<?xml version="1.0"?>
<div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">z</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">z1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">z</font> holds <br/> (  <a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">z1</font> is <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> &amp;  <a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">z1</font> is <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> )</span><br/><b>let </b><font color="Maroon" title="c2">z</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">z1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">z</font> holds <br/> (  <a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Olive" title="b1">z1</font> is <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> &amp;  <a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Olive" title="b1">z1</font> is <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> )</span><br/><b>let </b><font color="Maroon" title="c3">z1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">z</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font> is <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> &amp;  <a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font> is <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a> )</span><br/>





<b>set </b><font color="Maroon" title="c4">A</font> =  <a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font>;<br/>
<b>thus </b><a NAME="E1:79"/>
 <a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font> is <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font> is <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<a NAME="E1:79_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="osalg_1.html#K5" title="OSALG_1:func.5">ConstOSSet</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>
 
<b>by </b><i><a class="ref" href="osalg_1.html#D24" target="_self" title="OSALG_1:def.24">Def24</a></i>;<br/>
<a NAME="E2:79_1"/><b>then </b>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font></span>)</span> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 
<b>by </b><i><a class="ref" href="osalg_1.html#T15" target="_self" title="OSALG_1:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E3:79_1"/>
 <a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font> is <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>
 <b>by </b><i><a class="ref" href="msualg_1.html#D8" title="MSUALG_1:def.8">MSUALG_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>then reconsider </b><font color="Maroon" title="c5">A1</font> =  <a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E3:79"/>
for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</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> holds <br/> <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o1</font>,<font color="Maroon" title="c5">A1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b2">o2</font>,<font color="Maroon" title="c5">A1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">o1</font>, <font color="Maroon" title="c7">o2</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c7">o2</font> implies  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c5">A1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o2</font>,<font color="Maroon" title="c5">A1</font> )</span><br/>



<b>assume </b><a NAME="E1:79_2"/>
<font color="Maroon" title="c6">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c7">o2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c5">A1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o2</font>,<font color="Maroon" title="c5">A1</font></span><br/>

<a NAME="E2:79_2"/><b>then </b><i><font color="Green" title="E46">A1</font></i>: 
( <font color="Maroon" title="c6">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c7">o2</font> &amp;  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">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="c7">o2</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c6">o1</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="c7">o2</font> )
 
<b>by </b><i><a class="ref" href="osalg_1.html#D22" target="_self" title="OSALG_1:def.22">Def22</a></i>;<br/>
<a NAME="E3:79_2"/><i><font color="Green" title="E47">A2</font></i>: 
(  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<span class="p1">(<span class="default"><a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o1</font>,<span class="p2">(<span class="default"><a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font></span>)</span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c3">z1</font> &amp;  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o2</font>,<span class="p1">(<span class="default"><a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c7">o2</font>,<span class="p2">(<span class="default"><a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font></span>)</span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c3">z1</font> )
 
<b>by </b><i><a class="ref" href="osalg_1.html#D24" target="_self" title="OSALG_1:def.24">Def24</a></i>;<br/>
<i><font color="Green" title="E48">A3</font></i>:  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o1</font>,<span class="p1">(<span class="default"><a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p3">(<span class="default"><a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">o1</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D9" title="MSUALG_1:def.9">MSUALG_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">o1</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o1</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D6" title="MSUALG_1:def.6">MSUALG_1:def 6</a></i>
;<br/>
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c7">o2</font>,<span class="p1">(<span class="default"><a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p3">(<span class="default"><a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">o2</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D9" title="MSUALG_1:def.9">MSUALG_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">o2</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o2</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D6" title="MSUALG_1:def.6">MSUALG_1:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E6:79_2"/>
 <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c5">A1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o2</font>,<font color="Maroon" title="c5">A1</font>
 <b>by </b><i><a class="txt" href="#E2:79_2"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="#E3:79_2"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="#E4:79_2"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="osalg_1.html#T20" target="_self" title="OSALG_1:th.20">Th20</a>, <a class="ref" href="funct_4.html#T5" title="FUNCT_4:th.5">FUNCT_4:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:79"/>
 <a href="osalg_1.html#K8" title="OSALG_1:func.8">TrivialOSA</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">z1</font> is <a href="osalg_1.html#V13" title="OSALG_1:attr.13">monotone</a>
 <b>by </b><i><a class="ref" href="osalg_1.html#T27" target="_self" title="OSALG_1:th.27">Th27</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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