<?xml version="1.0"?>
<div class="add">

<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">E</font>, <font color="Olive" title="b3">T</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Olive" title="b1">M</font><br/> for <font color="Olive" title="b4">g</font> being  <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of <font color="Olive" title="b1">M</font><br/> for <font color="Olive" title="b5">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> <font color="Olive" title="b1">M</font>  st <font color="Olive" title="b5">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">E</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Olive" title="b3">T</font> &amp; <font color="Olive" title="b4">g</font> is <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a> holds <br/><font color="Olive" title="b4">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b5">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b2">E</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b3">T</font></span>)</span></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">E</font>, <font color="Olive" title="b2">T</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b3">g</font> being  <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b4">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> <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b4">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">E</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Olive" title="b2">T</font> &amp; <font color="Olive" title="b3">g</font> is <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a> holds <br/><font color="Olive" title="b3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b4">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">E</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b2">T</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">E</font>, <font color="Maroon" title="c4">T</font> be    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K2" title="CLOSURE2:func.2">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">g</font> being  <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of <font color="Maroon" title="c2">M</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> <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b2">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">E</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">T</font> &amp; <font color="Olive" title="b1">g</font> is <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a> holds <br/><font color="Olive" title="b1">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b2">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">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"> 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> <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b1">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">E</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">T</font> &amp; <font color="Maroon" title="c5">g</font> is <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a> holds <br/><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">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> <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="c6">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">E</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">T</font> &amp; <font color="Maroon" title="c5">g</font> is <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a> implies <font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span> )</span><br/>











<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">E</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">g</font> is <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a> or <font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E2:59"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c5">g</font> is <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span></span><br/>

<a NAME="E3:59"/>
<font color="Maroon" title="c3">E</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">T</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">E</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T17" title="PBOOLE:th.17">PBOOLE:17</a></i>;<br/>
<a NAME="E4:59"/><b>then </b><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font>
 
<b>by </b><i><a class="txt" href="#E1:59"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E2:59"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="closure2.html#D13" target="_self" title="CLOSURE2:def.13">Def13</a></i>;<br/>
<a NAME="E5:59"/>
<font color="Maroon" title="c3">E</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">T</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">T</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T17" title="PBOOLE:th.17">PBOOLE:17</a></i>;<br/>
<a NAME="E6:59"/><b>then </b>
<font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font>
 
<b>by </b><i><a class="txt" href="#E1:59"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E2:59"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="closure2.html#D13" target="_self" title="CLOSURE2:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E7:59"/>
<font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:59"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="pboole.html#T19" title="PBOOLE:th.19">PBOOLE:19</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>
