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<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">D</font> being  <a href="closure2.html#NM4" title="CLOSURE2:NM.4">ClosureSystem</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">f</font> being  <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 <font color="Olive" title="b1">D</font>  st ( 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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">D</font> holds  <font color="Olive" title="b2">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b3">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Olive" title="b3">x</font> ) holds <br/>( <font color="Olive" title="b2">f</font> is  <a href="closure2.html#V7" title="CLOSURE2:attr.7">reflexive</a>  &amp; <font color="Olive" title="b2">f</font> is  <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a>  &amp; <font color="Olive" title="b2">f</font> is  <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">D</font> be   <a href="closure2.html#NM4" title="CLOSURE2:NM.4">ClosureSystem</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <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 <font color="Maroon" title="c2">D</font>  st ( 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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> holds  <font color="Olive" title="b1">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b2">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Olive" title="b2">x</font> ) holds <br/>( <font color="Olive" title="b1">f</font> is  <a href="closure2.html#V7" title="CLOSURE2:attr.7">reflexive</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <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 <font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> holds  <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Olive" title="b1">x</font> ) implies ( <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>  ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:98"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> holds  <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Olive" title="b1">x</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>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">M</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>;<br/>
<a NAME="E2:98"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="closure2.html#V7" title="CLOSURE2:attr.7">reflexive</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</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="c5">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">x</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">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 <font color="Maroon" title="c2">D</font><span class="kw"> such that </span><br/><a NAME="E2:98_1"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
 <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Maroon" title="c5">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c6">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 <span class="kw">and </span><br/><a NAME="E3:98_1"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<font color="Maroon" title="c6">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 <font color="Maroon" title="c2">D</font> : ( <font color="Maroon" title="c5">x</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 <font color="Maroon" title="c2">D</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#D24" target="_self" title="CLOSURE2:def.24">Def24</a></span>;<br/>
<a NAME="E4:98_1"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
for <font color="Olive" title="b1">Z1</font> being  <a href="pboole.html#NM1" title="PBOOLE:NM.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>  st <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> holds <br/><font color="Maroon" title="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Olive" title="b1">Z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">Z1</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> implies <font color="Maroon" title="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c7">Z1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:98_1_1"/>
<font color="Maroon" title="c7">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">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="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c7">Z1</font></span><br/>

<a NAME="E2:98_1_1"/><span class="kw">then </span>
 ex <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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> st <br/>( <font color="Maroon" title="c7">Z1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</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 <font color="Maroon" title="c2">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:98_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c7">Z1</font>
 ; <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>
<a NAME="E5:98_1"/>
<font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c6">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E2:98_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:98_1"/>
<font color="Maroon" title="c5">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">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="closure2.html#T25" target="_self" title="CLOSURE2:th.25">Th25</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>
<a NAME="E3:98"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</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="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c6">y</font> implies <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">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="c6">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:98_2"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c6">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="c5">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="c6">y</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">Fy</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 <font color="Maroon" title="c2">D</font><span class="kw"> such that </span><br/><a NAME="E3:98_2"/><span class="lab"><font color="Green" title="E44">A8</font></span>: 
 <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Maroon" title="c6">y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c7">Fy</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 <span class="kw">and </span><br/><a NAME="E4:98_2"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
<font color="Maroon" title="c7">Fy</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 <font color="Maroon" title="c2">D</font> : ( <font color="Maroon" title="c6">y</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 <font color="Maroon" title="c2">D</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#D24" target="_self" title="CLOSURE2:def.24">Def24</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">Fx</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 <font color="Maroon" title="c2">D</font><span class="kw"> such that </span><br/><a NAME="E6:98_2"/><span class="lab"><font color="Green" title="E46">A10</font></span>: 
 <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Maroon" title="c5">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fx</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 <span class="kw">and </span><br/><a NAME="E7:98_2"/><span class="lab"><font color="Green" title="E47">A11</font></span>: 
<font color="Maroon" title="c8">Fx</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 <font color="Maroon" title="c2">D</font> : ( <font color="Maroon" title="c5">x</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 <font color="Maroon" title="c2">D</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#D24" target="_self" title="CLOSURE2:def.24">Def24</a></span>;<br/>
<a NAME="E8:98_2"/>
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c7">Fy</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fx</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">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> or <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c7">Fy</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="c9">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fx</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="c9">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:98_2_1"/><span class="lab"><font color="Green" title="E48">A12</font></span>: 
<font color="Maroon" title="c9">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c7">Fy</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="c9">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fx</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="c9">i</font></span><br/>

<span class="kw">thus </span><a NAME="E2:98_2_1"/>
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c7">Fy</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="c9">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fx</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="c9">i</font>
  <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="98_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c7">Fy</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="c9">i</font> or <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fx</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="c9">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:98_2_1_1"/><span class="lab"><font color="Green" title="E49">A13</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c7">Fy</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="c9">i</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="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fx</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="c9">i</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:98_2_1_1_1"/>
( <font color="Maroon" title="c7">Fy</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c7">Fy</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:98_2_1_1_1_1"/>
<font color="Maroon" title="c7">Fy</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fx</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="c9">i</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">Fy9</font> = <font color="Maroon" title="c7">Fy</font> as   <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 <font color="Maroon" title="c2">D</font> ;<br/><a NAME="E3:98_2_1_1_1_1"/>
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c11">Fy9</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="c9">i</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_2_1"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="ref" href="pboole.html#D15" title="PBOOLE:def.15">PBOOLE:def 15</a></span>;<br/><span class="kw">hence </span><a NAME="E4:98_2_1_1_1_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fx</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="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_2_1_1"><span class="lab"><font color="Green" title="E49">A13</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:98_2_1_1_1_2"/>
not <font color="Maroon" title="c7">Fy</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fx</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="c9">i</font></span><br/><div class="add"><a NAME="E2:98_2_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c7">Fy</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="c9">i</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">e</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> where <font color="Olive" title="b1">e</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 <font color="Maroon" title="c2">D</font> : <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Fy</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_2_1"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="ref" href="closure2.html#T15" target="_self" title="CLOSURE2:th.15">Th15</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">w</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 <font color="Maroon" title="c2">D</font><span class="kw"> such that </span><br/><a NAME="E4:98_2_1_1_1_2"/><span class="lab"><font color="Green" title="E50">A14</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E5:98_2_1_1_1_2"/><span class="lab"><font color="Green" title="E51">A15</font></span>: 
<font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Fy</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_2_1_1"><span class="lab"><font color="Green" title="E49">A13</font></span></a></span>;<br/><a NAME="E6:98_2_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A16</font></span>: 
 ex <font color="Olive" title="b1">r</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 <font color="Maroon" title="c2">D</font> st <br/>( <font color="Olive" title="b1">r</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c11">w</font> &amp; <font color="Maroon" title="c6">y</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</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 <font color="Maroon" title="c2">D</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_2"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E5:98_2_1_1_1_2"><span class="lab"><font color="Green" title="E51">A15</font></span></a></span>;<br/><a NAME="E7:98_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c11">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_2"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></span>;<br/><a NAME="E8:98_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A17</font></span>: 
<font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Fx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:98_2"><span class="lab"><font color="Green" title="E47">A11</font></span></a>, <a class="txt" href="#E6:98_2_1_1_1_2"><span class="lab"><font color="Green" title="E52">A16</font></span></a></span>;<br/><a NAME="E9:98_2_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fx</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="c9">i</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">e</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> where <font color="Olive" title="b1">e</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 <font color="Maroon" title="c2">D</font> : <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Fx</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_2_1"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="ref" href="closure2.html#T15" target="_self" title="CLOSURE2:th.15">Th15</a></span>;<br/><span class="kw">hence </span><a NAME="E10:98_2_1_1_1_2"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fx</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="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_2_1_1_1_2"><span class="lab"><font color="Green" title="E50">A14</font></span></a>, <a class="txt" href="#E8:98_2_1_1_1_2"><span class="lab"><font color="Green" title="E53">A17</font></span></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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E9:98_2"/><span class="kw">then </span>
 <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fx</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a>  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c7">Fy</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mssubfam.html#T46" title="MSSUBFAM:th.46">MSSUBFAM:46</a></span>;<br/>
<a NAME="E10:98_2"/><span class="kw">then </span>
 <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fx</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <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="c6">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E3:98_2"><span class="lab"><font color="Green" title="E44">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:98_2"/>
<font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">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="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E6:98_2"><span class="lab"><font color="Green" title="E46">A10</font></span></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>
<a NAME="E4:98"/>
<font color="Maroon" title="c3">f</font> is  <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</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="c5">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="c5">x</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">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 <font color="Maroon" title="c2">D</font><span class="kw"> such that </span><br/><a NAME="E2:98_3"/><span class="lab"><font color="Green" title="E43">A18</font></span>: 
 <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Maroon" title="c5">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c6">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 <span class="kw">and </span><br/><a NAME="E3:98_3"/><span class="lab"><font color="Green" title="E44">A19</font></span>: 
<font color="Maroon" title="c6">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 <font color="Maroon" title="c2">D</font> : ( <font color="Maroon" title="c5">x</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 <font color="Maroon" title="c2">D</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#D24" target="_self" title="CLOSURE2:def.24">Def24</a></span>;<br/>
<a NAME="E4:98_3"/>
<font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> or <font color="Maroon" title="c7">k</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 <font color="Maroon" title="c2">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:98_3_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">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="c7">k</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 <font color="Maroon" title="c2">D</font></span><br/>

<a NAME="E2:98_3_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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 <font color="Maroon" title="c2">D</font> st <br/>( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> &amp; <font color="Maroon" title="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Olive" title="b1">q</font> &amp; <font color="Olive" title="b1">q</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 <font color="Maroon" title="c2">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_3"><span class="lab"><font color="Green" title="E44">A19</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:98_3_1"/>
<font color="Maroon" title="c7">k</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 <font color="Maroon" title="c2">D</font>
 ; <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>
<a NAME="E5:98_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A20</font></span>: 
 <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c6">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></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 <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#D8" target="_self" title="CLOSURE2:def.8">Def8</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">x</font> = 
 <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c6">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E2:98_3"><span class="lab"><font color="Green" title="E43">A18</font></span></a></span>
<br/>.= 
<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"><a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p2"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c6">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E5:98_3"><span class="lab"><font color="Green" title="E45">A20</font></span></a>, <a class="ref" href="closure2.html#T40" target="_self" title="CLOSURE2:th.40">Th40</a></span>
<br/>.= 
<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="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E2:98_3"><span class="lab"><font color="Green" title="E43">A18</font></span></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">hence </span><a NAME="E5:98"/>
( <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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E42">A6</font></span></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>
