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<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">o3</font>, <font color="Olive" title="b2">o</font>, <font color="Olive" title="b3">o4</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="b4">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">SR</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Olive" title="b1">o3</font> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Olive" title="b2">o</font>,<font color="Olive" title="b4">w1</font> &amp; <font color="Olive" title="b3">o4</font> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Olive" title="b2">o</font>,<font color="Olive" title="b4">w1</font> holds <br/><font color="Olive" title="b1">o3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o4</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">o3</font>, <font color="Maroon" title="c3">o</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>; <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">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">SR</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Maroon" title="c2">o3</font> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c3">o</font>,<font color="Olive" title="b1">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="c3">o</font>,<font color="Olive" title="b1">w1</font> holds <br/><font color="Maroon" title="c2">o3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o4</font></span><br/><span class="kw">let </span><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> ; <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">o3</font> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c3">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="c3">o</font>,<font color="Maroon" title="c5">w1</font> implies <font color="Maroon" title="c2">o3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o4</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:46"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c2">o3</font> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">w1</font>
 <span class="kw">and </span><br/><a NAME="E2:46"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<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="c3">o</font>,<font color="Maroon" title="c5">w1</font>
 ; <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">o3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o4</font></span><br/>

<a NAME="E3:46"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c2">o3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a></span>;<br/>
<a NAME="E4:46"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c4">o4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a></span>;<br/>
<a NAME="E5:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<font color="Maroon" title="c2">o3</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c4">o4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="osalg_1.html#T2" target="_self" title="OSALG_1:th.2">Th2</a></span>;<br/>
<a NAME="E6:46"/>
<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">o3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a></span>;<br/>
<a NAME="E7:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A6</font></span>: 
 <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="c2">o3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a></span>;<br/>
<a NAME="E8:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A7</font></span>: 
 <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="c2">o3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="osalg_1.html#D8" target="_self" title="OSALG_1:def.8">Def8</a></span>;<br/>
<a NAME="E9:46"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="osalg_1.html#D10" target="_self" title="OSALG_1:def.10">Def10</a></span>;<br/>
<a NAME="E10:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A8</font></span>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E30">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="E11:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A9</font></span>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="osalg_1.html#T6" target="_self" title="OSALG_1:th.6">Th6</a></span>;<br/>
<a NAME="E12:46"/>
 <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="osalg_1.html#D8" target="_self" title="OSALG_1:def.8">Def8</a></span>;<br/>
<a NAME="E13:46"/><span class="kw">then </span>
 <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:46"><span class="lab"><font color="Green" title="E33">A7</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="E14:46"/>
<font color="Maroon" title="c2">o3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E11:46"><span class="lab"><font color="Green" title="E35">A9</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/>


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