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

<b>let </b><font color="Maroon" title="c1">t</font> be   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">C</font> <a href="trees_9.html#K7" title="TREES_9:func.7">-Subtrees</a> <font color="Maroon" title="c1">t</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff ( <font color="Maroon" title="c1">t</font> is <a href="trees_9.html#V1" title="TREES_9:attr.1">root</a> &amp; not <font color="Maroon" title="c1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">C</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">C</font> <a href="trees_9.html#K7" title="TREES_9:func.7">-Subtrees</a> <font color="Maroon" title="c1">t</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff ( <font color="Maroon" title="c1">t</font> is <a href="trees_9.html#V1" title="TREES_9:attr.1">root</a> &amp; not <font color="Maroon" title="c1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> ) )</span><br/>



<b>reconsider </b><font color="Maroon" title="c3">e</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">t</font> <b>by </b><i><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">t</font> is <a href="trees_9.html#V1" title="TREES_9:attr.1">root</a> &amp; not <font color="Maroon" title="c1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> implies <font color="Maroon" title="c2">C</font> <a href="trees_9.html#K7" title="TREES_9:func.7">-Subtrees</a> <font color="Maroon" title="c1">t</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon" title="c2">C</font> <a href="trees_9.html#K7" title="TREES_9:func.7">-Subtrees</a> <font color="Maroon" title="c1">t</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">t</font> is <a href="trees_9.html#V1" title="TREES_9:attr.1">root</a> &amp; not <font color="Maroon" title="c1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> )</span><br/><a NAME="E2:47_1"/><b>then </b>
not <font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> <a href="trees_9.html#K7" title="TREES_9:func.7">-Subtrees</a> <font color="Maroon" title="c1">t</font>
 ;<br/><a NAME="E3:47_1"/><b>then </b>
( <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">t</font></span>)</span> &amp; not <font color="Maroon" title="c1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> )
 ;<br/><b>hence </b><a NAME="E4:47_1"/>
( <font color="Maroon" title="c1">t</font> is <a href="trees_9.html#V1" title="TREES_9:attr.1">root</a> &amp; not <font color="Maroon" title="c1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> )
 <b>by </b><i><a class="ref" href="trees_9.html#T4" target="_self" title="TREES_9:th.4">Th4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:47"/><i><font color="Green" title="E23">A1</font></i>: 
( <font color="Maroon" title="c1">t</font> is <a href="trees_9.html#V1" title="TREES_9:attr.1">root</a> &amp; not <font color="Maroon" title="c1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">C</font> <a href="trees_9.html#K7" title="TREES_9:func.7">-Subtrees</a> <font color="Maroon" title="c1">t</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/>

<b>assume </b><a NAME="E4:47"/>
not <font color="Maroon" title="c2">C</font> <a href="trees_9.html#K7" title="TREES_9:func.7">-Subtrees</a> <font color="Maroon" title="c1">t</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then reconsider </b><font color="Maroon" title="c4">S</font> = <font color="Maroon" title="c2">C</font> <a href="trees_9.html#K7" title="TREES_9:func.7">-Subtrees</a> <font color="Maroon" title="c1">t</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_9.html#K3" title="TREES_9:func.3">Subtrees</a> <font color="Maroon" title="c1">t</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c5">s</font> being    <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c4">S</font>;<br/>
<b>consider </b><font color="Maroon" title="c6">n</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">t</font><b> such that </b><br/><a NAME="E8:47"/><i><font color="Green" title="E24">A2</font></i>: 
( <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c6">n</font> &amp; (  not <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">t</font></span>)</span> or <font color="Maroon" title="c1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> ) )
 <b>by </b><i><a class="ref" href="trees_9.html#T18" target="_self" title="TREES_9:th.18">Th18</a></i>;<br/>
<a NAME="E9:47"/><i><font color="Green" title="E25">A3</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="#E3:47"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="trees_9.html#D1" target="_self" title="TREES_9:def.1">Def1</a>, <a class="ref" href="trees_1.html#T56" title="TREES_1:th.56">TREES_1:56</a></i>;<br/>
<a NAME="E10:47"/><b>then </b>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:47"/><b>then </b>
( <font color="Maroon" title="c3">e</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">t</font> &amp; <font color="Maroon" title="c3">e</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="#E3:47"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E8:47"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a>, <a class="ref" href="trees_1.html#T91" title="TREES_1:th.91">TREES_1:91</a></i>;<br/>
<b>hence </b><a NAME="E12:47"/>
contradiction
 <b>by </b><i><a class="txt" href="#E9:47"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
