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

<span class="kw">let </span><font color="Maroon" title="c1">t</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>; <a class="txt" 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_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">t</font><br/>  for <font color="Olive" title="b2">q</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Olive" title="b1">p</font></span>)</span> holds   <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<font color="Maroon" title="c1">t</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">q</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Olive" title="b1">p</font></span>)</span>,<font color="Olive" title="b2">q</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">t</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">q</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span> holds   <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<font color="Maroon" title="c1">t</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">q</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Olive" title="b1">q</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">q</font> be   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<font color="Maroon" title="c1">t</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">q</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c3">q</font>)</span><br/>





<a NAME="E1:65"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">t</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#D10" title="TREES_2:def.10">TREES_2:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">pq</font> = <font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">q</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">t</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#D6" title="TREES_1:def.6">TREES_1:def 6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c3">q</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span> ;<br/>
<a NAME="E4:65"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">t</font></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">t</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#T6" title="TREES_2:th.6">TREES_2:6</a></span>;<br/>
<a NAME="E5:65"/><span class="kw">then </span>
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c4">pq</font>, <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="trees_2.html#T37" title="TREES_2:th.37">TREES_2:37</a></span>;<br/>
<a NAME="E6:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c4">pq</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T5" title="CARD_1:th.5">CARD_1:5</a></span>;<br/>
<a NAME="E7:65"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
 ex <font color="Olive" title="b1">r</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> &amp;  <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_9.html#D6" target="_self" title="TREES_9:def.6">Def6</a></span>;<br/>
<a NAME="E8:65"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span>
 
;<br/>
<a NAME="E9:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A4</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:65"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/>
<a NAME="E10:65"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
 ex <font color="Olive" title="b1">q</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">t</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pq</font> &amp;  <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<font color="Maroon" title="c1">t</font>,<font color="Maroon" title="c4">pq</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">t</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_9.html#D6" target="_self" title="TREES_9:def.6">Def6</a></span>;<br/>
<a NAME="E11:65"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pq</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">t</font>
 
;<br/>
<a NAME="E12:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A6</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<font color="Maroon" title="c1">t</font>,<font color="Maroon" title="c4">pq</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pq</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:65"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/>
<a NAME="E13:65"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c5">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_9.html#D5" target="_self" title="TREES_9:def.5">Def5</a></span>;<br/>
<a NAME="E14:65"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pq</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c4">pq</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_9.html#D5" target="_self" title="TREES_9:def.5">Def5</a></span>;<br/>
<a NAME="E15:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A9</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pq</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:65"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E6:65"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a></span>)</span> implies <span class="p1">(<span class="default"><a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<font color="Maroon" title="c1">t</font>,<font color="Maroon" title="c4">pq</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:65_1"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<font color="Maroon" title="c1">t</font>,<font color="Maroon" title="c4">pq</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:65_1"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E4:65_1"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="trees_9.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><a NAME="E5:65_1"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
<font color="Maroon" title="c5">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:65"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E4:65_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="trees_9.html#T7" target="_self" title="TREES_9:th.7">Th7</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<font color="Maroon" title="c1">t</font>,<font color="Maroon" title="c4">pq</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> = 
<font color="Maroon" title="c1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">pq</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:65"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E15:65"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E12:65"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E1:65_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<font color="Maroon" title="c1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pq</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:65"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E13:65"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E6:65"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E3:65_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E4:65_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="trees_9.html#D5" target="_self" title="TREES_9:def.5">Def5</a></span>
<br/>.= 
<font color="Maroon" title="c1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E5:65_1"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="ref" href="trees_2.html#D10" title="TREES_2:def.10">TREES_2:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E4:65_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="trees_9.html#D5" target="_self" title="TREES_9:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:65"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E9:65"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E1:65_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
; <a class="txt" 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="E17:65"/>
 <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<font color="Maroon" title="c1">t</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">q</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c3">q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:65"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E12:65"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E9:65"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
