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

<b>set </b><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <span class="p3">(<span class="default"><a href="osalg_2.html#K7" title="OSALG_2:func.7">@</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span> where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font> : verum </span>}</span> ;<br/>
<b>take </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <span class="p3">(<span class="default"><a href="osalg_2.html#K7" title="OSALG_2:func.7">@</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span> where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font> : verum </span>}</span> 
;<br/>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:69_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <span class="p3">(<span class="default"><a href="osalg_2.html#K7" title="OSALG_2:func.7">@</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span> where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font> : verum </span>}</span>  implies <font color="Maroon" title="c4">x</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">OU0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <span class="p3">(<span class="default"><a href="osalg_2.html#K7" title="OSALG_2:func.7">@</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span> where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">A</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font><b> such that </b><br/><a NAME="E3:69_1_1_1"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K7" title="OSALG_2:func.7">@</a> <font color="Maroon" title="c5">A</font></span>)</span>
 ;<br/>
<b>thus </b><a NAME="E4:69_1_1_1"/>
<font color="Maroon" title="c4">x</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">OU0</font>
 <b>by </b><i><a class="txt" href="osalg_2.html#E3:69_1_1_1"><i><font color="Green" title="E45">A1</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E2:69_1_1"/>
<font color="Maroon" title="c4">x</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">OU0</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">x</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">OU0</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">A</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c5">a</font> as    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">OU0</font> <b>by </b><i><a class="ref" href="osalg_2.html#T5" target="_self" title="OSALG_2:th.5">Th5</a></i>;<br/>
<a NAME="E5:69_1_1"/>
<font color="Maroon" title="c6">A</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>
 
<b>by </b><i><a class="ref" href="osalg_2.html#T5" target="_self" title="OSALG_2:th.5">Th5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">h</font> = <font color="Maroon" title="c6">A</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font> <b>by </b><i><a class="ref" href="osalg_2.html#T25" target="_self" title="OSALG_2:th.25">Th25</a></i>;<br/>
<a NAME="E7:69_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K7" title="OSALG_2:func.7">@</a> <font color="Maroon" title="c7">h</font></span>)</span>
 
<b>by </b><i><a class="ref" href="osalg_2.html#T40" target="_self" title="OSALG_2:th.40">Th40</a></i>;<br/>
<b>hence </b><a NAME="E8:69_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <span class="p3">(<span class="default"><a href="osalg_2.html#K7" title="OSALG_2:func.7">@</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span> where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font> : verum </span>}</span> 
 ;<br/>


</div>
