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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">g</font> being  <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of <font color="Olive" title="b1">M</font>  st <font color="Olive" title="b2">g</font> is <a href="closure2.html#V7" title="CLOSURE2:attr.7">reflexive</a> &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> <font color="Olive" title="b1">M</font> holds  <font color="Olive" title="b2">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b3">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b3">X</font> ) holds <br/><font color="Olive" title="b2">g</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a></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">g</font> being  <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b1">g</font> is <a href="closure2.html#V7" title="CLOSURE2:attr.7">reflexive</a> &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> <font color="Maroon" title="c2">M</font> holds  <font color="Olive" title="b1">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b2">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">X</font> ) holds <br/><font color="Olive" title="b1">g</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a></span><br/><b>let </b><font color="Maroon" title="c3">g</font> be   <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</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="c3">g</font> is <a href="closure2.html#V7" title="CLOSURE2:attr.7">reflexive</a> &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> <font color="Maroon" title="c2">M</font> holds  <font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">X</font> ) implies <font color="Maroon" title="c3">g</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:58"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c3">g</font> is <a href="closure2.html#V7" title="CLOSURE2:attr.7">reflexive</a>
 <b>and </b><br/><a NAME="E2:58"/><i><font color="Green" title="E30">A2</font></i>: 
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> <font color="Maroon" title="c2">M</font> holds  <font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">g</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a></span><br/>

<b>let </b><font color="Maroon" title="c4">X</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> <font color="Maroon" title="c2">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D14" target="_self" title="CLOSURE2:def.14">CLOSURE2:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font></span>)</span></span><br/>

<b>thus </b><a NAME="E3:58"/>
<font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:58"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="closure2.html#D12" target="_self" title="CLOSURE2:def.12">Def12</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="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font></span><br/>

<b>thus </b><a NAME="E4:58"/>
<font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="#E2:58"><i><font color="Green" title="E30">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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