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

<span class="kw">set </span><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/>
<span class="kw">take </span>
<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> 
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">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="b2">A</font></span>)</span></span>)</span> where <font color="Olive" title="b2">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>  iff <font color="Olive" title="b1">x</font> is   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>  <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">OU0</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  iff <font color="Maroon" title="c4">x</font> is   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>  <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">OU0</font> )</span><br/>

<span class="kw">thus </span><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#V3" title="MSUALG_1:attr.3">strict</a>  <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">OU0</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> is   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>  <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">OU0</font> implies <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>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="69_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> is   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>  <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">OU0</font></span><br/>

<a NAME="E2:69_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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> st <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="Olive" title="b1">A</font></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E3:69_1_1_1"/>
<font color="Maroon" title="c4">x</font> is   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>  <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">OU0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">assume </span><a NAME="E2:69_1_1"/>
<font color="Maroon" title="c4">x</font> is   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>  <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">OU0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">x</font> as   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>  <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">OU0</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">A</font> =  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#T5" target="_self" onmouseover="rs('osalg_2/T5')" onmouseout="rh()">Th5</a></span>;<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#T5" target="_self" onmouseover="rs('osalg_2/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#T20" target="_self" onmouseover="rs('osalg_2/T20')" onmouseout="rh()">Th20</a></span>;<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#T35" target="_self" onmouseover="rs('osalg_2/T35')" onmouseout="rh()">Th35</a></span>;<br/>
<span class="kw">hence </span><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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
