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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <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">  for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font>, <font color="Olive" title="b3">o3</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Olive" title="b2">o2</font> &amp; <font color="Olive" title="b2">o2</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Olive" title="b3">o3</font> holds <br/><font color="Olive" title="b1">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Olive" title="b3">o3</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">o1</font>, <font color="Maroon" title="c3">o2</font>, <font color="Maroon" title="c4">o3</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</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">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c3">o2</font> &amp; <font color="Maroon" title="c3">o2</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c4">o3</font> implies <font color="Maroon" title="c2">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c4">o3</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:70"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
( <font color="Maroon" title="c2">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c3">o2</font> &amp; <font color="Maroon" title="c3">o2</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c4">o3</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">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c4">o3</font></span><br/>

<a NAME="E2:70"/><span class="kw">then </span>
( <font color="Maroon" title="c2">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c3">o2</font> &amp; <font color="Maroon" title="c3">o2</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c4">o3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#D20" target="_self" title="OSALG_1:def.20">Def22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:70"/>
<font color="Maroon" title="c2">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c4">o3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#T2" target="_self" title="OSALG_1:th.2">Th2</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="osalg_1.html#D20" target="_self" title="OSALG_1:def.20">OSALG_1:def 20</a> <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">o1</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">o3</font> &amp;  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c2">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o3</font> )</span><br/>

<a NAME="E4:70"/>
(  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c2">o1</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">o2</font> &amp;  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c3">o2</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">o3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="osalg_1.html#D20" target="_self" title="OSALG_1:def.20">Def22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:70"/>
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c2">o1</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">o3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#T21" target="_self" title="OSALG_1:th.21">Th21</a></span>; <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#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c2">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o3</font></span><br/>

<a NAME="E6:70"/>
(  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c2">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c3">o2</font> &amp;  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c3">o2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="osalg_1.html#D20" target="_self" title="OSALG_1:def.20">Def22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:70"/>
 <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c2">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</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/>


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