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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">D</font> being  <a href="closure2.html#NM4" title="CLOSURE2:NM.4">ClosureSystem</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">a</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">D</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">D</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Olive" title="b1">D</font> &amp; ( for <font color="Olive" title="b4">x</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">D</font> holds  <font color="Olive" title="b3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b4">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Olive" title="b4">x</font> ) holds <br/><font color="Olive" title="b3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b2">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">D</font> be   <a href="closure2.html#NM4" title="CLOSURE2:NM.4">ClosureSystem</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> &amp; ( for <font color="Olive" title="b3">x</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> holds  <font color="Olive" title="b2">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b3">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Olive" title="b3">x</font> ) holds <br/><font color="Olive" title="b2">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>  st <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> holds  <font color="Olive" title="b1">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b2">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Olive" title="b2">x</font> ) holds <br/><font color="Olive" title="b1">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> holds  <font color="Maroon" title="c4">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Olive" title="b1">x</font> ) implies <font color="Maroon" title="c4">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">a</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:96"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font>
 <span class="kw">and </span><br/><a NAME="E2:96"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> holds  <font color="Maroon" title="c4">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Olive" title="b1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">a</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">F</font> being   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font><span class="kw"> such that </span><br/><a NAME="E4:96"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
 <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Maroon" title="c3">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c5">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 <span class="kw">and </span><br/><a NAME="E5:96"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> : ( <font color="Maroon" title="c3">a</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#D24" target="_self" title="CLOSURE2:def.24">Def24</a></span>;<br/>
<a NAME="E6:96"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c5">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E4:96"><span class="lab"><font color="Green" title="E41">A3</font></span></a></span>;<br/>
<a NAME="E7:96"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E5:96"><span class="lab"><font color="Green" title="E42">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:96"/>
<font color="Maroon" title="c4">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">a</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:96"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="closure2.html#T22" target="_self" title="CLOSURE2:th.22">Th22</a>, <a class="ref" href="mssubfam.html#T43" title="MSSUBFAM:th.43">MSSUBFAM:43</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pboole.html#D13" title="PBOOLE:def.13">PBOOLE:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c4">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">a</font></span><br/>

<a NAME="E9:96"/>
for <font color="Olive" title="b1">Z1</font> being  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> holds <br/><font color="Maroon" title="c3">a</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Olive" title="b1">Z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">Z1</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> implies <font color="Maroon" title="c3">a</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c6">Z1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:96_1"/>
<font color="Maroon" title="c6">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c6">Z1</font></span><br/>

<a NAME="E2:96_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">b</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> st <br/>( <font color="Maroon" title="c6">Z1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c3">a</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96"><span class="lab"><font color="Green" title="E42">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:96_1"/>
<font color="Maroon" title="c3">a</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c6">Z1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:96"/>
<font color="Maroon" title="c3">a</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c4">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:96"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="closure2.html#T25" target="_self" title="CLOSURE2:th.25">Th25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
