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<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>for <font color="Olive" title="b1">fT</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>  st  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Olive" title="b1">fT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">fT</font>];<br/>
<a NAME="E1:74_1"/><i><font color="Green" title="E44">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">k</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:74_1_1"/><i><font color="Green" title="E45">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/>for <font color="Olive" title="b2">fT</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>  st  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Olive" title="b2">fT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">fT</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c2">fT</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>;<br/>

<b>assume </b><a NAME="E2:74_1_1"/><i><font color="Green" title="E46">A4</font></i>: 
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c2">fT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">k</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="E1:74_1_1_1"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">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="c2">fT</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c4">k'</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">fT</font> ;<br/><a NAME="E3:74_1_1_1"/>
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">fT</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c4">k'</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c2">fT</font>
 <b>by </b><i><a class="ref" href="trees_1.html#T85" target="_self" title="TREES_1:th.85">Th85</a></i>;<br/><b>hence </b><a NAME="E4:74_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">fT</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 <b>by </b><i><a class="txt" href="trees_1.html#E1:74_1_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="trees_1.html#E2:74_1_1"><i><font color="Green" title="E46">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:74_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">fT</font>]
 <b>by </b><i><a class="txt" href="trees_1.html#E1:74"><i><font color="Green" title="E43">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:74_1"/><i><font color="Green" title="E45">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="trees_1.html#E1:74_1"><i><font color="Green" title="E44">A2</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c1">fT</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>;<br/>

<a NAME="E3:74_1"/>
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c1">fT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c1">fT</font>
 
;<br/>
<b>hence </b><a NAME="E4:74_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">fT</font>]
 <b>by </b><i><a class="txt" href="trees_1.html#E2:74_1"><i><font color="Green" title="E45">A5</font></i></a></i>;<br/>


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