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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  ex <font color="Olive" title="b1">n</font> being    <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>  ex <font color="Olive" title="b2">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">q</font> ) );<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">T</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:110_1_1"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> iff ( <font color="Olive" title="b1">y</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">y</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span>
<a NAME="E3:110_1_1"/>
 <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">T</font> = <font color="Maroon" title="c2">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a></span>;<br/>
<a NAME="E5:110_1_1"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font> is  <a href="trees_3.html#V1" title="TREES_3:attr.1">constituted-Trees</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="trees_3.html#D9" target="_self" title="TREES_3:def.9">Def9</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E45">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font> implies <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> )</span><br/><span class="kw">assume </span><a NAME="E1:110_1_1_1"/>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a></span><br/><a NAME="E2:110_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="trees_3.html#E59"><span class="lab"><font color="Green" title="E40">Lm3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:110_1_1_1"/>
<font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1"><span class="lab"><font color="Green" title="E44">A3</font></span></a></span>; <a class="txt" 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>
<a NAME="E7:110_1_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="110_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:110_1_1_2"/>
<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> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="trees_1.html#D3" title="TREES_1:def.3">TREES_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive">b<sub>1</sub></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>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> or  <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font> ) ) &amp; (  for <font color="Olive">b<sub>1</sub></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> <br/>  for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being    <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>   holds <br/>(  not <font color="Olive">b<sub>1</sub></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="Olive">b<sub>2</sub></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> or  not <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></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="Olive">b<sub>3</sub></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> ) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">y</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">y</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>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:110_1_1_2_1"/>
(  not <font color="Maroon" title="c4">y</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">y</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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a></span>; <a class="txt" 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">thus </span><a NAME="E2:110_1_1_2"/>
 for <font color="Olive" title="b1">w</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">w</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">w</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></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> <br/>  for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being    <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>   holds <br/>(  not <font color="Olive">b<sub>1</sub></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="Olive">b<sub>2</sub></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> or  not <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></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="Olive">b<sub>3</sub></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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">w</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> implies  <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c4">w</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:110_1_1_2_2"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c4">w</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_2_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:110_1_1_2_2_1"/>
<font color="Maroon" title="c4">w</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c4">w</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">n</font> being    <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> , <font color="Maroon" title="c6">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E3:110_1_1_2_2_1"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">and </span><br/><a NAME="E4:110_1_1_2_2_1"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:110_1_1_2_2_1"/><span class="lab"><font color="Green" title="E49">A8</font></span>: 
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E1:110_1_1_2_2"><span class="lab"><font color="Green" title="E46">A5</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">q</font> = <font color="Maroon" title="c6">q</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1_2_2_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></span>;<br/><span class="kw">thus </span><a NAME="E7:110_1_1_2_2_1"/>
 <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c4">w</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_2_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">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">w</font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:110_1_1_2_2_1_1"/>
<font color="Maroon" title="c8">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">w</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">r</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E3:110_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E50">A9</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r</font>
 <span class="kw">and </span><br/><a NAME="E4:110_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E51">A10</font></span>: 
<font color="Maroon" title="c9">r</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c4">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#D2" title="TREES_1:def.2">TREES_1:def 2</a></span>;<br/>
<a NAME="E5:110_1_1_2_2_1_1"/>
<font color="Maroon" title="c9">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c4">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E51">A10</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">k</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E7:110_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E52">A11</font></span>: 
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#D1" title="TREES_1:def.1">TREES_1:def 1</a></span>;<br/>
<a NAME="E8:110_1_1_2_2_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:110_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="ref" href="relat_1.html#T115" title="RELAT_1:th.115">RELAT_1:115</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">r</font> = <font color="Maroon" title="c9">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E10:110_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E53">A12</font></span>: 
<font color="Maroon" title="c11">r</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_2_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:110_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c11">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">r1</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> , <font color="Maroon" title="c13">i</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E3:110_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E54">A13</font></span>: 
<font color="Maroon" title="c11">r</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c13">i</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c12">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T130" title="FINSEQ_2:th.130">FINSEQ_2:130</a></span>;<br/><a NAME="E4:110_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E55">A14</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c13">i</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>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></span>;<br/><a NAME="E5:110_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E56">A15</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E6:110_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E57">A16</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="txt" href="#E4:110_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E55">A14</font></span></a>, <a class="ref" href="finseq_1.html#T26" title="FINSEQ_1:th.26">FINSEQ_1:26</a></span>;<br/><a NAME="E7:110_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E58">A17</font></span>: 
<font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></span>;<br/><a NAME="E8:110_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E59">A18</font></span>: 
<font color="Maroon" title="c4">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1_2_2_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></span>;<br/><a NAME="E9:110_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E60">A19</font></span>: 
<font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:110_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="txt" href="#E5:110_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="txt" href="#E6:110_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><a NAME="E10:110_1_1_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A20</font></span>: 
<font color="Maroon" title="c12">r1</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c7">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1_2_2_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="txt" href="#E4:110_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E51">A10</font></span></a>, <a class="txt" href="#E3:110_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="txt" href="#E7:110_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E58">A17</font></span></a>, <a class="txt" href="#E8:110_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="ref" href="trees_1.html#T49" title="TREES_1:th.49">TREES_1:49</a></span>;<br/><a NAME="E11:110_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E62">A21</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110_1_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E3:110_1_1_2_2_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a></span>;<br/><a NAME="E12:110_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c12">r1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c7">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:110_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E61">A20</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><a NAME="E13:110_1_1_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1_1_2_2_1"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="txt" href="#E11:110_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E62">A21</font></span></a>, <a class="ref" href="trees_1.html#T20" title="TREES_1:th.20">TREES_1:20</a></span>;<br/><span class="kw">hence </span><a NAME="E14:110_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E3:110_1_1_2_2_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="txt" href="#E3:110_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E10:110_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E53">A12</font></span></a>, <a class="txt" href="#E3:110_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="txt" href="#E7:110_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E58">A17</font></span></a>, <a class="txt" href="#E8:110_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="txt" href="#E9:110_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E60">A19</font></span></a></span>; <a class="txt" 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">hence </span><a NAME="E12:110_1_1_2_2_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E3:110_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E10:110_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E53">A12</font></span></a></span>; <a class="txt" 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></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:110_1_1_2_2"/>
 <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c4">w</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T15" title="TREES_1:th.15">TREES_1:15</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>; <a class="txt" 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">let </span><font color="Maroon" title="c4">w</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <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>   holds <br/>(  not <font color="Maroon" title="c4">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="Olive">b<sub>1</sub></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> or  not <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c4">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="Olive">b<sub>2</sub></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> )</span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">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="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> or  not <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> or <font color="Maroon" title="c4">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" title="c3">T</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:110_1_1_2"/><span class="lab"><font color="Green" title="E46">A22</font></span>: 
<font color="Maroon" title="c4">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="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>
 <span class="kw">and </span><br/><a NAME="E4:110_1_1_2"/><span class="lab"><font color="Green" title="E47">A23</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">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" title="c3">T</font></span><br/>

<div><span class="lab"><font color="Green" title="E48">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:110_1_1_2_3"/><span class="lab"><font color="Green" title="E49">A25</font></span>: 
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">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" title="c3">T</font></span><br/><a NAME="E2:110_1_1_2_3"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_2"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">m</font> being    <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> , <font color="Maroon" title="c8">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E4:110_1_1_2_3"/><span class="lab"><font color="Green" title="E50">A26</font></span>: 
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">and </span><br/><a NAME="E5:110_1_1_2_3"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:110_1_1_2_3"/><span class="lab"><font color="Green" title="E51">A27</font></span>: 
<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"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a></span>;<br/><a NAME="E7:110_1_1_2_3"/><span class="lab"><font color="Green" title="E52">A28</font></span>: 
<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>;<br/><a NAME="E8:110_1_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></span>;<br/><a NAME="E9:110_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A29</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1_1_2"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E4:110_1_1_2_3"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E6:110_1_1_2_3"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="txt" href="#E7:110_1_1_2_3"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:110_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="trees_3.html#E59"><span class="lab"><font color="Green" title="E40">Lm3</font></span></a></span>;<br/><a NAME="E11:110_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A30</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="trees_1.html#T22" title="TREES_1:th.22">TREES_1:22</a></span>;<br/><a NAME="E12:110_1_1_2_3"/><span class="lab"><font color="Green" title="E55">A31</font></span>: 
<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="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> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>;<br/><a NAME="E13:110_1_1_2_3"/><span class="lab"><font color="Green" title="E56">A32</font></span>: 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</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> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>;<br/><a NAME="E14:110_1_1_2_3"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/><span class="kw">hence </span><a NAME="E15:110_1_1_2_3"/>
<font color="Maroon" title="c4">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" title="c3">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E1:110_1_1_2_3"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="txt" href="#E9:110_1_1_2_3"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="txt" href="#E11:110_1_1_2_3"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E12:110_1_1_2_3"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="txt" href="#E13:110_1_1_2_3"><span class="lab"><font color="Green" title="E56">A32</font></span></a></span>; <a class="txt" 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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:110_1_1_2_4"/>
<font color="Maroon" title="c4">w</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">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" title="c3">T</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">q</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> , <font color="Maroon" title="c8">m</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E3:110_1_1_2_4"/><span class="lab"><font color="Green" title="E49">A33</font></span>: 
<font color="Maroon" title="c4">w</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T130" title="FINSEQ_2:th.130">FINSEQ_2:130</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">m9</font> being    <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> , <font color="Maroon" title="c10">r</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E5:110_1_1_2_4"/><span class="lab"><font color="Green" title="E50">A34</font></span>: 
<font color="Maroon" title="c9">m9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">and </span><br/><a NAME="E6:110_1_1_2_4"/><span class="lab"><font color="Green" title="E51">A35</font></span>: 
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:110_1_1_2_4"/><span class="lab"><font color="Green" title="E52">A36</font></span>: 
<font color="Maroon" title="c4">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="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"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">m9</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c10">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E3:110_1_1_2"><span class="lab"><font color="Green" title="E46">A22</font></span></a></span>;<br/><a NAME="E8:110_1_1_2_4"/><span class="lab"><font color="Green" title="E53">A37</font></span>: 
<font color="Maroon" title="c4">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="c5">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_2_4"><span class="lab"><font color="Green" title="E49">A33</font></span></a>, <a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></span>;<br/><a NAME="E9:110_1_1_2_4"/><span class="lab"><font color="Green" title="E54">A38</font></span>: 
<font color="Maroon" title="c4">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="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_2_4"><span class="lab"><font color="Green" title="E49">A33</font></span></a>, <a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></span>;<br/><a NAME="E10:110_1_1_2_4"/><span class="lab"><font color="Green" title="E55">A39</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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="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> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:110_1_1_2_4"><span class="lab"><font color="Green" title="E53">A37</font></span></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></span>;<br/><a NAME="E11:110_1_1_2_4"/><span class="lab"><font color="Green" title="E56">A40</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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="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> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:110_1_1_2_4"><span class="lab"><font color="Green" title="E52">A36</font></span></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></span>;<br/><a NAME="E12:110_1_1_2_4"/><span class="lab"><font color="Green" title="E57">A41</font></span>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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> <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> <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/><a NAME="E13:110_1_1_2_4"/><span class="lab"><font color="Green" title="E58">A42</font></span>: 
<font color="Maroon" title="c10">r</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> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:110_1_1_2_4"><span class="lab"><font color="Green" title="E52">A36</font></span></a>, <a class="txt" href="#E8:110_1_1_2_4"><span class="lab"><font color="Green" title="E53">A37</font></span></a>, <a class="txt" href="#E10:110_1_1_2_4"><span class="lab"><font color="Green" title="E55">A39</font></span></a>, <a class="txt" href="#E11:110_1_1_2_4"><span class="lab"><font color="Green" title="E56">A40</font></span></a>, <a class="ref" href="finseq_1.html#T33" title="FINSEQ_1:th.33">FINSEQ_1:33</a></span>;<br/><a NAME="E14:110_1_1_2_4"/>
<font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1_2_4"><span class="lab"><font color="Green" title="E50">A34</font></span></a>, <a class="txt" href="#E10:110_1_1_2_4"><span class="lab"><font color="Green" title="E55">A39</font></span></a>, <a class="txt" href="#E11:110_1_1_2_4"><span class="lab"><font color="Green" title="E56">A40</font></span></a>, <a class="txt" href="trees_3.html#E59"><span class="lab"><font color="Green" title="E40">Lm3</font></span></a></span>;<br/><a NAME="E15:110_1_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">q</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="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="txt" href="#E4:110_1_1_2"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E6:110_1_1_2_4"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E10:110_1_1_2_4"><span class="lab"><font color="Green" title="E55">A39</font></span></a>, <a class="txt" href="#E11:110_1_1_2_4"><span class="lab"><font color="Green" title="E56">A40</font></span></a>, <a class="txt" href="#E13:110_1_1_2_4"><span class="lab"><font color="Green" title="E58">A42</font></span></a>, <a class="ref" href="trees_1.html#D3" title="TREES_1:def.3">TREES_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E16:110_1_1_2_4"/>
<font color="Maroon" title="c4">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" title="c3">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E5:110_1_1_2_4"><span class="lab"><font color="Green" title="E50">A34</font></span></a>, <a class="txt" href="#E9:110_1_1_2_4"><span class="lab"><font color="Green" title="E54">A38</font></span></a>, <a class="txt" href="#E10:110_1_1_2_4"><span class="lab"><font color="Green" title="E55">A39</font></span></a>, <a class="txt" href="#E11:110_1_1_2_4"><span class="lab"><font color="Green" title="E56">A40</font></span></a>, <a class="txt" href="#E12:110_1_1_2_4"><span class="lab"><font color="Green" title="E57">A41</font></span></a></span>; <a class="txt" 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">hence </span><a NAME="E7:110_1_1_2"/>
<font color="Maroon" title="c4">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" title="c3">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1_2"><span class="lab"><font color="Green" title="E48">A24</font></span></a></span>; <a class="txt" 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="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/>
<span class="kw">take </span>
<font color="Maroon" title="c4">T</font>
; <a class="txt" 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">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="c4">T</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  ex <font color="Olive" title="b2">n</font> being    <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>  ex <font color="Olive" title="b3">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b3">q</font> ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  ex <font color="Olive" title="b1">n</font> being    <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>  ex <font color="Olive" title="b2">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">q</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E9:110_1_1"/>
(  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  ex <font color="Olive" title="b1">n</font> being    <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>  ex <font color="Olive" title="b2">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">q</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  ex <font color="Olive" title="b1">n</font> being    <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>  ex <font color="Olive" title="b2">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">q</font> ) ) implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font> )</span><br/>

<span class="kw">assume </span><a NAME="E10:110_1_1"/><span class="lab"><font color="Green" title="E46">A43</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  ex <font color="Olive" title="b1">n</font> being    <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>  ex <font color="Olive" title="b2">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">q</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font></span><br/>

<div><span class="lab"><font color="Green" title="E47">A44</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c6">n</font> being    <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> , <font color="Maroon" title="c7">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><a NAME="E2:110_1_1_3"/><span class="lab"><font color="Green" title="E48">A45</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">and </span><br/><a NAME="E3:110_1_1_3"/><span class="lab"><font color="Green" title="E49">A46</font></span>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:110_1_1_3"/><span class="lab"><font color="Green" title="E50">A47</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c7">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">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> </span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">T1</font> = <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110_1_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E2:110_1_1_3"><span class="lab"><font color="Green" title="E48">A45</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c7">q</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c8">T1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_3"><span class="lab"><font color="Green" title="E49">A46</font></span></a></span>;<br/><a NAME="E7:110_1_1_3"/>
<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="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/><span class="kw">hence </span><a NAME="E8:110_1_1_3"/>
<font color="Maroon" title="c5">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1_1_3"><span class="lab"><font color="Green" title="E50">A47</font></span></a></span>; <a class="txt" 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>
<a NAME="E12:110_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T49" title="FINSEQ_1:th.49">FINSEQ_1:49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:110_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E10:110_1_1"><span class="lab"><font color="Green" title="E46">A43</font></span></a>, <a class="txt" href="#E11:110_1_1"><span class="lab"><font color="Green" title="E47">A44</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
