies  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c2">om</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">om2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:41_1"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon" title="c2">om</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c4">om2</font>
 <span class="kw">and </span><br/><a NAME="E3:41_1"/>
<font color="Maroon" title="c3">wm1</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">om2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c2">om</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">om2</font></span><br/>

<span class="kw">thus </span><a NAME="E4:41_1"/>
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c2">om</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">om2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="osalg_1.html#T3" target="_self" title="OSALG_1:th.3">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:41"/>
 ex <font color="Olive" title="b1">o1</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">SM</font> st <font color="Olive" title="b1">o1</font> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c2">om</font>,<font color="Maroon" title="c3">wm1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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