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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</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>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>









<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K4">Trees</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:53"/><i><font color="Green">E22</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E3:53"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="trees_3.html#M5">Element</a> of  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="trees_3.html#K10">`1</a>  )
 <b>and </b><br/><a NAME="E4:53"/><i><font color="Green">E24</font></i>: 
<span class="p1">(<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> <a href="trees_4.html#K12" target="_self">-tree</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="trees_3.html#K10">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="trees_4.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E5:53"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="trees_3.html#K5">FinTrees</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<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="E1:53_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:53_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</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="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="trees_4.html#E3:53_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="trees_3.html#M5">Element</a> of  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E6:53_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="trees_3.html#K10">`1</a>  )
 <b>by </b><i><a class="txt" href="trees_4.html#E2:53"><i><font color="Green">E22</font></i></a>, <a class="txt" href="trees_4.html#E3:53"><i><font color="Green">E23</font></i></a>, <a class="txt" href="trees_4.html#E3:53_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E7:53_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="trees_3.html#K10">`1</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="ref" href="trees_4.html#T24" target="_self">Th24</a></i>;<br/>
<b>hence </b><a NAME="E8:53_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K5">FinTrees</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="trees_4.html#E3:53_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="trees_4.html#E6:53_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="trees_3.html#D8">TREES_3:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:53"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K5">FinTrees</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:53"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_3.html#K5">FinTrees</a> <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being   <a href="trees_3.html#M5">Element</a> of  <a href="trees_3.html#K5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="trees_3.html#K10">`1</a>  ) ) &amp; <span class="p1">(<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> <a href="trees_4.html#K12" target="_self">-tree</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="trees_3.html#K10">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K12" target="_self">-tree</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="trees_4.html#E2:53"><i><font color="Green">E22</font></i></a>, <a class="txt" href="trees_4.html#E3:53"><i><font color="Green">E23</font></i></a>, <a class="txt" href="trees_4.html#E4:53"><i><font color="Green">E24</font></i></a></i>;<br/>


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