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<span class="kw">let </span><font color="Maroon" title="c1">S1</font> be   <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</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">  for <font color="Olive" title="b1">OU0</font> being   <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font><br/>  for <font color="Olive" title="b2">B</font> being    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Olive" title="b1">OU0</font> holds <br/> ( <font color="Olive" title="b2">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Olive" title="b1">OU0</font> iff <font color="Olive" title="b2">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">OU0</font> be   <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</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">  for <font color="Olive" title="b1">B</font> being    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">OU0</font> holds <br/> ( <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font> iff <font color="Olive" title="b1">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">B</font> be    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</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="c3">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font> iff <font color="Maroon" title="c3">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>  )</span><br/>





<a NAME="E1:39"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
( <font color="Maroon" title="c3">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msualg_2.html#K6" title="MSUALG_2:func.6">SubSort</a> <font color="Maroon" title="c2">OU0</font> iff <font color="Maroon" title="c3">B</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="msualg_2.html#T14" onmouseover="rs('msualg_2/T14')" onmouseout="rh()">MSUALG_2:14</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:39"/>
( <font color="Maroon" title="c3">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font> implies <font color="Maroon" title="c3">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</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="c3">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>  implies <font color="Maroon" title="c3">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1"/>
<font color="Maroon" title="c3">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <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="c3">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> </span><br/>

<a NAME="E2:39_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">B1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K6" title="MSUALG_2:func.6">SubSort</a> <font color="Maroon" title="c2">OU0</font> st <br/>( <font color="Olive" title="b1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Olive" title="b1">B1</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:39_1"/>
<font color="Maroon" title="c3">B</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="msualg_2.html#T14" onmouseover="rs('msualg_2/T14')" onmouseout="rh()">MSUALG_2:14</a></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><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:39"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c3">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</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="c3">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font></span><br/>

<a NAME="E4:39"/>
<font color="Maroon" title="c3">B</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#D2" target="_self" onmouseover="rs('osalg_2/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:39"/>
<font color="Maroon" title="c3">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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="txt" href="#E1:39"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E23">A2</font></span></a></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/>


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