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

<b>set </b><font color="Maroon" title="c1">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">T</font> where <font color="Olive" title="b1">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span> : <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span>}</span> ;<br/>
<b>take </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">T</font> where <font color="Olive" title="b1">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span> : <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span>}</span> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">T</font> where <font color="Olive" title="b2">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span> : <font color="Olive" title="b2">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span>}</span>  iff <font color="Olive" title="b1">x</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> )</span><br/>

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">T</font> where <font color="Olive" title="b1">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span> : <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span>}</span>  iff <font color="Maroon" title="c2">x</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> )</span><br/>

<b>thus </b><a NAME="E1:2_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">T</font> where <font color="Olive" title="b1">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span> : <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span>}</span>  implies <font color="Maroon" title="c2">x</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">T</font> where <font color="Olive" title="b1">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span> : <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span>}</span>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">T</font> where <font color="Olive" title="b1">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span> : <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a></span><br/>

<a NAME="E2:2_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">T</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span> st <br/>( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">T</font> &amp; <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> )
 
;<br/>
<b>hence </b><a NAME="E3:2_1_1_1"/>
<font color="Maroon" title="c2">x</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:2_1_1"/>
<font color="Maroon" title="c2">x</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">T</font> where <font color="Olive" title="b1">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span> : <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span>}</span> </span><br/>

<b>then reconsider </b><font color="Maroon" title="c3">T</font> = <font color="Maroon" title="c2">x</font> as   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> ;<br/>
<a NAME="E4:2_1_1"/>
<font color="Maroon" title="c3">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="trees_1.html#D5" title="TREES_1:def.5">TREES_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:2_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">T</font> where <font color="Olive" title="b1">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span> : <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
