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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:34"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_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>thus </b><a NAME="E1:34_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K7" target="_self">Maps</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K7" target="_self">Maps</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K7" target="_self">Maps</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:34_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="ens_1.html#K7" target="_self">Maps</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="ens_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E3:34_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="ens_1.html#E1:34_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="ens_1.html#E1:34_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="ens_1.html#E2:34_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E5:34_1_1"/>
 <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="ens_1.html#T7" target="_self">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="ens_1.html#E3:34_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E7:34_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E1:34_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="ens_1.html#T8" target="_self">Th8</a>, <a class="ref" href="ens_1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E8:34_1_1"/><b>then </b>
(  <a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E10"><i><font color="Green">Lemma6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:34_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K7" target="_self">Maps</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="ens_1.html#E1:34"><i><font color="Green">E24</font></i></a>, <a class="ref" href="ens_1.html#T19" target="_self">Th19</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:34_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K7" target="_self">Maps</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E3:34_1"/>
 <a href="ens_1.html#K7" target="_self">Maps</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="ens_1.html#T17" target="_self">Th17</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="ens_1.html#E2:34_1"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E5:34_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="ens_1.html#E2:34_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="ens_1.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E6:34_1"/><b>then </b>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Maroon">c<sub>8</sub></font> <a href="mcart_1.html#K2">`2</a>  is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="ens_1.html#E2:34_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="ens_1.html#E29"><i><font color="Green">Lemma20</font></i></a></i>;<br/><b>hence </b><a NAME="E7:34_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K7" target="_self">Maps</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ens_1.html#E1:34"><i><font color="Green">E24</font></i></a>, <a class="txt" href="ens_1.html#E5:34_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="ens_1.html#T15" target="_self">Th15</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:34"/>
 <a href="ens_1.html#K7" target="_self">Maps</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K7" target="_self">Maps</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
