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<b>let </b><font color="Maroon" title="c1">S</font> be    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</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">A</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="closure2.html#NM5" title="CLOSURE2:NM.5">ClosureOperator</a> of <font color="Olive" title="b1">A</font> holds   <a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">f</font></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</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">f</font> being  <a href="closure2.html#NM5" title="CLOSURE2:NM.5">ClosureOperator</a> of <font color="Maroon" title="c2">A</font> holds   <a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="closure2.html#NM5" title="CLOSURE2:NM.5">ClosureOperator</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">f</font></span><br/>





<b>set </b><font color="Maroon" title="c4">D</font> =  <a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font>;<br/>
<b>set </b><font color="Maroon" title="c5">M</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">f1</font> =  <a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span>;<br/>
<a NAME="E1:99"/><i><font color="Green" title="E41">A1</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="closure2.html#D23" target="_self" title="CLOSURE2:def.23">Def23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">ff</font> = <font color="Maroon" title="c3">f</font> as  <a href="closure2.html#V7" title="CLOSURE2:attr.7">reflexive</a> <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a> <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a> <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span> ;<br/>
<a NAME="E3:99"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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">A</font> holds <br/><span class="p1">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <span class="p2">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</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"> ( <font color="Maroon" title="c8">x</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">A</font> implies <span class="p1">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <span class="p2">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> )</span><br/>

<b>assume </b><a NAME="E1:99_1"/>
<font color="Maroon" title="c8">x</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">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <span class="p2">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c9">x1</font> = <font color="Maroon" title="c8">x</font> as    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span> <b>by </b><i><a class="ref" href="closure2.html#D23" target="_self" title="CLOSURE2:def.23">Def23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">F</font> being   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span><b> such that </b><br/><a NAME="E4:99_1"/><i><font color="Green" title="E42">A2</font></i>: 
 <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Maroon" title="c9">x1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c10">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 <b>and </b><br/><a NAME="E5:99_1"/><i><font color="Green" title="E43">A3</font></i>: 
<font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span> : ( <font color="Maroon" title="c9">x1</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="closure2.html#D24" target="_self" title="CLOSURE2:def.24">Def24</a></i>;<br/>
<div><i><font color="Green" title="E44">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">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"> ( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> implies <span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p2"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c10">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> )</span><br/><b>assume </b><a NAME="E1:99_1_1"/><i><font color="Green" title="E45">A5</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p2"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c10">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span><br/><b>then consider </b><font color="Maroon" title="c12">Q</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span><b> such that </b><br/><a NAME="E3:99_1_1"/><i><font color="Green" title="E46">A6</font></i>: 
<font color="Maroon" title="c12">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c10">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <b>and </b><br/><a NAME="E4:99_1_1"/><i><font color="Green" title="E47">A7</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p2"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c10">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="ref" href="mssubfam.html#D2" title="MSSUBFAM:def.2">MSSUBFAM:def 2</a></i>;<br/><a NAME="E5:99_1_1"/><i><font color="Green" title="E48">A8</font></i>: 
<font color="Maroon" title="c9">x1</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/><a NAME="E6:99_1_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="closure2.html#D9" target="_self" title="CLOSURE2:def.9">Def9</a></i>;<br/><a NAME="E7:99_1_1"/><b>then </b>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F</font>
 <b>by </b><i><a class="txt" href="#E5:99_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="#E5:99_1_1"><i><font color="Green" title="E48">A8</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">F'</font> = <font color="Maroon" title="c10">F</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span> ;<br/><b>thus </b><a NAME="E9:99_1_1"/>
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p2"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c10">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</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="99_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:99_1_1_1"/><i><font color="Green" title="E49">A9</font></i>: 
<font color="Maroon" title="c12">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> where <font color="Olive" title="b1">q</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span> : <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">F'</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E1:99_1_1"><i><font color="Green" title="E45">A5</font></i></a>, <a class="txt" href="#E3:99_1_1"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="closure2.html#T15" target="_self" title="CLOSURE2:th.15">Th15</a></i>;<br/>
<a NAME="E2:99_1_1_1"/><i><font color="Green" title="E50">A10</font></i>: 
the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span> : <font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">q</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">q</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E1:99"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="closure2.html#D23" target="_self" title="CLOSURE2:def.23">Def23</a></i>;<br/>
<a NAME="E3:99_1_1_1"/>
<font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">x1</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">x1</font>
 
<b>by </b><i><a class="ref" href="closure2.html#D14" target="_self" title="CLOSURE2:def.14">Def14</a></i>;<br/>
<a NAME="E4:99_1_1_1"/><b>then </b><i><font color="Green" title="E51">A11</font></i>: 
<font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:99_1_1_1"><i><font color="Green" title="E50">A10</font></i></a></i>;<br/>
<a NAME="E5:99_1_1_1"/>
<font color="Maroon" title="c9">x1</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">x1</font>
 
<b>by </b><i><a class="ref" href="closure2.html#D12" target="_self" title="CLOSURE2:def.12">Def12</a></i>;<br/>
<a NAME="E6:99_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">F'</font>
 
<b>by </b><i><a class="txt" href="#E5:99_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="#E4:99_1_1_1"><i><font color="Green" title="E51">A11</font></i></a></i>;<br/>
<a NAME="E7:99_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 
<b>by </b><i><a class="txt" href="#E1:99_1_1_1"><i><font color="Green" title="E49">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:99_1_1_1"/>
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p2"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c10">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <b>by </b><i><a class="txt" href="#E4:99_1_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="mssubfam.html#T2" title="MSSUBFAM:th.2">MSSUBFAM:2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p2"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c10">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span><br/>

<a NAME="E9:99_1_1_1"/>
<font color="Maroon" title="c12">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c13">F'</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 
<b>by </b><i><a class="txt" href="#E3:99_1_1"><i><font color="Green" title="E46">A6</font></i></a></i>;<br/>
<a NAME="E10:99_1_1_1"/><b>then </b><i><font color="Green" title="E52">A12</font></i>: 
not <font color="Maroon" title="c12">Q</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="#E1:99_1_1"><i><font color="Green" title="E45">A5</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">z</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"> ( <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">z</font> )</span><br/><b>assume </b><a NAME="E1:99_1_1_1_1"/>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">z</font></span><br/><b>then consider </b><font color="Maroon" title="c15">q</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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span><b> such that </b><br/><a NAME="E3:99_1_1_1_1"/><i><font color="Green" title="E53">A13</font></i>: 
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <b>and </b><br/><a NAME="E4:99_1_1_1_1"/><i><font color="Green" title="E54">A14</font></i>: 
<font color="Maroon" title="c15">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">F'</font>
 <b>by </b><i><a class="txt" href="#E1:99_1_1_1"><i><font color="Green" title="E49">A9</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c16">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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span><b> such that </b><br/><a NAME="E6:99_1_1_1_1"/><i><font color="Green" title="E55">A15</font></i>: 
<font color="Maroon" title="c15">q</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c16">X</font>
 <b>and </b><br/><a NAME="E7:99_1_1_1_1"/><i><font color="Green" title="E56">A16</font></i>: 
<font color="Maroon" title="c9">x1</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c16">X</font>
 <b>and </b><br/><a NAME="E8:99_1_1_1_1"/><i><font color="Green" title="E57">A17</font></i>: 
<font color="Maroon" title="c16">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:99_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="#E4:99_1_1_1_1"><i><font color="Green" title="E54">A14</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c17">t</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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span><b> such that </b><br/><a NAME="E10:99_1_1_1_1"/><i><font color="Green" title="E58">A18</font></i>: 
<font color="Maroon" title="c16">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c17">t</font>
 <b>and </b><br/><a NAME="E11:99_1_1_1_1"/><i><font color="Green" title="E59">A19</font></i>: 
<font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c17">t</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c17">t</font>
 <b>by </b><i><a class="txt" href="#E2:99_1_1_1"><i><font color="Green" title="E50">A10</font></i></a>, <a class="txt" href="#E8:99_1_1_1_1"><i><font color="Green" title="E57">A17</font></i></a></i>;<br/><a NAME="E12:99_1_1_1_1"/>
<font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">x1</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c16">X</font>
 <b>by </b><i><a class="txt" href="#E7:99_1_1_1_1"><i><font color="Green" title="E56">A16</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="E13:99_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">z</font>
 <b>by </b><i><a class="txt" href="#E1:99_1_1"><i><font color="Green" title="E45">A5</font></i></a>, <a class="txt" href="#E3:99_1_1_1_1"><i><font color="Green" title="E53">A13</font></i></a>, <a class="txt" href="#E6:99_1_1_1_1"><i><font color="Green" title="E55">A15</font></i></a>, <a class="txt" href="#E10:99_1_1_1_1"><i><font color="Green" title="E58">A18</font></i></a>, <a class="txt" href="#E11:99_1_1_1_1"><i><font color="Green" title="E59">A19</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</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>
<b>hence </b><a NAME="E12:99_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">ff</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p2"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c10">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <b>by </b><i><a class="txt" href="#E4:99_1_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="txt" href="#E10:99_1_1_1"><i><font color="Green" title="E52">A12</font></i></a>, <a class="ref" href="mssubfam.html#T5" title="MSSUBFAM:th.5">MSSUBFAM:5</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><b>end;</b></div>
<b>thus </b><span class="p1">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <span class="p2">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> = 
 <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Maroon" title="c9">x1</font>
<b>by </b><i><a class="ref" href="closure2.html#D25" target="_self" title="CLOSURE2:def.25">Def25</a></i>
<br/>.= 
<font color="Maroon" title="c7">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="#E4:99_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="#E6:99_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</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>
<b>hence </b><a NAME="E4:99"/>
 <a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="#E1:99"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</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>
