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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>









<b>assume </b><a NAME="E1:13"/><i><font color="Green">E5</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>1</sub></font> iff ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>ex <font color="Olive">b<sub>5</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="mcart_1.html#K4">]</a></span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:13_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K3">:]</a></span>,<font color="Maroon">c<sub>5</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:13_1_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="mcart_1.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="domain_1.html#E1:13"><i><font color="Green">E5</font></i></a></i>;<br/>
<a NAME="E4:13_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1" target="_self">[</a><span class="default"><span class="p2"><a href="domain_1.html#K4" target="_self">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K4" target="_self">]</a></span>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="domain_1.html#E3:13_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="mcart_1.html#D4">MCART_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:13_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K3">:]</a></span>,<font color="Maroon">c<sub>5</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:13_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K3">:]</a></span>,<font color="Maroon">c<sub>5</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:13_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K3">:]</a></span>
 <b>and </b><br/><a NAME="E5:13_1"/><i><font color="Green">E7</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>and </b><br/><a NAME="E6:13_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="domain_1.html#E5:13_1"><i><font color="Green">E7</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E9:13_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4" target="_self">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K4" target="_self">]</a></span>
 <b>by </b><i><a class="txt" href="domain_1.html#E4:13_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="domain_1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E10:13_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="mcart_1.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="domain_1.html#E6:13_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="domain_1.html#E9:13_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="mcart_1.html#D4">MCART_1:def 4</a></i>;<br/><b>hence </b><a NAME="E11:13_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="domain_1.html#E1:13"><i><font color="Green">E5</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:13"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K3">:]</a></span>,<font color="Maroon">c<sub>5</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E4:13"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="zfmisc_1.html#K4">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D4">ZFMISC_1:def 4</a></i>;<br/>


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