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

<span class="kw">set </span><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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : <font color="Olive" title="b1">T</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span> } </span> ;<br/>
<a NAME="E1:4_1_1"/>
<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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : <font color="Olive" title="b1">T</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : <font color="Olive" title="b1">T</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span> } </span>  or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:4_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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : <font color="Olive" title="b1">T</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span> } </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="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a> </span><br/>

<a NAME="E2:4_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">T</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  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="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:4_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  : <font color="Olive" title="b1">T</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span> } </span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">X</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">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff <font color="Olive" title="b1">x</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">x</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff <font color="Maroon" title="c3">x</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> )</span><br/>

<span class="kw">thus </span><a NAME="E3:4_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> implies <font color="Maroon" title="c3">x</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <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"> ( <font color="Maroon" title="c3">x</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="4_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:4_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</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">x</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a></span><br/>

<a NAME="E2:4_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">T</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a>  st <br/>( <font color="Maroon" title="c3">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="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:4_1_1_2"/>
<font color="Maroon" title="c3">x</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <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"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E4:4_1_1"/>
<font color="Maroon" title="c3">x</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">T</font> = <font color="Maroon" title="c3">x</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> ;<br/>
<a NAME="E6:4_1_1"/>
<font color="Maroon" title="c4">T</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_3.html#K1" title="TREES_3:func.1">Trees</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#D1" target="_self" onmouseover="rs('trees_3/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:4_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</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"> verum</span><br/>


</div>
