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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> holds <br/> (  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  iff <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <font color="Maroon" title="c1">T</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">t</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  iff <font color="Maroon" title="c2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <font color="Maroon" title="c1">T</font> )</span><br/>



<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <font color="Maroon" title="c1">T</font> implies  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:3_1"/>
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <font color="Maroon" title="c1">T</font></span><br/><a NAME="E2:3_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c2">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c2">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">T</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#D5" onmouseover="rs('trees_2/D5')" onmouseout="rh()">TREES_2:def 5</a></span>;<br/><a NAME="E3:3_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c2">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">T</font>
 ;<br/><span class="kw">hence </span><a NAME="E4:3_1"/>
<font color="Maroon" title="c2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T54" onmouseover="rs('trees_1/T54')" onmouseout="rh()">TREES_1:54</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">set </span><font color="Maroon" title="c3">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c2">t</font>;<br/>
<span class="kw">assume </span><a NAME="E2:3"/>
<font color="Maroon" title="c2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <font color="Maroon" title="c1">T</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E3:3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E1">A1</font></span>: 
 not <font color="Maroon" title="c2">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T54" onmouseover="rs('trees_1/T54')" onmouseout="rh()">TREES_1:54</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:3"/>
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:3"/><span class="kw">then </span>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c2">t</font>
 
;<br/>
<a NAME="E6:3"/><span class="kw">then </span>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c2">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">T</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#D5" onmouseover="rs('trees_2/D5')" onmouseout="rh()">TREES_2:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:3"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default"><font color="Maroon" title="c4">n</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E9:3"/><span class="lab"><font color="Green" title="E2">A2</font></span>: 
<font color="Maroon" title="c2">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default"><font color="Maroon" title="c4">n</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">T</font>
 ;<br/>
<a NAME="E10:3"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" onmouseover="rs('nat_1/T2')" onmouseout="rh()">NAT_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="txt" href="#E9:3"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="ref" href="trees_1.html#D3" onmouseover="rs('trees_1/D3')" onmouseout="rh()">TREES_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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