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<b>let </b><font color="Maroon">T</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">T1</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="finseq_1.html#NM1">FinSequence</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : not <font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">t</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t1</font>) where <font color="Olive">t1</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t1</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">C</font> = <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t1</font>) where <font color="Olive">t1</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">t1</font>] </span>}</span> ;<br/>
<a NAME="E1:30"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">t1</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t1</font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">t1</font>] )
 
;<br/>
<a NAME="E2:30"/><i><font color="Green">E31</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t1</font>) where <font color="Olive">t1</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t1</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t1</font>) where <font color="Olive">t1</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">t1</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S3">FRAENKEL:sch 3</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:30_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : not <font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">t</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">t</font> being    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:30_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; not <font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">t</font> )
 ;<br/><a NAME="E4:30_1_1"/>
(  not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t</font> or <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E5:30_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:30_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><a NAME="E3:30_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or  ex <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t</font> ) )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">t</font> being    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:30_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E6:30_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or  not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t</font> )
 ;<br/><a NAME="E7:30_1"/>
not <font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E8:30_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : not <font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">t</font> </span>}</span> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:30"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : not <font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">t</font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:30"/>
(  <a href="xboole_0.html#K1">{}</a>  is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> &amp; <font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E6:30"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t1</font>) where <font color="Olive">t1</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">t1</font>] </span>}</span> 
 
;<br/>
<b>thus </b><font color="Maroon">T</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<font color="Maroon">T1</font> = 
<span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t1</font>) where <font color="Olive">t1</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">t1</font>] </span>}</span> 
<b>by </b><i>, , <a class="ref" href="trees_1.html#T64">TREES_1:64</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t1</font>) where <font color="Olive">t1</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">t1</font>] </span>}</span> </span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t1</font>) where <font color="Olive">t1</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">t1</font>] </span>}</span> 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/>


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