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<span class="kw">let </span><font color="Maroon" title="c1">T1</font>, <font color="Maroon" title="c2">T2</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">  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p2"><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>,<font color="Maroon" title="c1">T1</font>)</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>,<font color="Maroon" title="c2">T2</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="trees_2.html#D1" onmouseover="rs('trees_2/D1')" onmouseout="rh()">TREES_2:def 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"> ( (  not <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font>) or <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p2"><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>,<font color="Maroon" title="c1">T1</font>)</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>,<font color="Maroon" title="c2">T2</font>) ) &amp; (  not <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p2"><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>,<font color="Maroon" title="c1">T1</font>)</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>,<font color="Maroon" title="c2">T2</font>) or <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font>) ) )</span><br/>



<a NAME="E1:122"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T1</font>
 
;<br/>
<a NAME="E2:122"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T2</font>
 
;<br/>
<a NAME="E3:122"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>;<br/>
<a NAME="E4:122"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
;<br/>
<a NAME="E5:122"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
;<br/>
<a NAME="E6:122"/><span class="lab"><font color="Green" title="E54">A6</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">T1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T22" onmouseover="rs('trees_1/T22')" onmouseout="rh()">TREES_1:22</a></span>;<br/>
<a NAME="E7:122"/><span class="lab"><font color="Green" title="E55">A7</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">T2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T22" onmouseover="rs('trees_1/T22')" onmouseout="rh()">TREES_1:22</a></span>;<br/>
<a NAME="E8:122"/><span class="lab"><font color="Green" title="E56">A8</font></span>: 
<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>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a>, <a class="ref" href="trees_1.html#T53" onmouseover="rs('trees_1/T53')" onmouseout="rh()">TREES_1:53</a></span>;<br/>
<a NAME="E9:122"/><span class="lab"><font color="Green" title="E57">A9</font></span>: 
<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</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>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a>, <a class="ref" href="trees_1.html#T53" onmouseover="rs('trees_1/T53')" onmouseout="rh()">TREES_1:53</a></span>;<br/>
<a NAME="E10:122"/>
 not <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="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T52" onmouseover="rs('trees_1/T52')" onmouseout="rh()">TREES_1:52</a></span>;<br/>
<a NAME="E11:122"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A10</font></span>: 
<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</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"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</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>,<font color="Maroon" title="c1">T1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:122"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="txt" href="#E9:122"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="ref" href="trees_1.html#D9" onmouseover="rs('trees_1/D9')" onmouseout="rh()">TREES_1:def 9</a></span>;<br/>
<span class="kw">thus </span><a NAME="E12:122"/>
( <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font>) implies <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p2"><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>,<font color="Maroon" title="c1">T1</font>)</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>,<font color="Maroon" title="c2">T2</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"> (  not <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p2"><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>,<font color="Maroon" title="c1">T1</font>)</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>,<font color="Maroon" title="c2">T2</font>) or <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font>) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="122_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:122_1"/><span class="lab"><font color="Green" title="E59">A11</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</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"> <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p2"><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>,<font color="Maroon" title="c1">T1</font>)</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>,<font color="Maroon" title="c2">T2</font>)</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="122_1_1"><span class="kw">now </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="c3">p</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>  implies <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p2"><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>,<font color="Maroon" title="c1">T1</font>)</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>,<font color="Maroon" title="c2">T2</font>) )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:122_1_1"/>
<font color="Maroon" title="c3">p</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"> <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p2"><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>,<font color="Maroon" title="c1">T1</font>)</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>,<font color="Maroon" title="c2">T2</font>)</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>, <font color="Maroon" title="c5">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E3:122_1_1"/><span class="lab"><font color="Green" title="E60">A12</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E4:122_1_1"/><span class="lab"><font color="Green" title="E61">A13</font></span>: 
<font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:122_1_1"/><span class="lab"><font color="Green" title="E62">A14</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:122_1"><span class="lab"><font color="Green" title="E59">A11</font></span></a>, <a class="ref" href="trees_3.html#D15" target="_self" onmouseover="rs('trees_3/D15')" onmouseout="rh()">Def15</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">q</font> = <font color="Maroon" title="c5">q</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:122_1_1"><span class="lab"><font color="Green" title="E62">A14</font></span></a>, <a class="ref" href="finseq_1.html#T36" onmouseover="rs('finseq_1/T36')" onmouseout="rh()">FINSEQ_1:36</a></span>;<br/><a NAME="E7:122_1_1"/><span class="lab"><font color="Green" title="E63">A15</font></span>: 
 not <span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</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="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T50" onmouseover="rs('trees_1/T50')" onmouseout="rh()">TREES_1:50</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c4">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><div><span class="lab"><font color="Green" title="E64">A16</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="122_1_1_1"><span class="kw">now </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="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p2"><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>,<font color="Maroon" title="c1">T1</font>)</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>,<font color="Maroon" title="c2">T2</font>) )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:122_1_1_1"/><span class="lab"><font color="Green" title="E65">A17</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p2"><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>,<font color="Maroon" title="c1">T1</font>)</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>,<font color="Maroon" title="c2">T2</font>)</span><br/><a NAME="E2:122_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A18</font></span>: 
 not <span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default"><font color="Maroon" title="c7">n</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:122_1_1"><span class="lab"><font color="Green" title="E63">A15</font></span></a></span>;<br/><a NAME="E3:122_1_1_1"/>
<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default"><font color="Maroon" title="c7">n</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</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>,<font color="Maroon" title="c1">T1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:122"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="txt" href="#E4:122_1_1"><span class="lab"><font color="Green" title="E61">A13</font></span></a>, <a class="txt" href="#E1:122_1_1_1"><span class="lab"><font color="Green" title="E65">A17</font></span></a>, <a class="ref" href="trees_1.html#D9" onmouseover="rs('trees_1/D9')" onmouseout="rh()">TREES_1:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E4:122_1_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p2"><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>,<font color="Maroon" title="c1">T1</font>)</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>,<font color="Maroon" title="c2">T2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:122"><span class="lab"><font color="Green" title="E58">A10</font></span></a>, <a class="txt" href="#E5:122_1_1"><span class="lab"><font color="Green" title="E62">A14</font></span></a>, <a class="txt" href="#E2:122_1_1_1"><span class="lab"><font color="Green" title="E66">A18</font></span></a>, <a class="ref" href="trees_1.html#D9" onmouseover="rs('trees_1/D9')" onmouseout="rh()">TREES_1:def 9</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><a NAME="E10:122_1_1"/>
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:122"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="txt" href="#E4:122"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E3:122_1_1"><span class="lab"><font color="Green" title="E60">A12</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E11:122_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; ( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default"><font color="Maroon" title="c7">n</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p2"><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>,<font color="Maroon" title="c1">T1</font>)</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>,<font color="Maroon" title="c2">T2</font>) ) ) or ( <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:122"><span class="lab"><font color="Green" title="E58">A10</font></span></a>, <a class="txt" href="#E4:122_1_1"><span class="lab"><font color="Green" title="E61">A13</font></span></a>, <a class="ref" href="nat_1.html#T8" onmouseover="rs('nat_1/T8')" onmouseout="rh()">NAT_1:8</a>, <a class="ref" href="trees_1.html#D9" onmouseover="rs('trees_1/D9')" onmouseout="rh()">TREES_1:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E12:122_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p2"><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>,<font color="Maroon" title="c1">T1</font>)</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>,<font color="Maroon" title="c2">T2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:122_1_1"><span class="lab"><font color="Green" title="E62">A14</font></span></a>, <a class="txt" href="#E9:122_1_1"><span class="lab"><font color="Green" title="E64">A16</font></span></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">hence </span><a NAME="E3:122_1"/>
<font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p2"><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>,<font color="Maroon" title="c1">T1</font>)</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>,<font color="Maroon" title="c2">T2</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T22" onmouseover="rs('trees_1/T22')" onmouseout="rh()">TREES_1:22</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">assume </span><a NAME="E13:122"/>
<font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p2"><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>,<font color="Maroon" title="c1">T1</font>)</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span>,<font color="Maroon" title="c2">T2</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"> <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font>)</span><br/>

<a NAME="E14:122"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A19</font></span>: 
( ( <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</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>,<font color="Maroon" title="c1">T1</font>) &amp;  not <span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c3">p</font> ) or  ex <font color="Olive" title="b1">r</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">T2</font> &amp; <font color="Maroon" title="c3">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">r</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:122"><span class="lab"><font color="Green" title="E58">A10</font></span></a>, <a class="ref" href="trees_1.html#D9" onmouseover="rs('trees_1/D9')" onmouseout="rh()">TREES_1:def 9</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="122_2"><span class="kw">now </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="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</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>,<font color="Maroon" title="c1">T1</font>) implies <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font>) )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:122_2"/>
<font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</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>,<font color="Maroon" title="c1">T1</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"> <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font>)</span><br/><a NAME="E2:122_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A20</font></span>: 
( ( <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2 &amp;  not <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="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c3">p</font> ) or  ex <font color="Olive" title="b1">r</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">T1</font> &amp; <font color="Maroon" title="c3">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</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="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">r</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:122"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="ref" href="trees_1.html#D9" onmouseover="rs('trees_1/D9')" onmouseout="rh()">TREES_1:def 9</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="122_2_1"><span class="kw">now </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="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2 implies <font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font>) )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:122_2_1"/>
<font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2
 ; <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="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font>)</span><br/><a NAME="E2:122_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A21</font></span>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</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> or <font color="Maroon" title="c3">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a>, <a class="ref" href="trees_1.html#T53" onmouseover="rs('trees_1/T53')" onmouseout="rh()">TREES_1:53</a></span>;<br/><a NAME="E3:122_2_1"/>
<font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a></span>;<br/><span class="kw">hence </span><a NAME="E4:122_2_1"/>
<font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:122"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E2:122"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E3:122"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="txt" href="#E4:122"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E5:122"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E6:122"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="txt" href="#E7:122"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="txt" href="#E2:122_2_1"><span class="lab"><font color="Green" title="E61">A21</font></span></a>, <a class="ref" href="trees_3.html#D15" target="_self" onmouseover="rs('trees_3/D15')" onmouseout="rh()">Def15</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">hence </span><a NAME="E4:122_2"/>
<font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:122"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E3:122"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="txt" href="#E5:122"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E2:122_2"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="ref" href="trees_3.html#D15" target="_self" onmouseover="rs('trees_3/D15')" onmouseout="rh()">Def15</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">hence </span><a NAME="E16:122"/>
<font color="Maroon" title="c3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:122"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E3:122"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="txt" href="#E4:122"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E14:122"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="ref" href="trees_3.html#D15" target="_self" onmouseover="rs('trees_3/D15')" onmouseout="rh()">Def15</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>
