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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:35_1_1"/><i><font color="Green">E21</font></i>: 
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>3</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E3:35_1_1"/><i><font color="Green">E22</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D1">ZFMISC_1:def 1</a></i>;<br/>
<a NAME="E4:35_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</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="group_4.html#E2:35_1_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E5:35_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="group_4.html#E2:35_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="group_4.html#E3:35_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:35_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E4:35_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E2:35_1_1"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E5:35_1_1_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T55">GROUP_2:55</a></i>;<br/><b>hence </b><a NAME="E6:35_1_1_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<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="txt" href="group_4.html#E3:35_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:35_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="group_4.html#E5:35_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E8:35_1_1"/>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:35_1_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>5</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="group_4.html#E4:35_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:35_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E5:35_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E2:35_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="group_4.html#E2:35_1_1_2"><i><font color="Green">E26</font></i></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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E6:35_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E7:35_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</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="group_4.html#E2:35_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="group_4.html#E4:35_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:35_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:35_1_1_3"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1_1_3_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:35_1_1_3_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E4:35_1_1_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E2:35_1_1"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E5:35_1_1_3_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E1:35_1_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="group_4.html#E1:35_1_1_3_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="group_4.html#E3:35_1_1_3_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E6:35_1_1_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E3:35_1_1_3_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="group_2.html#T89">GROUP_2:89</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:35_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</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="group_4.html#E5:35_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:35_1_1_4"/><i><font color="Green">E27</font></i>: 
<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="hs2(this)" href="javascript:()" title="35_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1_1_4_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:35_1_1_4_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E4:35_1_1_4_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E2:35_1_1"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E5:35_1_1_4_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E1:35_1_1_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="group_4.html#E1:35_1_1_4_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="group_4.html#E3:35_1_1_4_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E6:35_1_1_4_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E3:35_1_1_4_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="group_2.html#T90">GROUP_2:90</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:35_1_1_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E5:35_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:35_1_1"/><i><font color="Green">E27</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E7:35_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="group_4.html#E10:35_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="group_2.html#T61">GROUP_2:61</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1_1_5"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:35_1_1_5"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E2:35_1_1"><i><font color="Green">E21</font></i></a></i>;<br/><b>hence </b><a NAME="E3:35_1_1_5"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:35_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E5:35_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="group_4.html#E13:35_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E16:35_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E17:35_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>7</sub></font>
 
;<br/>
<a NAME="E18:35_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</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="group_4.html#E2:35_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="group_4.html#E16:35_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E19:35_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<b>hence </b><a NAME="E20:35_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E13:35_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="group_2.html#T66">GROUP_2:66</a></i>;<br/>


</div>
