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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="trees_3.html#V3" title="TREES_3:attr.3">constituted-DTrees</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Maroon" title="c1">C</font> <a href="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Olive" title="b1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff for <font color="Olive" title="b2">t</font> being   <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Olive" title="b1">X</font> holds <br/> ( <font color="Olive" title="b2">t</font> is  <a href="trees_9.html#NV1" title="TREES_9:NV.1">root</a>  &amp; not <font color="Olive" title="b2">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="c1">C</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="trees_3.html#V3" title="TREES_3:attr.3">constituted-DTrees</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">C</font> <a href="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c2">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff for <font color="Olive" title="b1">t</font> being   <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c2">X</font> holds <br/> ( <font color="Olive" title="b1">t</font> is  <a href="trees_9.html#NV1" title="TREES_9:NV.1">root</a>  &amp; not <font color="Olive" title="b1">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="c1">C</font> ) )</span><br/>



<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">t</font> being   <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c2">X</font> holds <br/> ( <font color="Olive" title="b1">t</font> is  <a href="trees_9.html#NV1" title="TREES_9:NV.1">root</a>  &amp; not <font color="Olive" title="b1">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="c1">C</font> ) ) implies <font color="Maroon" title="c1">C</font> <a href="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c2">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:58_1"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c1">C</font> <a href="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c2">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">t</font> being   <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c2">X</font> holds <br/> ( <font color="Olive" title="b1">t</font> is  <a href="trees_9.html#NV1" title="TREES_9:NV.1">root</a>  &amp; not <font color="Olive" title="b1">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="c1">C</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">t</font> be    <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">t</font> is  <a href="trees_9.html#NV1" title="TREES_9:NV.1">root</a>  &amp; not <font color="Maroon" title="c3">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="c1">C</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c4">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="c3">t</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></span>;<br/><a NAME="E3:58_1"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
not <font color="Maroon" title="c3">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> <a href="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1"><span class="lab"><font color="Green" title="E25">A1</font></span></a></span>;<br/><a NAME="E4:58_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">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="c3">t</font></span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E5:58_1"/>
( <font color="Maroon" title="c3">t</font> is  <a href="trees_9.html#NV1" title="TREES_9:NV.1">root</a>  &amp; not <font color="Maroon" title="c3">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="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="trees_9.html#T4" target="_self" title="TREES_9:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:58"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
for <font color="Olive" title="b1">t</font> being   <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c2">X</font> holds <br/> ( <font color="Olive" title="b1">t</font> is  <a href="trees_9.html#NV1" title="TREES_9:NV.1">root</a>  &amp; not <font color="Olive" title="b1">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="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c2">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/>

<span class="kw">assume </span><a NAME="E3:58"/>
not <font color="Maroon" title="c1">C</font> <a href="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c2">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">S</font> = <font color="Maroon" title="c1">C</font> <a href="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c2">X</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#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c2">X</font></span>)</span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">s</font> being    <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c3">S</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">t</font> being    <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c2">X</font>, <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="c5">t</font><span class="kw"> such that </span><br/><a NAME="E7:58"/>
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">and </span><br/><a NAME="E8:58"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
(  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="c5">t</font></span>)</span> or <font color="Maroon" title="c5">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="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_9.html#T25" target="_self" title="TREES_9:th.25">Th25</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">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="c5">t</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></span>;<br/>
<a NAME="E10:58"/>
<font color="Maroon" title="c5">t</font> is  <a href="trees_9.html#NV1" title="TREES_9:NV.1">root</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E25">A3</font></span></a></span>;<br/>
<a NAME="E11:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A5</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">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>
 
<span class="kw">by</span> <span class="lab"><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></span>;<br/>
<a NAME="E12:58"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E13:58"/><span class="kw">then </span>
<font color="Maroon" title="c7">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="c5">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E8:58"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="trees_1.html#T91" title="TREES_1:th.91">TREES_1:91</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:58"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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