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<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" 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  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font>) holds <br/> ( ( <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>  implies  <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> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">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"><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></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">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">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &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="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="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="b2">p</font> holds <br/> for <font color="Olive" title="b3">sp</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds <br/> ( <font color="Olive" title="b3">sp</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="Olive" title="b2">p</font> iff <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b3">sp</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="Olive" title="b1">t</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="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="b2">p</font> holds <br/> for <font color="Olive" title="b3">sp</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds <br/> ( <font color="Olive" title="b3">sp</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="Olive" title="b2">p</font> iff <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b3">sp</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="Olive" title="b1">t</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#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">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">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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"><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></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &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="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="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">p</font> holds <br/> for <font color="Olive" title="b2">sp</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds <br/> ( <font color="Olive" title="b2">sp</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="Olive" title="b1">p</font> iff <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">sp</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="c3">t</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="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">p</font> holds <br/> for <font color="Olive" title="b2">sp</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds <br/> ( <font color="Olive" title="b2">sp</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="Olive" title="b1">p</font> iff <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">sp</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="c3">t</font> ) ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">RT</font> =  <a href="trees_3.html#K13" title="TREES_3:func.13">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" 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">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="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">p</font> holds <br/> for <font color="Olive" title="b2">sp</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds <br/> ( <font color="Olive" title="b2">sp</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="Olive" title="b1">p</font> iff <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">sp</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="c3">t</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="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">p</font> holds <br/> for <font color="Olive" title="b2">sp</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds <br/> ( <font color="Olive" title="b2">sp</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="Olive" title="b1">p</font> iff <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">sp</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="c3">t</font> ) ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:11_1"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c3">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">  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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"><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></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><a NAME="E2:11_1"/>
(  <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">T1</font> &amp; <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="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">1</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> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> )
 <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>, <a class="ref" href="trees_1.html#T22" onmouseover="rs('trees_1/T22')" onmouseout="rh()">TREES_1:22</a></span>;<br/><a NAME="E3:11_1"/><span class="kw">then </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_3.html#K13" title="TREES_3:func.13">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_3.html#T68" onmouseover="rs('trees_3/T68')" onmouseout="rh()">TREES_3:68</a></span>;<br/><a NAME="E4:11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<font color="Maroon" title="c3">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">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_3.html#K13" title="TREES_3:func.13">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="#E1:11_1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a></span>;<br/><a NAME="E5:11_1"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">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="c3">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>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</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="E6:11_1"/>
(  <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">T0</font> &amp; <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="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"><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#K7" title="FINSEQ_1:func.7">^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a>, <a class="ref" href="trees_1.html#T22" onmouseover="rs('trees_1/T22')" onmouseout="rh()">TREES_1:22</a></span>;<br/><a NAME="E7:11_1"/><span class="kw">then </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_3.html#K13" title="TREES_3:func.13">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_3.html#T68" onmouseover="rs('trees_3/T68')" onmouseout="rh()">TREES_3:68</a></span>;<br/><a NAME="E8:11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c3">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>  <a href="trees_3.html#K13" title="TREES_3:func.13">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="#E1:11_1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="11_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">  for <font color="Olive" title="b1">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( ( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c3">t</font> implies <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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"><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></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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"><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></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c3">t</font> ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c3">t</font> implies <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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"><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></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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"><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></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c3">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="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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"><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></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c3">t</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:11_1_1_1"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c3">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"> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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"><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></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:11_1_1_1"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c6">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="E4:11_1_1_1"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
<font color="Maroon" title="c3">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="c6">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>  <a href="trees_3.html#K13" title="TREES_3:func.13">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="#E5:11_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c5">x1</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_1_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a></span>;<br/><a NAME="E6:11_1_1_1"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( ( <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">T0</font> &amp; <font color="Maroon" title="c7">x</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"><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#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">p</font> ) or ( <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">T1</font> &amp; <font color="Maroon" title="c7">x</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">1</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="Olive" title="b1">p</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_1_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E4:11_1_1_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="trees_3.html#T68" onmouseover="rs('trees_3/T68')" onmouseout="rh()">TREES_3:68</a></span>;<br/><a NAME="E7:11_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A7</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T41" onmouseover="rs('finseq_1/T41')" onmouseout="rh()">FINSEQ_1:41</a></span>;<br/><a NAME="E8:11_1_1_1"/>
<font color="Maroon" title="c5">x1</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="c6">n</font></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="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E3:11_1_1_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a></span>;<br/><a NAME="E9:11_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">x</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"><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="c7">x</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">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="txt" href="#E7:11_1_1_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a></span>;<br/><a NAME="E10:11_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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">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="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <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="E11:11_1_1_1"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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"><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></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</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="E2:11_1_1"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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"><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></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</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"> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c3">t</font></span><br/><a NAME="E3:11_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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> or <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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">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="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:11_1_1"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c3">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E8:11_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E5">A2</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="E10:11_1"/>
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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"><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></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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">1</span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</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>

<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">  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="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">p</font> holds <br/> for <font color="Olive" title="b2">sp</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds <br/> ( <font color="Olive" title="b2">sp</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="Olive" title="b1">p</font> iff <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">sp</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="c3">t</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" 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">t</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"><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="c5">p</font> implies  for <font color="Olive" title="b1">sp</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds <br/> ( ( <font color="Olive" title="b1">sp</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="c5">p</font> implies <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">sp</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="c3">t</font> ) &amp; ( <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">sp</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="c3">t</font> implies <font color="Olive" title="b1">sp</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="c5">p</font> ) ) )</span><br/><span class="kw">assume </span><a NAME="E1:11_2"/><span class="lab"><font color="Green" title="E4">A8</font></span>: 
<font color="Maroon" title="c3">t</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"><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="c5">p</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">  for <font color="Olive" title="b1">sp</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds <br/> ( ( <font color="Olive" title="b1">sp</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="c5">p</font> implies <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">sp</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="c3">t</font> ) &amp; ( <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">sp</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="c3">t</font> implies <font color="Olive" title="b1">sp</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="c5">p</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">sp</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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="c6">sp</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="c5">p</font> implies <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</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="c3">t</font> ) &amp; ( <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</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="c3">t</font> implies <font color="Maroon" title="c6">sp</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="c5">p</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"> ( <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</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="c3">t</font> implies <font color="Maroon" title="c6">sp</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="c5">p</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:11_2_1"/>
<font color="Maroon" title="c6">sp</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="c5">p</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"> <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</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="c3">t</font></span><br/><a NAME="E2:11_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">sp</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="c5">p</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="c5">p</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">T0</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="c7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:11_2_1"/><span class="lab"><font color="Green" title="E5">A9</font></span>: 
<font color="Maroon" title="c6">sp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</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="c7">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="E5:11_2_1"/><span class="lab"><font color="Green" title="E6">A10</font></span>: 
<font color="Maroon" title="c5">p</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="c7">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">T0</font>
 ;<br/><a NAME="E6:11_2_1"/>
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><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></span>)</span> <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">T0</font>,<font color="Maroon" title="c2">T1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11_2_1"><span class="lab"><font color="Green" title="E6">A10</font></span></a>, <a class="ref" href="trees_3.html#T69" onmouseover="rs('trees_3/T69')" onmouseout="rh()">TREES_3:69</a></span>;<br/><a NAME="E7:11_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">p</font></span>)</span> <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="c7">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>  <a href="trees_3.html#K13" title="TREES_3:func.13">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#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>;<br/><a NAME="E8:11_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="c7">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> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">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">k</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c3">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">k</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>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font>) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_2"><span class="lab"><font color="Green" title="E4">A8</font></span></a></span>;<br/><a NAME="E9:11_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="c7">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>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c3">t</font>
 <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">hence </span><a NAME="E10:11_2_1"/>
<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#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</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="c3">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_2"><span class="lab"><font color="Green" title="E4">A8</font></span></a>, <a class="txt" href="#E4:11_2_1"><span class="lab"><font color="Green" title="E5">A9</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</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="c7">zsp</font> = <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</font>;<br/><span class="kw">assume </span><a NAME="E3:11_2"/>
<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#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</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="c3">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"> <font color="Maroon" title="c6">sp</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="c5">p</font></span><br/><a NAME="E4:11_2"/><span class="kw">then </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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</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="c3">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="c3">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>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</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="c8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:11_2"/><span class="lab"><font color="Green" title="E5">A11</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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c8">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="E7:11_2"/><span class="lab"><font color="Green" title="E6">A12</font></span>: 
<font color="Maroon" title="c3">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="c8">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>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font>)
 ;<br/><a NAME="E8:11_2"/>
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default"><font color="Maroon" title="c8">n</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span> <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">T0</font>,<font color="Maroon" title="c2">T1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_2"><span class="lab"><font color="Green" title="E4">A8</font></span></a>, <a class="txt" href="#E7:11_2"><span class="lab"><font color="Green" title="E6">A12</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>;<br/><a NAME="E9:11_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</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="c8">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">T0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#T69" onmouseover="rs('trees_3/T69')" onmouseout="rh()">TREES_3:69</a></span>;<br/><a NAME="E10:11_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</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="c8">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> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</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">k</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c5">p</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">k</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">T0</font> </span> } </span> 
 ;<br/><a NAME="E11:11_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A13</font></span>: 
<font color="Maroon" title="c5">p</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="c8">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>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c5">p</font>
 <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="E12:11_2"/>
<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#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</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"><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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default"><font color="Maroon" title="c8">n</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_2"><span class="lab"><font color="Green" title="E4">A8</font></span></a>, <a class="txt" href="#E6:11_2"><span class="lab"><font color="Green" title="E5">A11</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>;<br/><span class="kw">hence </span><a NAME="E13:11_2"/>
<font color="Maroon" title="c6">sp</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="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:11_2"><span class="lab"><font color="Green" title="E7">A13</font></span></a>, <a class="ref" href="finseq_1.html#T33" onmouseover="rs('finseq_1/T33')" onmouseout="rh()">FINSEQ_1:33</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">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" 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">t</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">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="Maroon" title="c5">p</font> implies  for <font color="Olive" title="b1">sp</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds <br/> ( <font color="Olive" title="b1">sp</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="c5">p</font> iff <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">sp</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="c3">t</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:11"/><span class="lab"><font color="Green" title="E4">A14</font></span>: 
<font color="Maroon" title="c3">t</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">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="Maroon" title="c5">p</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">  for <font color="Olive" title="b1">sp</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds <br/> ( <font color="Olive" title="b1">sp</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="c5">p</font> iff <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">sp</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="c3">t</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">sp</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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="c6">sp</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="c5">p</font> iff <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</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="c3">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"> ( <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</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="c3">t</font> implies <font color="Maroon" title="c6">sp</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="c5">p</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:11_3"/>
<font color="Maroon" title="c6">sp</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="c5">p</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"> <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</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="c3">t</font></span><br/><a NAME="E2:11_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">sp</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="c5">p</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="c5">p</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="c2">T1</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="c7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:11_3"/><span class="lab"><font color="Green" title="E5">A15</font></span>: 
<font color="Maroon" title="c6">sp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</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="c7">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="E5:11_3"/><span class="lab"><font color="Green" title="E6">A16</font></span>: 
<font color="Maroon" title="c5">p</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="c7">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="c2">T1</font>
 ;<br/><a NAME="E6:11_3"/>
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><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></span>)</span> <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">T0</font>,<font color="Maroon" title="c2">T1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11_3"><span class="lab"><font color="Green" title="E6">A16</font></span></a>, <a class="ref" href="trees_3.html#T70" onmouseover="rs('trees_3/T70')" onmouseout="rh()">TREES_3:70</a></span>;<br/><a NAME="E7:11_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><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="Maroon" title="c5">p</font></span>)</span> <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="c7">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>  <a href="trees_3.html#K13" title="TREES_3:func.13">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#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>;<br/><a NAME="E8:11_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="c7">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> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">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">k</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c3">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">k</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>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font>) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E4">A14</font></span></a></span>;<br/><a NAME="E9:11_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="c7">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>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c3">t</font>
 <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">hence </span><a NAME="E10:11_3"/>
<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#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</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="c3">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E4">A14</font></span></a>, <a class="txt" href="#E4:11_3"><span class="lab"><font color="Green" title="E5">A15</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</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="c7">zsp</font> = <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#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</font>;<br/>
<span class="kw">assume </span><a NAME="E5:11"/>
<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#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</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="c3">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"> <font color="Maroon" title="c6">sp</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="c5">p</font></span><br/>

<a NAME="E6:11"/><span class="kw">then </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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</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="c3">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="c3">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>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</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="c8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:11"/><span class="lab"><font color="Green" title="E5">A17</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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c8">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="E9:11"/><span class="lab"><font color="Green" title="E6">A18</font></span>: 
<font color="Maroon" title="c3">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="c8">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>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> (<font color="Maroon" title="c1">T0</font>,<font color="Maroon" title="c2">T1</font>)
 ;<br/>
<a NAME="E10:11"/>
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default"><font color="Maroon" title="c8">n</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span> <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">T0</font>,<font color="Maroon" title="c2">T1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E4">A14</font></span></a>, <a class="txt" href="#E9:11"><span class="lab"><font color="Green" title="E6">A18</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>;<br/>
<a NAME="E11:11"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</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="c8">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="c2">T1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#T70" onmouseover="rs('trees_3/T70')" onmouseout="rh()">TREES_3:70</a></span>;<br/>
<a NAME="E12:11"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</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="c8">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> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</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">k</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c5">p</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">k</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="c2">T1</font> </span> } </span> 
 
;<br/>
<a NAME="E13:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A19</font></span>: 
<font color="Maroon" title="c5">p</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="c8">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>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c5">p</font>
 
<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="E14:11"/>
<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#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">sp</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">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="trees_3.html#K14" title="TREES_3:func.14">&lt;*</a><span class="default"><font color="Maroon" title="c8">n</font></span><a href="trees_3.html#K14" title="TREES_3:func.14">*&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E4">A14</font></span></a>, <a class="txt" href="#E8:11"><span class="lab"><font color="Green" title="E5">A17</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:11"/>
<font color="Maroon" title="c6">sp</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="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:11"><span class="lab"><font color="Green" title="E7">A19</font></span></a>, <a class="ref" href="finseq_1.html#T33" onmouseover="rs('finseq_1/T33')" onmouseout="rh()">FINSEQ_1:33</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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