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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> ;<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>;<br/><b>let </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">R</font> be   <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</a> of <font color="Maroon" title="c2">M</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:32"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c3">P</font> is <a href="closure1.html#V2" title="CLOSURE1:attr.2">monotonic</a>
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c4">R</font> is <a href="closure1.html#V2" title="CLOSURE1:attr.2">monotonic</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">X</font>, <font color="Maroon" title="c6">Y</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure1.html#D3" target="_self" title="CLOSURE1:def.3">CLOSURE1:def 3</a><br/>

<b>assume </b><a NAME="E3:32"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c5">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">Y</font>
 ;<br/>

<a NAME="E4:32"/><i><font color="Green" title="E15">A4</font></i>: 
 <a href="mssubfam.html#K1" title="MSSUBFAM:func.1">doms</a> <font color="Maroon" title="c4">R</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T17" title="MSSUBFAM:th.17">MSSUBFAM:17</a></i>;<br/>
<a NAME="E5:32"/>
<font color="Maroon" title="c4">R</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">R</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c6">Y</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E2:32"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="closure1.html#E3:32"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="closure1.html#D3" target="_self" title="CLOSURE1:def.3">Def3</a></i>;<br/>
<a NAME="E6:32"/><b>then </b>
<font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c6">Y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:32"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="closure1.html#D3" target="_self" title="CLOSURE1:def.3">Def3</a></i>;<br/>
<a NAME="E7:32"/><b>then </b>
<font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K4" title="CLOSURE1:func.4">**</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c6">Y</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E4:32"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="closure1.html#T4" target="_self" title="CLOSURE1:th.4">Th4</a>, <a class="ref" href="mssubfam.html#T12" title="MSSUBFAM:th.12">MSSUBFAM:12</a></i>;<br/>
<b>hence </b><a NAME="E8:32"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K4" title="CLOSURE1:func.4">**</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K4" title="CLOSURE1:func.4">**</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c6">Y</font>
 <b>by </b><i><a class="txt" href="closure1.html#E4:32"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="closure1.html#T4" target="_self" title="CLOSURE1:th.4">Th4</a>, <a class="ref" href="mssubfam.html#T12" title="MSSUBFAM:th.12">MSSUBFAM:12</a></i>;<br/>


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