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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="trees_3.html#NM7">DecoratedTree</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<a NAME="E1:94"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="trees_3.html#K10" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="trees_3.html#K8" target="_self">pr1</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_3.html#K7" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="trees_3.html#K11" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="trees_3.html#K9" target="_self">pr2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_3.html#K7" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="trees_3.html#K8" target="_self">pr1</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</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> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="trees_3.html#K9" target="_self">pr2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</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> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</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>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="trees_2.html#D9">TREES_2:def 9</a></i>;<br/>
<a NAME="E2:94"/><b>then </b><i><font color="Green">E42</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_3.html#K10" target="_self">`1</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_3.html#K11" target="_self">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E3:94"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="trees_3.html#K6" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_3.html#K10" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_3.html#K11" target="_self">`2</a> </span>)</span></span><a href="trees_3.html#K6" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_3.html#T70">FUNCT_3:70</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:94_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> ;<br/><b>thus </b><span class="p1"><a href="trees_3.html#K6" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_3.html#K10" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_3.html#K11" target="_self">`2</a> </span>)</span></span><a href="trees_3.html#K6" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_3.html#K10" target="_self">`1</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_3.html#K11" target="_self">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="trees_3.html#E2:94"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_3.html#T69">FUNCT_3:69</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_3.html#K11" target="_self">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="trees_3.html#T41" target="_self">Th41</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="trees_3.html#T41" target="_self">Th41</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:94"/>
<span class="p1"><a href="trees_3.html#K6" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_3.html#K10" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_3.html#K11" target="_self">`2</a> </span>)</span></span><a href="trees_3.html#K6" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="trees_3.html#E3:94"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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