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<b>let </b><font color="Maroon" title="c1">I</font> be    <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">M</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b2">D</font> being <a href="mssubfam.html#V5" title="MSSUBFAM:attr.5">properly-upper-bound</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Olive" title="b1">M</font><br/> for <font color="Olive" title="b3">A</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Olive" title="b1">M</font><br/> for <font color="Olive" title="b4">J</font> being  <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</a> of <font color="Olive" title="b1">M</font>  st <font color="Olive" title="b3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b2">D</font> &amp; ( for <font color="Olive" title="b5">X</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Olive" title="b1">M</font><br/> for <font color="Olive" title="b6">SF</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Olive" title="b1">M</font>  st ( for <font color="Olive" title="b7">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b7">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b6">SF</font> iff ( <font color="Olive" title="b7">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b2">D</font> &amp; <font color="Olive" title="b5">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b7">Y</font> ) ) ) holds <br/><font color="Olive" title="b4">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b5">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Olive" title="b6">SF</font> ) holds <br/><font color="Olive" title="b4">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b3">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">A</font></span><br/><b>let </b><font color="Maroon" title="c2">M</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <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">D</font> being <a href="mssubfam.html#V5" title="MSSUBFAM:attr.5">properly-upper-bound</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b2">A</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b3">J</font> being  <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</a> of <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b2">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b1">D</font> &amp; ( for <font color="Olive" title="b4">X</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b5">SF</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>  st ( for <font color="Olive" title="b6">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b6">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b5">SF</font> iff ( <font color="Olive" title="b6">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b4">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b6">Y</font> ) ) ) holds <br/><font color="Olive" title="b3">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b4">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Olive" title="b5">SF</font> ) holds <br/><font color="Olive" title="b3">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</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/><b>let </b><font color="Maroon" title="c3">D</font> be  <a href="mssubfam.html#V5" title="MSSUBFAM:attr.5">properly-upper-bound</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>; <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">A</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b2">J</font> being  <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</a> of <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b1">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">D</font> &amp; ( for <font color="Olive" title="b3">X</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b4">SF</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>  st ( for <font color="Olive" title="b5">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b5">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b4">SF</font> iff ( <font color="Olive" title="b5">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b3">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b5">Y</font> ) ) ) holds <br/><font color="Olive" title="b2">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b3">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Olive" title="b4">SF</font> ) holds <br/><font color="Olive" title="b2">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</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/><b>let </b><font color="Maroon" title="c4">A</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font>; <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">J</font> being  <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</a> of <font color="Maroon" title="c2">M</font>  st <font color="Maroon" title="c4">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">D</font> &amp; ( for <font color="Olive" title="b2">X</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b3">SF</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>  st ( for <font color="Olive" title="b4">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b4">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b3">SF</font> iff ( <font color="Olive" title="b4">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b2">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b4">Y</font> ) ) ) holds <br/><font color="Olive" title="b1">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b2">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Olive" title="b3">SF</font> ) holds <br/><font color="Olive" title="b1">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c4">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">A</font></span><br/><b>let </b><font color="Maroon" title="c5">J</font> be   <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</a> of <font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">D</font> &amp; ( for <font color="Olive" title="b1">X</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b2">SF</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>  st ( for <font color="Olive" title="b3">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b3">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b2">SF</font> iff ( <font color="Olive" title="b3">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b1">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b3">Y</font> ) ) ) holds <br/><font color="Maroon" title="c5">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b1">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Olive" title="b2">SF</font> ) implies <font color="Maroon" title="c5">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c4">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">A</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:73"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">D</font>
 <b>and </b><br/><a NAME="E2:73"/><i><font color="Green" title="E30">A2</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b2">SF</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>  st ( for <font color="Olive" title="b3">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b3">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b2">SF</font> iff ( <font color="Olive" title="b3">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b1">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b3">Y</font> ) ) ) holds <br/><font color="Maroon" title="c5">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b1">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Olive" title="b2">SF</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c4">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">A</font></span><br/>

<b>consider </b><font color="Maroon" title="c6">SF</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E4:73"/><i><font color="Green" title="E31">A3</font></i>: 
for <font color="Olive" title="b1">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c6">SF</font> iff ( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c4">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">Y</font> ) )
 <b>by </b><i><a class="ref" href="closure1.html#T31" target="_self" title="CLOSURE1:th.31">Th31</a></i>;<br/>
<b>thus </b><a NAME="E5:73"/>
<font color="Maroon" title="c5">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c4">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">A</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<a NAME="E1:73_1"/>
<font color="Maroon" title="c4">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c6">SF</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:73"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="closure1.html#E4:73"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/>
<a NAME="E2:73_1"/><b>then </b>
 <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">SF</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T43" title="MSSUBFAM:th.43">MSSUBFAM:43</a></i>;<br/>
<b>hence </b><a NAME="E3:73_1"/>
<font color="Maroon" title="c5">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c4">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="closure1.html#E2:73"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="closure1.html#E4:73"><i><font color="Green" title="E31">A3</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D13" title="PBOOLE:def.13">PBOOLE:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c5">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c4">A</font></span><br/>

<a NAME="E4:73_1"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c5">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c4">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">SF</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E2:73"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="closure1.html#E4:73"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/>
<a NAME="E5:73_1"/>
for <font color="Olive" title="b1">Z1</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">Z1</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c6">SF</font> holds <br/><font color="Maroon" title="c4">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">Z1</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E4:73"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:73_1"/>
<font color="Maroon" title="c4">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c5">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="closure1.html#E4:73_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="mssubfam.html#T45" title="MSSUBFAM:th.45">MSSUBFAM:45</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>


</div>
