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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="trees_3.html#V3">constituted-DTrees</a>  <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:16_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9">Subtrees</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:16_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="trees_2.html#K5">|</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="trees_9.html#T20">TREES_9:20</a></i>;<br/><a NAME="E4:16_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:16_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="circtrm1.html#E3:16_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="trees_9.html#T20">TREES_9:20</a></i>;<br/><b>hence </b><a NAME="E6:16_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>3</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="E2:16"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="circtrm1.html#E2:16"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1"><b>suppose </b></a><a NAME="E1:16_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:16_2_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="trees_2.html#K5">|</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="trees_9.html#T20">TREES_9:20</a></i>;<br/><a NAME="E4:16_2_1"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:16_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9">Subtrees</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circtrm1.html#E3:16_2_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="trees_9.html#T20">TREES_9:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2"><b>suppose </b></a><a NAME="E1:16_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9">Subtrees</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:16_2_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="trees_2.html#K5">|</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="trees_9.html#T20">TREES_9:20</a></i>;<br/><a NAME="E4:16_2_2"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:16_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9">Subtrees</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circtrm1.html#E3:16_2_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="trees_9.html#T20">TREES_9:20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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