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

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E1:73_1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="fsm_1.html#V4" target="_self">final</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="fsm_1.html#V4" target="_self">final</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:73_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:73_1_2_1"/><b>then </b>
not for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:73_1_2_1"/><i><font color="Green">E52</font></i>: 
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> ) or ( not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> ) )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_2_1_1"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> ) or ( not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="fsm_1.html#E4:73_1_2_1"><i><font color="Green">E52</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:73_1_2_1_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:73_1_2_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="fsm_1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E5:73_1_2_1_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:73_1_2_1_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E7:73_1_2_1_1_1"><i><font color="Green">E55</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>10</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>;<br/><a NAME="E9:73_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:73_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/>
<a NAME="E3:73_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E4:73_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="fsm_1.html#E7:73_1_2_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E11:73_1_2_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2_1_1_1"><i><font color="Green">E53</font></i></a></i>;<br/><div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:73_1_2_1_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:73_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:73_1_2_1_1_1_2"/>
not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1_2_1_1_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:73_1_2_1_1_1_2_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self">Def14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_2_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:73_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:73_1_2_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="fsm_1.html#E3:73_1_2_1_1_1_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:73_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><a NAME="E2:73_1_2_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E7:73_1_2_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><a NAME="E3:73_1_2_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:73_1_2_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E7:73_1_2_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="fsm_1.html#E3:73_1_2_1_1_1_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E4:73_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="fsm_1.html#E4:73_1_2_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="fsm_1.html#E2:73_1_2_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self">Def14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:73_1_2_1_1_1_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E2:73_1_2_1_1_1_3"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:73_1_2_1_1_1_3"/>
not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1_2_1_1_1_3_1"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:73_1_2_1_1_1_3_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self">Def14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_2_1_1_1_3_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:73_1_2_1_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:73_1_2_1_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2_1_1_1_3"><i><font color="Green">E58</font></i></a>, <a class="txt" href="fsm_1.html#E3:73_1_2_1_1_1_3_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:73_1_2_1_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><a NAME="E2:73_1_2_1_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><a NAME="E3:73_1_2_1_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:73_1_2_1_1_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:73_1_2_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="fsm_1.html#E3:73_1_2_1_1_1_3_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E4:73_1_2_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="fsm_1.html#E7:73_1_2_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="fsm_1.html#E2:73_1_2_1_1_1_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self">Def14</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:73_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E11:73_1_2_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="fsm_1.html#E13:73_1_2_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:73_1_2_1_1_2"/><i><font color="Green">E53</font></i>: 
( not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:73_1_2_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="fsm_1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E5:73_1_2_1_1_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:73_1_2_1_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E7:73_1_2_1_1_2"><i><font color="Green">E55</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>10</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>;<br/><a NAME="E9:73_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:73_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/>
<a NAME="E3:73_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E4:73_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="fsm_1.html#E7:73_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E11:73_1_2_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2_1_1_2"><i><font color="Green">E53</font></i></a></i>;<br/><div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:73_1_2_1_1_2_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:73_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:73_1_2_1_1_2_2"/>
not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1_2_1_1_2_2_1"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:73_1_2_1_1_2_2_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self">Def14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_2_1_1_2_2_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:73_1_2_1_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:73_1_2_1_1_2_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2_1_1_2_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="fsm_1.html#E3:73_1_2_1_1_2_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:73_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><a NAME="E2:73_1_2_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E7:73_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><a NAME="E3:73_1_2_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:73_1_2_1_1_2_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E7:73_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="fsm_1.html#E3:73_1_2_1_1_2_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E4:73_1_2_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="fsm_1.html#E4:73_1_2_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="fsm_1.html#E2:73_1_2_1_1_2_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self">Def14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:73_1_2_1_1_2_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E2:73_1_2_1_1_2_3"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:73_1_2_1_1_2_3"/>
not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1_2_1_1_2_3_1"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:73_1_2_1_1_2_3_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self">Def14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_2_1_1_2_3_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:73_1_2_1_1_2_3_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:73_1_2_1_1_2_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2_1_1_2_3"><i><font color="Green">E58</font></i></a>, <a class="txt" href="fsm_1.html#E3:73_1_2_1_1_2_3_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:73_1_2_1_1_2_3_1_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><a NAME="E2:73_1_2_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><a NAME="E3:73_1_2_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:73_1_2_1_1_2_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:73_1_2_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="fsm_1.html#E3:73_1_2_1_1_2_3_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E4:73_1_2_1_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:73_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="fsm_1.html#E7:73_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="fsm_1.html#E2:73_1_2_1_1_2_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self">Def14</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:73_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E11:73_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="fsm_1.html#E13:73_1_2_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:73_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


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