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<span class="kw">reconsider </span><font color="Maroon" title="c3">f</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font></span>)</span> as   <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of <font color="Maroon" title="c2">M</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">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">f</font> is  <a href="closure2.html#V7" title="CLOSURE2:attr.7">reflexive</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="closure2.html#V10" title="CLOSURE2:attr.10">topological</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:53_1_1"/>
<font color="Maroon" title="c3">f</font> is  <a href="closure2.html#V7" title="CLOSURE2:attr.7">reflexive</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">f</font> is  <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="closure2.html#V10" title="CLOSURE2:attr.10">topological</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="closure2.html#D12" target="_self" title="CLOSURE2:def.12">CLOSURE2:def 12</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="c4">X</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font></span><br/>

<span class="kw">thus </span><a NAME="E1:53_1_1_1"/>
<font color="Maroon" title="c4">X</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</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><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:53_1_1"/>
<font color="Maroon" title="c3">f</font> is  <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</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">f</font> is  <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="closure2.html#V10" title="CLOSURE2:attr.10">topological</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c5">Y</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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="closure2.html#D13" target="_self" title="CLOSURE2:def.13">CLOSURE2: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="c4">X</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c5">Y</font> implies <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:53_1_1_2"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c5">Y</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">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">Y</font></span><br/>

<a NAME="E2:53_1_1_2"/>
<font color="Maroon" title="c3">f</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="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:53_1_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1_1_2"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</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><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:53_1_1"/>
<font color="Maroon" title="c3">f</font> is  <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</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">f</font> is  <a href="closure2.html#V10" title="CLOSURE2:attr.10">topological</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</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">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E1:53_1_1_3"/>
<font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</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><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E5:53_1_1"/>
<font color="Maroon" title="c3">f</font> is  <a href="closure2.html#V10" title="CLOSURE2:attr.10">topological</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c5">Y</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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="closure2.html#D15" target="_self" title="CLOSURE2:def.15">CLOSURE2:def 15</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">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">Y</font></span>)</span></span><br/>

<a NAME="E1:53_1_1_4"/>
( <font color="Maroon" title="c4">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></span>;<br/>
<a NAME="E2:53_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T18" title="PBOOLE:th.18">PBOOLE:18</a></span>;<br/>
<a NAME="E3:53_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></span>;<br/>
<a NAME="E4:53_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#D1" target="_self" title="CLOSURE2:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font></span>)</span> = 
<font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">Y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>
;<br/> <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>


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