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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:81"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:81"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span> ;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:81"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:81"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E5:81"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E3:81"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E6:81"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E2:81"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E7:81"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E1:81"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rcomp_3.html#E2:81"><i><font color="Green">E58</font></i></a>, <a class="txt" href="rcomp_3.html#E3:81"><i><font color="Green">E59</font></i></a>, <a class="ref" href="rcomp_3.html#T59" target="_self">Th59</a></i>;<br/>
<a NAME="E8:81"/><i><font color="Green">E63</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E9:81"/><i><font color="Green">E64</font></i>: 
 <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>5</sub></font> is <a href="relat_2.html#V1">reflexive</a>
 
<b>by </b><i><a class="ref" href="wellord2.html#T2">WELLORD2:2</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D10">ORDERS_1:def 10</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E10:81"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E11:81"/><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>7</sub></font> <a href="orders_1.html#NR3" target="_self">is_linear-order</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1"/>
(  not <font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1"><b>suppose </b></a><a NAME="E1:81_1_1"/><i><font color="Green">E67</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>8</sub></font> =  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>7</sub></font>;<br/><b>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="E3:81_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>9</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="rcomp_3.html#E1:81_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:81_1_1"/>
<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="txt" href="rcomp_3.html#E10:81"><i><font color="Green">E65</font></i></a>, <a class="txt" href="rcomp_3.html#E3:81_1_1"><i><font color="Green">E68</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:81_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E7:81_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E8:81_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E4:81"><i><font color="Green">E60</font></i></a></i>;<br/><a NAME="E9:81_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E3:81_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="rcomp_3.html#E6:81_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="setfam_1.html#T8">SETFAM_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E11:81_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E6:81_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="rcomp_3.html#E8:81_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="rcomp_3.html#E3:81_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="setfam_1.html#T8">SETFAM_1:8</a></i>;<br/><a NAME="E12:81_1_1"/><i><font color="Green">E72</font></i>: 
 <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E6:81"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finset_1.html#T24">FINSET_1:24</a></i>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> ;<br/><a NAME="E13:81_1_1"/><i><font color="Green">E73</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="txt" href="rcomp_3.html#E12:81_1_1"><i><font color="Green">E72</font></i></a>);<br/></i><a NAME="E14:81_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:81_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:81_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E4:81"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:81_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div><a NAME="E15:81_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E13:81_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E16:81_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E10:81"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] means ( not <font color="Maroon">a<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> implies  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> );<br/><a NAME="E17:81_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 ;<br/><a NAME="E18:81_1_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:81_1_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E2:81_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E3:81_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>14</sub></font>]
 <b>and </b><br/><a NAME="E4:81_1_1_2"/>
not <font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1_1_2_1"/>
( <font color="Maroon">c<sub>14</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>14</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:81_1_1_2_1_1"/>
<font color="Maroon">c<sub>14</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:81_1_1_2_1_1"/>
 <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:81_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:81_1_1_2_1_2"/><i><font color="Green">E79</font></i>: 
not <font color="Maroon">c<sub>14</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E80</font></i>:  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="setfam_1.html#T10">SETFAM_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>
;<br/><a NAME="E3:81_1_1_2_1_2"/><i><font color="Green">E81</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E8:81"><i><font color="Green">E63</font></i></a>, <a class="txt" href="rcomp_3.html#E9:81"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rcomp_3.html#E10:81"><i><font color="Green">E65</font></i></a>, <a class="ref" href="orders_1.html#T181">ORDERS_1:181</a></i>;<br/><a NAME="E4:81_1_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>7</sub></font> is <a href="relat_2.html#V6">connected</a>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E11:81"><i><font color="Green">E66</font></i></a>, <a class="ref" href="orders_1.html#D5">ORDERS_1:def 5</a></i>;<br/><a NAME="E5:81_1_1_2_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>7</sub></font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a></i>;<br/><a NAME="E6:81_1_1_2_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>7</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:81_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="rcomp_3.html#E3:81_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="rcomp_3.html#E1:81_1_1_2_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="rcomp_3.html#E3:81_1_1_2_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="rcomp_3.html#E5:81_1_1_2_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/><a NAME="E7:81_1_1_2_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>5</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:81_1_1_2_1_2"/><b>then </b>
(  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:81_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="rcomp_3.html#E10:81"><i><font color="Green">E65</font></i></a>, <a class="txt" href="rcomp_3.html#E3:81_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="rcomp_3.html#E1:81_1_1_2_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><b>hence </b><a NAME="E9:81_1_1_2_1_2"/>
 <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:81_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="rcomp_3.html#E2:81_1_1_2_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="rcomp_3.html#E3:81_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="rcomp_3.html#E1:81_1_1_2_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E19:81_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>7</sub></font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2">FINSET_1:sch 2</a>(<a class="txt" href="rcomp_3.html#E16:81_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="rcomp_3.html#E17:81_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="rcomp_3.html#E18:81_1_1"><i><font color="Green">E76</font></i></a>);<br/></i><a NAME="E20:81_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:81_1_1_3"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>11</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:81_1_1_3_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:81_1_1_3_1"/>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E19:81_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="rcomp_3.html#E10:81"><i><font color="Green">E65</font></i></a>, <a class="txt" href="rcomp_3.html#E1:81_1_1"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E3:81_1_1_3_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E4:81"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:81_1_1_3_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:81_1_1_3_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:81_1_1_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:81_1_1_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E5:81_1_1_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:81_1_1_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E3:81_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="rcomp_3.html#E5:81_1_1_3"><i><font color="Green">E79</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>thus </b><a NAME="E7:81_1_1_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E4:81_1_1_3"><i><font color="Green">E78</font></i></a>, <a class="txt" href="rcomp_3.html#E6:81_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="rcomp_3.html#E6:81_1_1_3"><i><font color="Green">E80</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E21:81_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E11:81_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="rcomp_3.html#E4:81"><i><font color="Green">E60</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E22:81_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E23:81_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T8">SETFAM_1:8</a></i>;<br/><b>hence </b><a NAME="E24:81_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E10:81"><i><font color="Green">E65</font></i></a>, <a class="txt" href="rcomp_3.html#E21:81_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="rcomp_3.html#E22:81_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_2"><b>suppose </b></a><a NAME="E1:81_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/><b>thus </b><a NAME="E2:81_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>5</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E7:81"><i><font color="Green">E62</font></i></a>, <a class="txt" href="rcomp_3.html#E1:81_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="rcomp_3.html#E4:81"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
