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<b>let </b><font color="Maroon">S1</font> be   <a href="trees_1.html#NM1">Tree</a>, <font color="Maroon">S2</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:2_1_2"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font> iff ( ( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> not <font color="Olive">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">q</font> ) ) or  ex <font color="Olive">p</font>, <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> ) ) )
 <b>and </b><br/><a NAME="E2:2_1_2"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font> iff ( ( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> not <font color="Olive">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">q</font> ) ) or  ex <font color="Olive">p</font>, <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> ) ) )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_2.html#D1">TREES_2:def 1</a><br/>

<b>thus </b><a NAME="E3:2_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_1_2_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">x</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:2_1_2_1"/>
( ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> not <font color="Olive">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">q</font> ) ) or  ex <font color="Olive">p</font>, <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:2_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E4:2_1_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">x</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E6:2_1_2"/>
( ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> not <font color="Olive">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">q</font> ) ) or  ex <font color="Olive">p</font>, <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font>
 <b>by </b><i/>;<br/>


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