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

<span class="kw">let </span><font color="Maroon" title="c1">T0</font>, <font color="Maroon" title="c2">T1</font> be   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</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_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font> holds <br/> ( ( for <font color="Olive" title="b2">p</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T0</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">p</font> holds <br/>( <font color="Olive" title="b1">t</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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span> iff <font color="Olive" title="b2">p</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> <font color="Maroon" title="c1">T0</font> ) ) &amp; ( for <font color="Olive" title="b2">p</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">p</font> holds <br/>( <font color="Olive" title="b1">t</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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span> iff <font color="Olive" title="b2">p</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> <font color="Maroon" title="c2">T1</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">t</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font>; <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">p</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T0</font>  st <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">p</font> holds <br/>( <font color="Maroon" title="c3">t</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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span> iff <font color="Olive" title="b1">p</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> <font color="Maroon" title="c1">T0</font> ) ) &amp; ( for <font color="Olive" title="b1">p</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font>  st <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">p</font> holds <br/>( <font color="Maroon" title="c3">t</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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span> iff <font color="Olive" title="b1">p</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> <font color="Maroon" title="c2">T1</font> ) ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">RT</font> =  <a href="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font>;<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">p</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font>  st <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">p</font> holds <br/>( <font color="Maroon" title="c3">t</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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span> iff <font color="Olive" title="b1">p</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> <font color="Maroon" title="c2">T1</font> )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c5">p</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T0</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">p</font> implies ( ( <font color="Maroon" title="c3">t</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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span> implies <font color="Maroon" title="c5">p</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> <font color="Maroon" title="c1">T0</font> ) &amp; ( <font color="Maroon" title="c5">p</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> <font color="Maroon" title="c1">T0</font> implies <font color="Maroon" title="c3">t</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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span> ) ) )</span><br/><span class="kw">assume </span><a NAME="E1:13_1"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">p</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> <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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span> implies <font color="Maroon" title="c5">p</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> <font color="Maroon" title="c1">T0</font> ) &amp; ( <font color="Maroon" title="c5">p</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> <font color="Maroon" title="c1">T0</font> implies <font color="Maroon" title="c3">t</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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span> ) )</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"> ( <font color="Maroon" title="c5">p</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> <font color="Maroon" title="c1">T0</font> implies <font color="Maroon" title="c3">t</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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:13_1_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c3">t</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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span>
 ; <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="c5">p</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> <font color="Maroon" title="c1">T0</font></span><br/><span class="kw">assume </span><a NAME="E2:13_1_1"/>
not <font color="Maroon" title="c5">p</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> <font color="Maroon" title="c1">T0</font>
 ; <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">consider </span><font color="Maroon" title="c6">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:13_1_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T92" title="TREES_1:th.92">TREES_1:92</a></span>;<br/><a NAME="E5:13_1_1"/>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="trees_3.html#T72" title="TREES_3:th.72">TREES_3:72</a></span>;<br/><a NAME="E6:13_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/><span class="kw">hence </span><a NAME="E7:13_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E1:13_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="trees_1.html#T92" title="TREES_1:th.92">TREES_1:92</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="E3:13_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<font color="Maroon" title="c5">p</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> <font color="Maroon" title="c1">T0</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> <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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span></span><br/><span class="kw">assume </span><a NAME="E4:13_1"/>
not <font color="Maroon" title="c3">t</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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span>
 ; <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">consider </span><font color="Maroon" title="c6">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:13_1"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<font color="Maroon" title="c3">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T92" title="TREES_1:th.92">TREES_1:92</a></span>;<br/><a NAME="E7:13_1"/>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E6:13_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/><a NAME="E8:13_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#T72" title="TREES_3:th.72">TREES_3:72</a></span>;<br/><span class="kw">hence </span><a NAME="E9:13_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="trees_1.html#T92" title="TREES_1:th.92">TREES_1:92</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">let </span><font color="Maroon" title="c5">p</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">p</font> implies ( <font color="Maroon" title="c3">t</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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span> iff <font color="Maroon" title="c5">p</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> <font color="Maroon" title="c2">T1</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:13"/><span class="lab"><font color="Green" title="E5">A6</font></span>: 
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">p</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> <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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span> iff <font color="Maroon" title="c5">p</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> <font color="Maroon" title="c2">T1</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"> ( <font color="Maroon" title="c5">p</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> <font color="Maroon" title="c2">T1</font> implies <font color="Maroon" title="c3">t</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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:13_2"/><span class="lab"><font color="Green" title="E6">A7</font></span>: 
<font color="Maroon" title="c3">t</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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span>
 ; <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="c5">p</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> <font color="Maroon" title="c2">T1</font></span><br/><span class="kw">assume </span><a NAME="E2:13_2"/>
not <font color="Maroon" title="c5">p</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> <font color="Maroon" title="c2">T1</font>
 ; <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">consider </span><font color="Maroon" title="c6">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:13_2"/><span class="lab"><font color="Green" title="E7">A8</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T92" title="TREES_1:th.92">TREES_1:92</a></span>;<br/><a NAME="E5:13_2"/>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13_2"><span class="lab"><font color="Green" title="E7">A8</font></span></a>, <a class="ref" href="trees_3.html#T73" title="TREES_3:th.73">TREES_3:73</a></span>;<br/><a NAME="E6:13_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/><span class="kw">hence </span><a NAME="E7:13_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E5">A6</font></span></a>, <a class="txt" href="#E1:13_2"><span class="lab"><font color="Green" title="E6">A7</font></span></a>, <a class="ref" href="trees_1.html#T92" title="TREES_1:th.92">TREES_1:92</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="E4:13"/><span class="lab"><font color="Green" title="E6">A9</font></span>: 
<font color="Maroon" title="c5">p</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> <font color="Maroon" title="c2">T1</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> <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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span></span><br/>

<span class="kw">assume </span><a NAME="E5:13"/>
not <font color="Maroon" title="c3">t</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="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font></span>)</span>
 ; <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">consider </span><font color="Maroon" title="c6">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:13"/><span class="lab"><font color="Green" title="E7">A10</font></span>: 
<font color="Maroon" title="c3">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T92" title="TREES_1:th.92">TREES_1:92</a></span>;<br/>
<a NAME="E8:13"/>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E5">A6</font></span></a>, <a class="txt" href="#E7:13"><span class="lab"><font color="Green" title="E7">A10</font></span></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/>
<a NAME="E9:13"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#T73" title="TREES_3:th.73">TREES_3:73</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:13"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13"><span class="lab"><font color="Green" title="E6">A9</font></span></a>, <a class="ref" href="trees_1.html#T92" title="TREES_1:th.92">TREES_1:92</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>
