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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="closure2.html#NM1">SubsetFamily</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="closure3.html#K2" target="_self">MSUnion</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure3.html#K3" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> as    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">c<sub>3</sub></font> as    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">c<sub>4</sub></font> as    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E7:18"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:18_1"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="closure3.html#K3" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="closure3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:18_1"/><i><font color="Green">E11</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="closure3.html#E1:18_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="pboole.html#D7">PBOOLE:def 7</a></i>;<br/>
<a NAME="E4:18_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E1:18_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="closure3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:18_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E1:18_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="closure3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E6:18_1"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:18_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E4:18_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="closure3.html#K3" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="closure3.html#E2:18_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:18_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E7:18_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="closure3.html#K3" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="closure3.html#E4:18_1_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E8:18_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="closure3.html#E7:18_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E9:18_1_1"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="closure3.html#E6:18_1_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E10:18_1_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="closure3.html#E3:18_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E11:18_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="closure3.html#E3:18_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="closure3.html#E4:18_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="closure3.html#E5:18_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:18_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:18_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E2:18_1_2"/><b>then </b><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="closure3.html#E3:18_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_2_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="closure3.html#E1:18_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="closure3.html#E2:18_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="closure3.html#D4" target="_self">Def4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_1"><b>suppose </b></a><a NAME="E1:18_1_2_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_1_2_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E4:18_1_2_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:18_1_2_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E7:18_1_2_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="closure3.html#E4:18_1_2_1_1"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E8:18_1_2_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="closure3.html#K3" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="closure3.html#E7:18_1_2_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:18_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="closure3.html#K3" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="closure3.html#E6:18_1_2_1_1"><i><font color="Green">E18</font></i></a></i>;<br/><b>hence </b><a NAME="E10:18_1_2_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="closure3.html#E2:18_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="closure3.html#E3:18_1_2_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_2"><b>suppose </b></a><a NAME="E1:18_1_2_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_1_2_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E4:18_1_2_1_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:18_1_2_1_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E7:18_1_2_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="closure3.html#E4:18_1_2_1_2"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E8:18_1_2_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="closure3.html#K3" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="closure3.html#E7:18_1_2_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:18_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="closure3.html#K3" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="closure3.html#E6:18_1_2_1_2"><i><font color="Green">E18</font></i></a></i>;<br/><b>hence </b><a NAME="E10:18_1_2_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="closure3.html#E2:18_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="closure3.html#E3:18_1_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E8:18_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="closure3.html#E6:18_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:18"/>
 <a href="closure3.html#K2" target="_self">MSUnion</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure3.html#K3" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K2">\/</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
