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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() or  ex <font color="Olive" title="b1">q</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ex <font color="Olive" title="b2">r</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">q</font>] &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> ) );<br/>
<b>consider </b><font color="Maroon" title="c1">T</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:42_1"/><i><font color="Green" title="E15">A1</font></i>: 
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> <font color="Maroon" title="c1">T</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c2">t</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<a NAME="E4:42_1"/>
<font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">T</font> = <font color="Maroon" title="c1">T</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="trees_4.html#E2:42_1"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/>
<a NAME="E6:42_1"/>
<font color="Maroon" title="c3">T</font> is <a href="trees_1.html#V1" title="TREES_1:attr.1">Tree-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:42_1_1"/>
<font color="Maroon" title="c3">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="trees_1.html#D5" title="TREES_1:def.5">TREES_1:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>thus </b><a NAME="E1:42_1_1_1"/>
(  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )
 <b>by </b><i><a class="txt" href="trees_4.html#E2:42_1"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:42_1_1"/>
for <font color="Olive" title="b1">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> holds <br/> <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Olive" title="b1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:42_1_1_2"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:42_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c4">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E4:42_1_1_2"/><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> &amp; <font color="Maroon" title="c6">q</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c4">p</font> )
 <b>by </b><i><a class="ref" href="trees_1.html#D4" title="TREES_1:def.4">TREES_1:def 4</a></i>;<br/>
<b>assume </b><a NAME="E5:42_1_1_2"/><i><font color="Green" title="E18">A4</font></i>: 
not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font>
 ;<br/>

<a NAME="E6:42_1_1_2"/>
( <font color="Maroon" title="c6">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> )
 
<b>by </b><i><a class="txt" href="trees_4.html#E4:42_1_1_2"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">r</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E8:42_1_1_2"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="ref" href="trees_1.html#T8" title="TREES_1:th.8">TREES_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c6">q</font>, <font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c7">r</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="trees_4.html#E8:42_1_1_2"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/>
<a NAME="E10:42_1_1_2"/>
( ( <font color="Maroon" title="c8">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; ( <font color="Maroon" title="c8">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() ) ) or  ex <font color="Olive" title="b1">q</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ex <font color="Olive" title="b2">r</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">q</font>] &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> ) )
 
<b>by </b><i><a class="txt" href="trees_4.html#E2:42_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="trees_4.html#E1:42_1_1_2"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="trees_4.html#E8:42_1_1_2"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a>, <a class="ref" href="trees_1.html#T46" title="TREES_1:th.46">TREES_1:46</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">q'</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c11">r'</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E12:42_1_1_2"/><i><font color="Green" title="E20">A6</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c10">q'</font>] &amp; <font color="Maroon" title="c8">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q'</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">r'</font> )
 <b>by </b><i><a class="txt" href="trees_4.html#E2:42_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="trees_4.html#E4:42_1_1_2"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="trees_4.html#E5:42_1_1_2"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="trees_4.html#E8:42_1_1_2"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">q'</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c12">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E3:42_1_1_2_1"/><i><font color="Green" title="E21">A7</font></i>: 
<font color="Maroon" title="c8">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q'</font>
 <b>by </b><i><a class="txt" href="trees_4.html#E12:42_1_1_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="finseq_1.html#T64" title="FINSEQ_1:th.64">FINSEQ_1:64</a></i>;<br/><a NAME="E4:42_1_1_2_1"/>
( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )
 <b>by </b><i><a class="txt" href="trees_4.html#E3:42_1_1_2_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a>, <a class="ref" href="trees_1.html#T46" title="TREES_1:th.46">TREES_1:46</a></i>;<br/><b>hence </b><a NAME="E5:42_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="trees_4.html#E2:42_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="trees_4.html#E4:42_1_1_2"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="trees_4.html#E5:42_1_1_2"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c12">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E15:42_1_1_2"/><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c10">q'</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font>
 <b>by </b><i><a class="txt" href="trees_4.html#E12:42_1_1_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="finseq_1.html#T64" title="FINSEQ_1:th.64">FINSEQ_1:64</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">s</font> = <font color="Maroon" title="c12">s</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="trees_4.html#E15:42_1_1_2"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/>
<a NAME="E17:42_1_1_2"/>
<font color="Maroon" title="c10">q'</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">r'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q'</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">s</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="trees_4.html#E12:42_1_1_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="trees_4.html#E15:42_1_1_2"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E18:42_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">s</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r'</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T46" title="FINSEQ_1:th.46">FINSEQ_1:46</a></i>;<br/>
<a NAME="E19:42_1_1_2"/><b>then </b>
( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <font color="Maroon" title="c13">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a>, <a class="ref" href="trees_1.html#T46" title="TREES_1:th.46">TREES_1:46</a></i>;<br/>
<b>hence </b><a NAME="E20:42_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="trees_4.html#E2:42_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="trees_4.html#E4:42_1_1_2"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="trees_4.html#E5:42_1_1_2"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="trees_4.html#E12:42_1_1_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="trees_4.html#E15:42_1_1_2"><i><font color="Green" title="E21">A8</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c4">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c5">k</font>, <font color="Maroon" title="c6">n</font> be    <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> ;<br/>



<b>assume </b><a NAME="E3:42_1_1"/><i><font color="Green" title="E16">A9</font></i>: 
( <font color="Maroon" title="c4">p</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="c5">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> <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> )
 ;<br/>

<div><i><font color="Green" title="E17">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_3"/>
<font color="Maroon" title="c4">p</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="c5">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> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><a NAME="E2:42_1_1_3"/><b>then </b>
( <font color="Maroon" title="c4">p</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="c6">n</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> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c4">p</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="c6">n</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="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )
 <b>by </b><i><a class="txt" href="trees_4.html#E3:42_1_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a>, <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="E3:42_1_1_3"/>
<font color="Maroon" title="c4">p</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="c6">n</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> <font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="txt" href="trees_4.html#E2:42_1"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_4"/><i><font color="Green" title="E18">A11</font></i>: 
not <font color="Maroon" title="c4">p</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="c5">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> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><b>then consider </b><font color="Maroon" title="c7">q</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c8">r</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:42_1_1_4"/><i><font color="Green" title="E19">A12</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">q</font>] &amp; <font color="Maroon" title="c4">p</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="c5">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">r</font> )
 <b>by </b><i><a class="txt" href="trees_4.html#E2:42_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="trees_4.html#E3:42_1_1"><i><font color="Green" title="E16">A9</font></i></a></i>;<br/><a NAME="E4:42_1_1_4"/>
<font color="Maroon" title="c7">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>;<br/><a NAME="E5:42_1_1_4"/><b>then </b>
<font color="Maroon" title="c8">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="trees_4.html#E1:42_1_1_4"><i><font color="Green" title="E18">A11</font></i></a>, <a class="txt" href="trees_4.html#E3:42_1_1_4"><i><font color="Green" title="E19">A12</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">w</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>, <font color="Maroon" title="c10">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:42_1_1_4"/><i><font color="Green" title="E20">A13</font></i>: 
<font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">z</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T63" title="FINSEQ_1:th.63">FINSEQ_1:63</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">w</font> = <font color="Maroon" title="c9">w</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="trees_4.html#E7:42_1_1_4"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/><a NAME="E9:42_1_1_4"/><i><font color="Green" title="E21">A14</font></i>: 
<font color="Maroon" title="c4">p</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="c5">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">z</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="trees_4.html#E3:42_1_1_4"><i><font color="Green" title="E19">A12</font></i></a>, <a class="txt" href="trees_4.html#E7:42_1_1_4"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>;<br/><a NAME="E10:42_1_1_4"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</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="c5">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">z</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">w</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">z</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</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="c5">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">z</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">w</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">z</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>;<br/><a NAME="E11:42_1_1_4"/><b>then </b>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">w</font> &amp; <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z</font> )
 <b>by </b><i><a class="txt" href="trees_4.html#E9:42_1_1_4"><i><font color="Green" title="E21">A14</font></i></a>, <a class="ref" href="finseq_1.html#T46" title="FINSEQ_1:th.46">FINSEQ_1:46</a></i>;<br/><a NAME="E12:42_1_1_4"/><b>then </b>
( <font color="Maroon" title="c11">w</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="c6">n</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> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon" title="c4">p</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="c6">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</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="c6">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c4">p</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="c6">n</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="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )
 <b>by </b><i><a class="txt" href="trees_4.html#E3:42_1_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="trees_4.html#E7:42_1_1_4"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a>, <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="E13:42_1_1_4"/>
<font color="Maroon" title="c4">p</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="c6">n</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> <font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="txt" href="trees_4.html#E2:42_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="trees_4.html#E3:42_1_1_4"><i><font color="Green" title="E19">A12</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:42_1_1"/>
<font color="Maroon" title="c4">p</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="c6">n</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> <font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="txt" href="trees_4.html#E4:42_1_1"><i><font color="Green" title="E17">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">T</font> = <font color="Maroon" title="c3">T</font> as   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c4">T</font>
;<br/>

<b>let </b><font color="Maroon" title="c5">p</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>

<a NAME="E8:42_1"/>
( ( <font color="Maroon" title="c5">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() or  ex <font color="Olive" title="b1">q</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ex <font color="Olive" title="b2">r</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">q</font>] &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> ) ) implies <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E9:42_1"/>
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font> iff ( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() or  ex <font color="Olive" title="b1">q</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ex <font color="Olive" title="b2">r</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">q</font>] &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> ) ) )
 <b>by </b><i><a class="txt" href="trees_4.html#E2:42_1"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/>


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