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<span class="kw">let </span><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><br/>  for <font color="Olive" title="b2">AA</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st (  for <font color="Olive" title="b3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">AA</font> holds <br/><font color="Olive" title="b3">x</font> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Olive" title="b1">M</font> ) holds <br/> for <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font> being   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Olive" title="b1">M</font>  st <font color="Olive" title="b4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b5">X</font></span>)</span> where <font color="Olive" title="b5">X</font> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Olive" title="b1">M</font> : <font color="Olive" title="b5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">AA</font> </span> } </span>  &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b2">AA</font> holds <br/> <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b4">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b3">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">AA</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">AA</font> holds <br/><font color="Olive" title="b2">x</font> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font> ) holds <br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b4">X</font></span>)</span> where <font color="Olive" title="b4">X</font> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">AA</font> </span> } </span>  &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">AA</font> holds <br/> <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b3">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b2">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">AA</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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> <font color="Maroon" title="c3">AA</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font> ) implies  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b3">X</font></span>)</span> where <font color="Olive" title="b3">X</font> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">AA</font> </span> } </span>  &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">AA</font> holds <br/> <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b2">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b1">A</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:18"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 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> <font color="Maroon" title="c3">AA</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b3">X</font></span>)</span> where <font color="Olive" title="b3">X</font> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">AA</font> </span> } </span>  &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">AA</font> holds <br/> <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b2">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b1">A</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> be   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">AA</font> </span> } </span>  &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">AA</font> implies  <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:18"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">AA</font> </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E3:18"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">AA</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span><br/>

<a NAME="E4:18"/>
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:18_1"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> implies <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/><span class="kw">thus </span><a NAME="E1:18_1_1"/>
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> implies <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_1_1_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/>

<a NAME="E2:18_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">f</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> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="closure3.html#D2" target="_self" title="CLOSURE3:def.2">Def4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:18_1_1_1"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <span class="kw">and </span><br/><a NAME="E5:18_1_1_1"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<font color="Maroon" title="c8">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">f</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> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">f</font> being    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font><span class="kw"> such that </span><br/><a NAME="E7:18_1_1_1"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Y</font>
 <span class="kw">and </span><br/><a NAME="E8:18_1_1_1"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_1_1_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">Q</font> being   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font><span class="kw"> such that </span><br/><a NAME="E10:18_1_1_1"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c10">Q</font>
 <span class="kw">and </span><br/><a NAME="E11:18_1_1_1"/><span class="lab"><font color="Green" title="E15">A10</font></span>: 
<font color="Maroon" title="c10">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">AA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E8:18_1_1_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a></span>;<br/>
<a NAME="E12:18_1_1_1"/>
<span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c10">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">g</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> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="closure3.html#D2" target="_self" title="CLOSURE3:def.2">Def4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E14:18_1_1_1"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">d</font>
 <span class="kw">and </span><br/><a NAME="E15:18_1_1_1"/><span class="lab"><font color="Green" title="E17">A12</font></span>: 
<font color="Maroon" title="c11">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">g</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> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_1_1_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E7:18_1_1_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E10:18_1_1_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">g</font> being    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font><span class="kw"> such that </span><br/><a NAME="E17:18_1_1_1"/><span class="lab"><font color="Green" title="E18">A13</font></span>: 
<font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E18:18_1_1_1"/><span class="lab"><font color="Green" title="E19">A14</font></span>: 
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:18_1_1_1"><span class="lab"><font color="Green" title="E17">A12</font></span></a></span>;<br/>
<a NAME="E19:18_1_1_1"/>
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">AA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:18_1_1_1"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="txt" href="#E18:18_1_1_1"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E20:18_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">h</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> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">AA</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:18_1_1_1"><span class="lab"><font color="Green" title="E18">A13</font></span></a></span>;<br/>
<a NAME="E21:18_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">h</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> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">AA</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18_1_1_1"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:18_1_1_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="closure3.html#D2" target="_self" title="CLOSURE3:def.2">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:18_1"/>
<span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:18"/>
 <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pboole.html#D10" title="PBOOLE:def.10">PBOOLE:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span><br/>

<a NAME="E6:18"/>
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:18_2"/><span class="lab"><font color="Green" title="E9">A15</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> or <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/>

<a NAME="E2:18_2"/>
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> implies <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_2_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/>

<a NAME="E2:18_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">f</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> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_2"><span class="lab"><font color="Green" title="E9">A15</font></span></a>, <a class="ref" href="closure3.html#D2" target="_self" title="CLOSURE3:def.2">Def4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:18_2_1"/><span class="lab"><font color="Green" title="E10">A16</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <span class="kw">and </span><br/><a NAME="E5:18_2_1"/><span class="lab"><font color="Green" title="E11">A17</font></span>: 
<font color="Maroon" title="c8">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">f</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> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">f</font> being    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font><span class="kw"> such that </span><br/><a NAME="E7:18_2_1"/><span class="lab"><font color="Green" title="E12">A18</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Y</font>
 <span class="kw">and </span><br/><a NAME="E8:18_2_1"/><span class="lab"><font color="Green" title="E13">A19</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_2_1"><span class="lab"><font color="Green" title="E11">A17</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:18_2_1"/><span class="lab"><font color="Green" title="E14">A20</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">and </span><br/><a NAME="E11:18_2_1"/><span class="lab"><font color="Green" title="E15">A21</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">AA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E8:18_2_1"><span class="lab"><font color="Green" title="E13">A19</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">c</font> = <font color="Maroon" title="c10">c</font> as   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E11:18_2_1"><span class="lab"><font color="Green" title="E15">A21</font></span></a></span>;<br/>
<a NAME="E13:18_2_1"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">v</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> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">c</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:18_2_1"><span class="lab"><font color="Green" title="E14">A20</font></span></a></span>;<br/>
<a NAME="E14:18_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A22</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">v</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> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">c</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_2_1"><span class="lab"><font color="Green" title="E10">A16</font></span></a>, <a class="txt" href="#E7:18_2_1"><span class="lab"><font color="Green" title="E12">A18</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E15:18_2_1"/>
<span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">v</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> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">c</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_2"><span class="lab"><font color="Green" title="E9">A15</font></span></a>, <a class="ref" href="closure3.html#D2" target="_self" title="CLOSURE3:def.2">Def4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">cos</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E17:18_2_1"/><span class="lab"><font color="Green" title="E17">A23</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">cos</font>
 <span class="kw">and </span><br/><a NAME="E18:18_2_1"/><span class="lab"><font color="Green" title="E18">A24</font></span>: 
<font color="Maroon" title="c12">cos</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18_2_1"><span class="lab"><font color="Green" title="E16">A22</font></span></a></span>;<br/>
<a NAME="E19:18_2_1"/>
 <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">AA</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:18_2_1"><span class="lab"><font color="Green" title="E15">A21</font></span></a></span>;<br/>
<a NAME="E20:18_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">cos</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">t</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> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E18:18_2_1"><span class="lab"><font color="Green" title="E18">A24</font></span></a></span>;<br/>
<a NAME="E21:18_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">t</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> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:18_2_1"><span class="lab"><font color="Green" title="E17">A23</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:18_2_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_2"><span class="lab"><font color="Green" title="E9">A15</font></span></a>, <a class="ref" href="closure3.html#D2" target="_self" title="CLOSURE3:def.2">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:18_2"/>
(  not <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> or <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:18"/>
 <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
